A Sharp Bound for the Ratio of the First Two Dirichlet Eigenvalues of a Domain in a Hemisphere of 𝕊^{n}

Mark S. Ashbaugh^{∗} Department of Mathematics University of Missouri Columbia, Missouri 65211–0001, USA mark@math.missouri.edu

Rafael D. Benguria^{†} Departamento de Física P. Universidad Católica de Chile Casilla 306, Santiago 22, CHILE rbenguri@fis.puc.cl

September 20, 2000

1991 Mathematics Subject Classification: Primary 58G25, Secondary 35P15, 49Rxx, 33C55.

Keywords and phrases: eigenvalues of the Laplacian, Dirichlet problem for domains on spheres, Payne–Pólya–Weinberger conjecture, Sperner’s inequality, ratios of eigenvalues, isoperimetric inequalities for eigenvalues.

Short title: The PPW Conjecture in 𝕊^{n}

Abstract

For a domain Ω contained in a hemisphere of the n–dimensional sphere 𝕊^{n} we prove the optimal result λ_{2}/λ_{1}(Ω) ≤ λ_{2}/λ_{1}(Ω^{⋆}) for the ratio of its first two Dirichlet eigenvalues where Ω^{⋆}, the symmetric rearrangement of Ω in 𝕊^{n}, is a geodesic ball in 𝕊^{n} having the same n–volume as Ω. We also show that λ_{2}/λ_{1} for geodesic balls of geodesic radius θ_{1} less than or equal to π/2 is an increasing function of θ_{1} which runs between the value (j_{n/2,1}/j_{n/2−1,1})^{2} for θ_{1} = 0 (this is the Euclidean value) and 2(n + 1)/n for θ_{1} = π/2. Here j_{ν,k} denotes the k^{th} positive zero of the Bessel function J_{ν}(t). This result generalizes the Payne–Pólya–Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of 𝕊^{n} and having a fixed value of λ_{1} the one with the maximal value of λ_{2} is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for λ_{2}/λ_{1}. Various other results for λ_{1} and λ_{2} of geodesic balls in 𝕊^{n} are proved in the course of our work.

∗Partially supported by National Science Foundation (USA) grants DMS–9114162, INT–9123481, DMS– 9500968, and DMS–9870156.

†Partially supported by FONDECYT (Chile) project number 196–0462 and a Cátedra Presidencial en Ciencias (Chile).

1 Introduction

With our earlier proof [5], [6], [7] of the Payne–Pólya–Weinberger conjecture [42], [43] the bound

λ_{2}/λ_{1}(Ω) ≤ λ_{2}/λ_{1}(Ω^{⋆}) = j_{n/2,1}^{2}/j_{n/2−1,1}^{2} (1.1)

was established for the ratio of the first two eigenvalues of the Laplacian −Δ on a bounded domain Ω ⊂ ℝ^{n} with Dirichlet boundary conditions imposed on ∂Ω. Here Ω^{⋆} represents the n–dimensional ball having the same volume as Ω (but, in fact, by scaling any ball will do) and j_{ν,k} represents the k^{th} positive zero of the Bessel function J_{ν}(t) [1]. Equality obtains in (1.1) if and only if Ω is a ball to begin with. In this paper we prove the analog of this result for domains in a hemisphere of 𝕊^{n} (−Δ is now, of course, the Laplacian on 𝕊^{n}).

It turns out to be better to view (1.1) as

λ_{2}(Ω) ≤ λ_{2}(B_{λ_{1}}) (1.2)

where B_{λ_{1}} represents the n–dimensional ball that has the value λ_{1}(Ω) as its first Dirichlet eigenvalue. This is, in fact, the way our proof proceeded ([5], [6], [8], see also [27]). Of course, the choice of ball here just involves choosing an appropriate radius and this choice is always unique since the first eigenvalue of a ball is a strictly decreasing function of its radius and goes from infinity to zero as the radius goes from zero to infinity. In words, (1.2) says that among all n–dimensional domains having the same first eigenvalue the n–dimensional ball has maximal second eigenvalue. One might compare this statement with the statement of the Faber–Krahn inequality [30], [37], [38]: among all n–dimensional domains having the same volume the n–dimensional ball has minimal first (Dirichlet) eigenvalue. Also of interest is the Szegő–Weinberger inequality [50], [52]: among all n–dimensional domains having the same volume the n–dimensional ball has maximal first nonzero Neumann eigenvalue. Both of these other inequalities are relevant here; the first because it figures in our proof and the second because in many ways the proof of this result is analogous to (though considerably simpler than) our proof of the Payne–Pólya–Weinberger conjecture.

What we do in this paper is transfer the strategy outlined in the last paragraph over to bounded domains in 𝕊^{n} contained in hemispheres. It turns out that, properly interpreted, everything that we have said above concerning domains in Euclidean space also holds in 𝕊^{n}. Thus we prove that (1.2) holds if Ω is a domain in a hemisphere in 𝕊^{n} and B_{λ_{1}} is the geodesic ball in 𝕊^{n} having λ_{1}(Ω) as its first eigenvalue. There is a Faber–Krahn result in 𝕊^{n} [49] (see also [31]), too, and λ_{1} of a geodesic ball is still strictly monotone decreasing and goes to infinity at zero so a unique B_{λ_{1}} which is a hemisphere or less will exist. For the Faber–Krahn and Szegő–Weinberger inequalities in 𝕊^{n} one need only read “volume” as “canonical n–dimensional volume in 𝕊^{n}” and “ball” as “geodesic ball”. The generalization of the Szegő–Weinberger result to domains in hemispheres of 𝕊^{n} is a recent result of ours [9] (see also prior work of Chavel [22]). Our proof here parallels the one in [9] as well as our proof of the Payne–Pólya–Weinberger conjecture [5], [6], [7] (see especially our proof in [7]) for the Euclidean case.

The biggest difference between our work on λ_{2}/λ_{1} for domains in 𝕊^{n} versus those in ℝ^{n} revolves around the difference between (1.1) and (1.2). In ℝ^{n} (1.1) and (1.2) are equivalent since λ_{2}/λ_{1} is the same for any ball, whatever its size. This follows from the fact that in that case the eigenvalues scale with the radius, in particular, for a ball of radius R in ℝ^{n} λ_{1} = j_{n/2−1,1}^{2}/R^{2}, λ_{2} = j_{n/2,1}^{2}/R^{2}, and hence λ_{2}/λ_{1} = j_{n/2,1}^{2}/j_{n/2−1,1}^{2} and is independent of R. When one passes to 𝕊^{n} this is no longer the case. If we let θ_{1} denote the radius of our geodesic ball then the first and second eigenvalues of that ball, which we denote by λ_{1}(θ_{1}) and λ_{2}(θ_{1}), are in general more complicated functions of θ_{1}. By domain monotonicity (see, for example, [14], [23], [28], [44]) these are, of course, strictly decreasing functions but more precise knowledge of them (or combinations thereof) requires considerable effort. For example, for the case of 𝕊^{n}, to pass from (1.2) back to the first part of (1.1) (i.e., λ_{2}/λ_{1}(Ω) ≤ λ_{2}/λ_{1}(Ω^{⋆}), where Ω^{⋆} is the symmetric rearrangement of Ω on 𝕊^{n}) one needs to know that λ_{2}/λ_{1} for geodesic balls is an increasing function of θ_{1}. We do this below in Section 3. In general, most of what could almost be taken for granted for the case of balls in ℝ^{n} expands to some problem about how λ_{1}, λ_{2}, or a combination of the two behaves as a function of θ_{1}. Another instance of this is that while for a ball in ℝ^{n} it is easy to see that λ_{2} corresponds to an ℓ = 1 eigenfunction (i.e., an eigenfunction associated with an ℓ = 1 spherical harmonic) in 𝕊^{n} the analog of this must be proved for all θ_{1} ∈ (0, π/2]. The proof of this fact, while not difficult, is given in Section 3. In fact, in Section 3 (see Lemma 3.1) we prove the result for all θ_{1} ∈ (0, π) (note that m replaces ℓ there).

For orientation we outline the elements of our proof here. Aside from item 4 (which we have just discussed) these elements were also present in our proof in the Euclidean case.

1. Rayleigh–Ritz inequality for estimating λ_{2}:

λ_{2} − λ_{1} ≤ \frac{∫_{Ω} |∇P |^{2}u_{1}^{2} dμ}{∫_{Ω} P^{2}u_{1}^{2} dμ}

if ∫_{Ω} P u_{1}^{2}dμ = 0 and P ≢ 0. Here u_{1} denotes the normalized eigenfunction for λ_{1}. This inequality applies on manifolds as in Euclidean space; one only has to view |∇P | as a norm with respect to the metric on the manifold (see, e.g., [23], pp. 50–51). Also dμ represents the intrinsic volume element for the manifold. The inequality given here follows from the usual Rayleigh-Ritz inequality for λ_{2} by taking P u_{1} as trial function and integrating by parts appropriately.

2. A Brouwer fixed point theorem argument that allows us to insure that the condition ∫_{Ω} P u_{1}^{2}dμ = 0 is satisfied for n specific choices of the function P. In our recent paper [9] we gave a version of this argument for 𝕊^{n} which also applies here. In Section 2 of this paper we give an improved version of this argument (using degree theory rather than the Brouwer fixed point theorem) which yields a slightly stronger result. The original argument of this type (to our knowledge) was given by Weinberger in [52]. For future reference, we note that all such results will be referred to as “center of mass results”.

3. Rearrangement results for functions and domains. These results are measure theoretic in nature and easily extend to problems on manifolds. There is also an easily effected preliminary rearrangement in 𝕊^{n} (see also [8], [9]).

4. Properties of the eigenvalues and eigenfunctions of geodesic balls in 𝕊^{n}. These we establish in Section 3 below. The details depend on Legendre and associated Legendre functions though we manage to keep these functions (also expressible in terms of hypergeometric functions) in the background. In ℝ^{n}, of course, the special functions that occur are Bessel functions.

5. Monotonicity properties of certain special combinations of the eigenfunctions for λ_{1} and λ_{2} for geodesic balls contained in hemispheres. These functions are special to the problem of maximizing λ_{2}/λ_{1} (or λ_{2} subject to λ_{1} = const.) and in a technical sense their properties proved here are the most difficult part of our overall proof. These properties are proved in Section 4 below.

6. Chiti’s comparison argument. This is a specialized comparison result which establishes a crossing property of the symmetric–decreasing rearrangement of the eigenfunction u_{1} vis–à–vis the first eigenfunction of the geodesic ball B_{λ_{1}}. This result of Chiti [24], [25], [26], [27] for the Euclidean case is based upon a rearrangement technique for partial differential equations due to Talenti [51] which in turn is based on the classical isoperimetric inequality in ℝ^{n} [21], [23], [41]. All of these results generalize to 𝕊^{n}. We give these arguments in detail in Section 5.

Our main results are summarized in the following theorems:

Theorem 1.1. Let Ω be contained in a hemisphere of 𝕊^{n} and let B_{λ_{1}} denote the geodesic ball in 𝕊^{n} having the same value of λ_{1} as Ω (i.e., λ_{1}(Ω) = λ_{1}(B_{λ_{1}})). Then

λ_{2}(Ω) ≤ λ_{2}(B_{λ_{1}})

with equality if and only if Ω is itself a geodesic ball in 𝕊^{n}.

Theorem 1.2. The first Dirichlet eigenvalue for a geodesic ball in 𝕊^{n} of geodesic radius θ_{1}, λ_{1}(θ_{1}), is such that θ_{1}^{2}λ_{1}(θ_{1}) is a decreasing function of θ_{1} for 0 < θ_{1} ≤ π.

Theorem 1.3. The quotient between the first two Dirichlet eigenvalues for a geodesic ball in 𝕊^{n}, of geodesic radius θ_{1}, λ_{2}/λ_{1}, is an increasing function of θ_{1} for 0 < θ_{1} ≤ π/2.

Finally, we come to the PPW conjecture for domains in hemispheres of 𝕊^{n}:

Theorem 1.4. Let Ω be contained in a hemisphere of 𝕊^{n}. Then

λ_{2}/λ_{1}(Ω) ≤ λ_{2}/λ_{1}(Ω^{⋆})

with equality if and only if Ω is itself a geodesic ball in 𝕊^{n}.

This theorem follows from Theorems 1.1 and 1.3 above. If B_{λ_{1}} is a ball having the same λ_{1} as Ω we have

λ_{2}/λ_{1}(Ω) ≤ λ_{2}/λ_{1}(B_{λ_{1}}) = λ_{2}(θ_{1})/λ_{1}(θ_{1}),

where θ_{1} is the geodesic radius of B_{λ_{1}}. Theorem 1.4 now follows from Theorem 1.3 and Sperner’s optimal Faber–Krahn-type result [49] for domains in 𝕊^{n}, which implies that θ_{1}(Ω^{⋆}) ≥ θ_{1}(B_{λ_{1}}) (via domain monotonicity).

The results given in Theorems 1.1–1.4 were announced earlier in [8] and [10] (note, however, that Theorem 1.2 is not mentioned explicitly in [8], nor is Theorem 1.3 stated formally there). Moreover, [10] contains alternative proofs of some of the results proved here. These may be of independent interest, since they may point the way to generalizing the results presented here to other settings. We note, though, that the proofs contained herein are those by which we first proved the results stated above. In fact, there is a close parallel between the proofs given here and those of our “second” proof of the Euclidean PPW conjecture given in [7] and of our 𝕊^{n}–version of the Szegő–Weinberger inequality given in [9]. One of our objectives in [8] was to bring these similarities to the fore.

Beyond this, and as alluded to in passing in the previous paragraph, one might ask to what extent our results above are optimal, and, further, whether or not they are amenable to generalization (for example, to other spaces, the most obvious being ℍ^{n}). In particular, one might ask if it might be possible to remove our restriction to domains contained in a hemisphere of 𝕊^{n}. It will be apparent to anyone who studies our proofs that the success of our approach is very much dependent on this condition. Certainly it should take little convincing to see that geometrically things become quite different “beyond the hemisphere”: for example, up to the hemisphere both the volume and surface area of a geodesic ball are increasing functions, but beyond the hemisphere the volume continues to grow while the surface area actually shrinks. On the other hand, we do not have any counterexamples to the conjecture that our theorems above (aside from Theorem 1.2, which is already established for all geodesic balls, and not just those contained in a hemisphere) continue to hold in the absence of the hemisphere condition. In fact, we have been able to establish that Theorem 1.3 does hold for all 0 < θ_{1} < π for n = 2 and 3. We might also note that it is common to encounter some sort of impediment at the hemisphere when dealing with the eigenvalues (specifically, Dirichlet or Neumann) of the Laplacian for domains in 𝕊^{n}. For example, one might note that while for domains in Euclidean space and for domains strictly contained in a hemisphere of 𝕊^{n} the first nonzero Neumann eigenvalue is always less than the first Dirichlet eigenvalue, the situation is reversed for geodesic balls which are larger than a hemisphere (equality obtains at the hemisphere). For more discussion along these lines, the reader might consult [11], [8], [9], [14], and references therein.

As for extensions to bounded domains contained in ℍ^{n}, the situation is still unsettled, but we make the following comments and observations. The analog of Theorem 1.4 (the “naive” PPW conjecture for ℍ^{n}) cannot possibly hold, since one can imagine having a small disk (ball) with very narrow tentacles extending from it in such a way that the first two Dirichlet eigenvalues of this domain are very nearly those of the disk (ball), while the volume of the domain is as large as one wants. This means that Ω^{⋆} can be an arbitrarily large ball, and the rub now comes from the fact that both eigenvalues of the large ball can be made arbitrarily close to the bottom of the spectrum of the Laplacian on all of ℍ^{n}, which is a positive value (specifically (n − 1)^{2}/4), by making the ball sufficiently large. This implies that λ_{2}/λ_{1}(Ω^{⋆}) can be made as close as we want to 1, while λ_{2}/λ_{1}(Ω) will stay close to its value for a small disk (ball). Since this latter value is nearly the Euclidean value (any value larger than 1 will do as well), we are faced with a contradiction. In fact, it is likely that λ_{2}/λ_{1} for a geodesic ball in ℍ^{n} is a decreasing function of the radius, i.e., that the counterpart to Theorem 1.3 goes the other way (this certainly appears to be the case for ℍ^{2}, based on numerical studies we have done). This, of course, would be an interesting fact to prove in its own right (and perhaps especially for its geometric implications), even if it doesn’t lead into a proof of the “PPW conjecture for ℍ^{n}”. It would also be, in a certain sense, the “natural” result, since λ_{2}/λ_{1} is increasing for geodesic balls contained in a hemisphere of 𝕊^{n} (and quite possibly for all geodesic balls in 𝕊^{n}; we note in this connection that λ_{2}/λ_{1} goes to infinity as the full sphere is approached since in that limit λ_{1} → 0^{+} while λ_{2} → n^{+}), while it is constant for all balls in ℝ^{n}. As for an ℍ^{n}-analog of Theorem 1.1 (the “sophisticated” PPW conjecture for ℍ^{n}), this may well be true, but as yet it is not proved. One might also speculate that for bounded domains in ℍ^{n} λ_{2}/λ_{1} is always less than the Euclidean bound (i.e., the value of λ_{2}/λ_{1} for a Euclidean ball). This conjecture is certainly supported by the behavior of λ_{2}/λ_{1} for geodesic balls in ℍ^{n}, and in general by the fact that “large” domains can be expected to have λ_{2}/λ_{1} near 1. It is also supported by a result of Harrell and Michel [33], which gives a finite upper bound to λ_{2}/λ_{1} for bounded domains contained in ℍ^{2}. While their bound is almost certainly not optimal, it is at least in the right ballpark: it is 17, while the Euclidean value in two dimensions is approximately 2.5387.

2 The “center of mass” result for domains in spheres

In this section we present a general Center of Mass Theorem which follows from general topological arguments. This Center of Mass Theorem guarantees the orthogonality of certain functions which is needed in later sections. In a previous paper [9] we gave such a theorem for domains contained in a hemisphere of 𝕊^{n}. The proof of that theorem was fairly involved since we had to identify a hemisphere of 𝕊^{n} with the ball B^{n} and use the Brouwer fixed point theorem. We also required a limiting argument since there were certain problems which arose for domains extending to the equator that we could not handle directly. These arguments would suffice to yield a version of our Theorem 2.1 below for domains contained in a hemisphere (see Theorem 2.1 of [9] and Remark 4 following it to see how to make this extension). Such a result would be enough to allow us to prove only slightly weaker versions of the theorems found in the remainder of this paper. We have chosen to present the more general result here since its proof is both simpler and more natural than our former argument. In the new proof attention is confined to mappings from 𝕊^{n} to 𝕊^{n} and to certain natural geometrical conditions. A modest knowledge of degree theory is needed to conclude the proof.

Theorem 2.1. (Center of Mass Theorem) Let Ω be a domain in 𝕊^{n} and let G̃ be a continuous function on [0, π] which is positive on (0, π) and symmetric about π/2. Also, let dμ be any positive measure on Ω. Then there is a choice of Cartesian coordinates x′_{1}, x′_{2}, …, x′_{n+1} for ℝ^{n+1} with the origin at the center of 𝕊^{n} such that

∫_{Ω} x′_{i} G̃(θ) dμ = 0 for i = 1, 2, …, n, (2.1)

where θ represents the angle from the positive x′_{n+1}–axis.

Remarks. In our later applications dμ will be u_{1}^{2} dσ where u_{1} is the first eigenfunction of −Δ on Ω with Dirichlet boundary conditions imposed on ∂Ω and dσ is the standard volume element for 𝕊^{n}. Similarly, in our applications the function G̃ will be related to the function g(θ) defined in equation (4.3) below (extended appropriately to [0, π]) by G̃(θ) = g(θ)/ sin θ. Note that Theorem 2.1 makes no statement about the integral in (2.1) for the case i = n+1.

Proof. We begin by considering the vector function v⃗ : 𝕊^{n} → ℝ^{n+1} defined for y⃗ ∈ 𝕊^{n} by

v⃗(y⃗) = ∫_{Ω} x⃗ G̃(θ) dμ (2.2)

where θ represents the angle between y⃗ and the integration variable x⃗ ∈ Ω and x⃗ = (x_{1}, x_{2}, …, x_{n+1}) where x_{1}, x_{2}, …, x_{n+1} represent some (initial) set of Cartesian coordinates. The vector v⃗(y⃗) simply gives the center of mass in ℝ^{n+1} of the hypersurface distribution on Ω ⊂ 𝕊^{n} with mass density given by G̃(θ) dμ. Note that its dependence on the point y⃗ is entirely through the function G̃(θ); indeed, if not for this function we would have only a single center of mass vector v⃗.

First we argue that what we should look for are points y⃗_{0} ∈ 𝕊^{n} such that

v⃗(y⃗_{0}) = αy⃗_{0}, (2.3)

i.e., such that y⃗_{0} and v⃗(y⃗_{0}) are linearly dependent. To see that finding such a point will suffice to prove our theorem, suppose y⃗_{0} is a point where (2.3) holds and let R be an (n + 1) by (n + 1) rotation matrix with y⃗_{0} as its last row. Defining new Cartesian coordinates x′_{1}, x′_{2}, …, x′_{n+1} via

x′_{i} = ∑_{j=1}^{n+1} R_{ij}x_{j} (2.4)

we have (with θ measured from y⃗_{0})

∫_{Ω} x′_{i}G̃(θ) dμ = ∑_{j=1}^{n+1} R_{ij} ∫_{Ω} x_{j}G̃(θ) dμ = [Rv⃗(y⃗_{0})]_{i} =
(2.5)  = [αRy⃗_{0}]_{i} = [αê_{n+1}]_{i} = 0 for i = 1, …, n,

since R is an orthogonal matrix and hence its first n rows are orthogonal to its last row. Thus, the conclusion to the theorem will follow from finding a solution to (2.3) and from here on we concentrate on finding such a solution.

Now if v⃗(y⃗) ever vanished for some y⃗ ∈ 𝕊^{n} the conclusion (2.1) would follow immediately with no need to rotate coordinates. So we may as well assume that v⃗ never vanishes on 𝕊^{n}. Under this assumption we can pass to consideration of the mapping w⃗ : 𝕊^{n} → 𝕊^{n} defined by

w⃗(y⃗) = \frac{v⃗(y⃗)}{|v⃗(y⃗)|}. (2.6)

The dependence condition (2.3) then reduces to

w⃗(y⃗_{0}) = ±y⃗_{0} (2.7)

so that we are seeking a fixed point or an “anti–fixed point” of w⃗. Under the assumptions of the theorem it suffices to seek only fixed points, i.e., solutions to

w⃗(y⃗_{0}) = y⃗_{0}. (2.8)

This follows from the symmetry of G̃(θ) about θ = π/2 which implies that v⃗(−y⃗) = v⃗(y⃗) for all y⃗ ∈ 𝕊^{n} and thus that w⃗(−y⃗) = w⃗(y⃗) for all y⃗ ∈ 𝕊^{n}. Hence if y⃗_{0} is a solution to (2.7), then either y⃗_{0} or −y⃗_{0} must be a solution to (2.8) and we can therefore concentrate solely on finding solutions to (2.8). (Geometrically, too, it is more natural to view a point y⃗_{0} satisfying (2.8) as a center of mass of Ω than the point −y⃗_{0} satisfying w⃗(−y⃗_{0}) = y⃗_{0}, even though there is no difference between the two as far as fulfilling the conditions of the theorem goes.)

Finally we are at a point where we can use degree theory to conclude that w⃗ must have a fixed point, i.e., a solution y⃗_{0} to (2.8). The definition of v⃗ shows that it is continuous and our assumption that v⃗ does not vanish on 𝕊^{n} guarantees the continuity of w⃗ as defined by (2.6). Therefore, by standard theory (see, for example, [2], p. 195; [29], Chapter 16, Section 1; [35], p. 263; [40], p. 116; [47], Chapter 16; [53], p. 807) w⃗ has a degree as a map from 𝕊^{n} to 𝕊^{n}. The fact that w⃗(−y⃗) = w⃗(y⃗) for all y⃗ ∈ 𝕊^{n} implies that deg(w⃗) is even (simply observe that for an image point z⃗ ∈ 𝕊^{n} the points in the preimage come in pairs ±y⃗). Now if w⃗ has no fixed points, i.e., if w⃗(y⃗) ≠ y⃗ for all y⃗ ∈ 𝕊^{n}, then w⃗ would be homotopic to the antipodal map A⃗ defined by A⃗(y⃗) = −y⃗ via the homotopy

h(t, y⃗) = \frac{−ty⃗ + (1 − t)w⃗(y⃗)}{| − ty⃗ + (1 − t)w⃗(y⃗)|} for 0 ≤ t ≤ 1. (2.9)

But the degree of a mapping is a homotopy invariant and deg(A⃗) = ±1 ≠ deg(w⃗), a contradiction. Hence w⃗ must have some fixed point y⃗_{0} ∈ 𝕊^{n} and the proof is complete.

Remarks. (1) If Ω is contained in a hemisphere of 𝕊^{n} then another way to complete the proof of the theorem is to set things up initially in a Cartesian frame such that Ω lies in the northern hemisphere of 𝕊^{n} and observe that for any y⃗ ∈ 𝕊^{n}

v_{n+1}(y⃗) = ∫_{Ω} x_{n+1}G̃(θ) dμ > 0. (2.10)

Hence v⃗ never vanishes and we may regard w⃗ = v⃗/|v⃗| as a mapping from the closed northern hemisphere into itself. Since this space is homeomorphic to the ball B^{n} we can apply the Brouwer fixed point theorem to conclude that w⃗ has a fixed point y⃗_{0} in the northern hemisphere (see, for example, [39], Section 8-10; [47], p. 406; [48], pp. 151, 194). This proof is similar to, but simpler than, the proof of the restricted version of Theorem 2.1 that we gave in [9].

(2) The alternative proof just given does not use the symmetry of G̃(θ) about θ = π/2 stated in the theorem. Only positivity of G̃(θ) on (0, π) is used. There are certainly situations, in particular for domains which are in some sense larger than a hemisphere, where one might not want to require that G̃ be symmetric about π/2. In such situations one possible route to a center of mass result is to show that the mapping w⃗ misses at least one point of 𝕊^{n}, following the spirit of the alternative proof given above. Brouwer’s fixed point theorem can then be applied (to 𝕊^{n} less a sufficiently small neighborhood of a point that w⃗ misses) to yield a fixed point. Or in the language of degree theory, the case where a map misses a point is the simplest case of a map which has degree 0. This can be seen directly or by observing that such a map w⃗ is homotopic to a constant map, i.e., contractible to a point, or inessential (see [35], p. 154; [39], p. 357; or [48], p. 23) via the homotopy

g(t, y⃗) = \frac{−tz⃗_{0} + (1 − t)w⃗(y⃗)}{| − tz⃗_{0} + (1 − t)w⃗(y⃗)|} for 0 ≤ t ≤ 1 (2.11)

if z⃗_{0} is a point that w⃗ misses.

(3) If Ω has a center of symmetry and dμ is either u_{1}^{2} dσ (as occurs in Dirichlet problems for −Δ on Ω) or dσ (as occurs in Neumann problems for −Δ on Ω), then this point will certainly serve as a center of mass in the sense of Theorem 2.1. Here dσ represents the standard volume element for 𝕊^{n} and u_{1} represents the first Dirichlet eigenfunction of −Δ on Ω. One has only to note that u_{1}, being unique up to a constant factor and of one sign, must share the symmetries of Ω. Also, if dμ = dσ and Ω is an arbitrary domain such that its complement has a center of mass in the sense of Theorem 2.1 (as could be concluded, for example, via any of the conditions discussed so far, or by other means) then Ω shares this center of mass since it is clear that

∫_{𝕊^{n}} x′_{i}G̃(θ) dσ = 0 for i = 1, 2 …, n (2.12)

if x′_{1}, …, x′_{n+1} represent Cartesian coordinates and θ is measured from the positive x′_{n+1}– axis. In particular, Ω certainly has a center of mass in this sense if the complement of Ω is contained in a hemisphere. All the observations made in this remark apply whether or not G̃(θ) is even with respect to θ = π/2. For another result that holds in the absence of symmetry of G̃ about π/2 see Theorem 2.2 below.

(4) With Theorem 2.1 in hand we can obtain a modest improvement of our main theorem in [9] (see Theorem 5.1). In particular, in Remark 2 following the proof of Theorem 5.1 in [9] we now have no need to invoke condition (2.4). We thus obtain the result μ_{1}(Ω) ≤ μ_{1}(Ω^{⋆}) comparing the first nonzero Neumann eigenvalue of the domain Ω with that of the spherical cap Ω^{⋆} having the same volume for any domain Ω such that Ω ∩ (−Ω) = ∅ (equivalently, −Ω ⊂ 𝕊^{n} \ Ω). This inequality is an equality if and only if Ω is itself a geodesic ball. More generally, the same result holds if, when the north pole is a center of mass in the sense of Theorem 2.1, Ω has the property that for each k ∈ [0, 1]

|{y⃗ ∈ Ω | y_{n+1} ≤ −k}| ≤ |{y⃗ ∈ 𝕊^{n} \ Ω | y_{n+1} ≥ k}| (2.13)

(cf. Remark 2 following the proof of Theorem 5.1 in [9]). Here |X| denotes the measure of X with respect to the canonical measure (standard volume element) on 𝕊^{n} where X is any measurable subset of 𝕊^{n}. We refer to condition (2.13) as the “excess less than or equal to deficit property”. For some further comments relating to this property, see our remarks at the end of Section 6. These give the most general conditions known at this time.

Finally, for possible future use (see also Remarks 2 and 3 above) we state the following variant of our Center of Mass Theorem which holds in even dimension (i.e., for Ω ⊂ 𝕊^{n} with n even) in the absence of symmetry of G̃ about π/2:

Theorem 2.2. Let Ω be a domain in 𝕊^{n} for n even and let G̃ be continuous on [0, π] and positive on (0, π). Then Ω has a center of mass in the sense of Theorem 2.1. That is, there exists a choice of Cartesian coordinates such that (2.1) holds for i = 1, 2, …, n.

Proof. Defining v⃗ as above (equation (2.2)) either v⃗ vanishes somewhere and we are done or we can pass to w⃗ = v⃗/|v⃗|. Continuing with the latter case, if w⃗ has neither a fixed point nor an anti–fixed point (i.e., there are no solutions y⃗ ∈ 𝕊^{n} to w⃗(y⃗) = ±y⃗) then as above we can show that w⃗ is homotopic to the antipodal map and also to the identity map. But the antipodal map has degree (−1)^{n+1} (see, for example, [2], p. 197, Theorem 9.2; [29], p. 339, Exercise 4; [40], p. 118, Theorem 21.3; [47], p. 403, Theorem 4.3; [53], p. 809) and for n even (−1)^{n+1} = −1 ≠ 1 = degree of the identity map. This is a contradiction since degree is a homotopy invariant (see [2], p. 195; [29], p. 339; [35], p. 266; [40], p. 117; [47], p. 401; or [53], p. 809), hence w⃗ must have either a fixed point or an anti–fixed point, and the conclusion of the theorem follows.

3 Properties of the first two Dirichlet eigenvalues of geodesic balls in 𝕊^{n}

We consider the Dirichlet problem on a geodesic ball of radius θ_{1} in 𝕊^{n} (where θ_{1} ∈ (0, π)) which we view as a polar cap centered at the north pole (i.e., the point ê_{n+1} ∈ 𝕊^{n} is taken as the center of our geodesic ball). By using the O(n) symmetry of the polar cap (this O(n) is the subgroup of O(n + 1) which leaves the point ê_{n+1} fixed), one can separate variables in the usual way obtaining the family of ordinary differential equations in the “radial” variable θ

(3.1) −y′′ − (n − 1) cot θ y′ + m(m + n − 2) csc^{2} θ y = λy on (0, θ_{1})

for m = 0, 1, 2, . . .. The boundary conditions to be applied for (3.1) are y(0) finite and y(θ_{1}) = 0. In particular, we shall be concerned with the lowest eigenvalues of the m = 0 and m = 1 cases of this equation, but first we develop some general properties of the solutions to these equations and some of their interrelationships.

We begin by considering λ as a positive parameter (all the eigenvalues that we consider here are easily seen to be positive by consideration of the Rayleigh quotients that characterize them) and defining u_{m}(θ; λ) for m = 0, 1, 2, … as that solution to (3.1) which has the behavior

u_{m}(θ; λ) = c_{m}θ^{m} + O(θ^{m+2}) (3.2)

where the constants c_{m} will be specified below. This behavior is consistent with equation (3.1) as can be seen from Frobenius theory (θ = 0 is a regular singular point of (3.1)). In particular, the eigenfunctions to (3.1) will all be found among the u_{m}’s defined in (3.2) assuming c_{m} ≠ 0 since finiteness at θ = 0 forces this behavior (up to multiplicative factors). Moreover, it is not difficult to verify that if u_{m} solves (3.1) then

u′_{m} − m cot θ u_{m}

satisfies (3.1) for m replaced by m + 1 and also

u′_{m} + (m + n − 2) cot θ u_{m}

satisfies (3.1) for m replaced by m−1. From these facts and Frobenius theory again it follows that

u_{m+1} = −u′_{m} + m cot θ u_{m} (3.3)

and

[λ − (m − 1)(m + n − 2)] u_{m−1} = u′_{m} + (m + n − 2) cot θ u_{m} (3.4)

if we agree to set c_{0} = 1 and define successive c_{m}’s via

c_{m+1} = \frac{λ − m(m + n − 1)}{2m + n}c_{m} for m = 0, 1, 2, . . .. (3.5)

These constitute the raising and lowering relations for the functions u_{m}. Also, elimination of u′_{m} between (3.3) and (3.4) yields the pure recursion relation in m

u_{m+1} − (2m + n − 2) cot θ u_{m} + [λ − (m − 1)(m + n − 2)] u_{m−1} = 0. (3.6)

Since (sin θ)^{m+n−2}u_{m}(θ) is 0 at θ = 0 one can integrate (3.4) and replace m by m + 1 to obtain

(sin θ)^{m+n−1}u_{m+1}(θ) = [λ − m(m + n − 1)] ∫_{0}^{θ}(sin t)^{m+n−1}u_{m}(t) dt. (3.7)

What we have developed so far could be considered the algebraic properties of the functions u_{m}. One should think of the u_{m}’s as the analogs of associated Legendre functions in n dimensions. In particular, when n = 2 one finds that u_{m}(θ; ν(ν + 1)) = (−1)^{m}P_{ν}^{m}(cos θ) following [1] (in general this should be correct up to a constant factor, typically a factor of (−1)^{m}, depending on the precise convention adopted; cf. [46], for example, where the convention differs by (−1)^{m}); here we have replaced the eigenvalue parameter λ by ν(ν + 1), as is traditional in dealing with Legendre functions. In this case (3.1) reduces to the associated Legendre equation and (3.3), (3.4), and (3.6) all reduce to standard relations between the associated Legendre functions. Almost certainly this generalization to n dimensions of associated Legendre functions and their basic relations has been developed before, though we do not know of a reference where the details needed here are developed explicitly. (Cf. also [31] which deals with the m = 0 case of (3.1) in n dimensions but in Schrödinger normal form.) In any event, it is a relatively simple matter to pass to the n–dimensional case once the situation in two dimensions is known. In fact, our functions u_{m} can be expressed in terms of associated Legendre functions no matter what the dimension but this connection is not particularly useful in this context so we do not elaborate upon it here (but see the equations in Section 6 following equation (6.4), or Remark 1 below). We note that our convention on the constants c_{m} allows them to vanish from a certain value of m on for specific values of the parameter λ (cf. (3.5)). (Recall, for example, that when dealing with the full sphere 𝕊^{2} one needs P_{ℓ}^{m} only for ℓ = 0, 1, … and m = 0, 1, …, ℓ.) This is not a problem for our purposes here since we will be most interested in passing from u_{m} to u_{m+1} via either (3.3) or (3.7).

We come back now to the eigenvalue problem for (3.1). If one defines the left-hand side of equation (3.1) as the operator h_{m} applied to y (with boundary conditions incorporated in the definition of h_{m}), then it is easily seen that h_{m′} > h_{m} in the sense of quadratic forms if m′ > m. Thus λ_{1} of −Δ for the polar cap is λ_{1}(h_{0}) while λ_{2} of −Δ for the polar cap must be either λ_{1}(h_{1}) or λ_{2}(h_{0}). We now show that the former is the case.

Lemma 3.1. The first eigenvalue of the Dirichlet Laplacian on a polar cap is the first eigenvalue of (3.1) with m = 0 while the second eigenvalue of the Dirichlet Laplacian is the first eigenvalue of (3.1) with m = 1. The second eigenvalue of the cap occurs with multiplicity n. These results hold for all polar caps, i.e., for all θ_{1} ∈ (0, π).

Proof. In the notation of the preceding paragraph we must show that λ_{2}(h_{0}) > λ_{1}(h_{1}). The argument proceeds via a simple use of Rolle’s theorem as applied to (3.3) and (3.4) rewritten in the forms

[(sin θ)^{−m}u_{m}]′ = −(sin θ)^{−m}u_{m+1} (3.8)

and

[(sin θ)^{m+n−2}u_{m}]′ = [λ − (m − 1)(m + n − 2)] (sin θ)^{m+n−2}u_{m−1}. (3.9)

In particular, with m = 0 in the first of these we have

u′_{0} = −u_{1} (3.10)

and with m = 1 in the second we have

[(sin θ)^{n−1}u_{1}]′ = λ(sin θ)^{n−1}u_{0}. (3.11)

By Rolle’s theorem, between any two zeros of u_{0} there is a zero of u′_{0} and hence of u_{1}, since (3.10) holds. Similarly, between two zeros of (sin θ)^{n−1}u_{1} there is a zero of its derivative and hence, by (3.11), of u_{0}. Thus for fixed λ > 0 the zeros of u_{0} and (sin θ)^{n−1} u_{1} on [0, π) interlace.

Now consider u_{0} and u_{1} for λ = λ_{1}(h_{1}). Since this makes θ_{1} the first positive zero of u_{1} it is clear by what we have just proved that u_{0} has exactly one zero in (0, θ_{1}) and that θ_{1} is not a zero of u_{0}. This then implies, by the fact that the positive zeros of any u_{m} are decreasing functions of the parameter λ (see, for example, [20], p. 315, or [28], p. 454), that λ_{2}(h_{0}) > λ_{1}(h_{1}).

That the multiplicity of λ_{1}(h_{1}) as an eigenvalue of −Δ on the polar cap (= geodesic ball) with Dirichlet boundary conditions is n follows from the details of separation of variables. It can be shown that the corresponding eigenfunctions can be taken as (x_{i}/ sin θ)y(θ) (restricted to 𝕊^{n}) where i = 1, 2, …, n, x_{n+1} = cos θ, and y(θ) is the eigenfunction of (3.1) for the eigenvalue λ_{1}(h_{1}). These functions form an orthogonal basis for the eigenspace of −Δ corresponding to the eigenvalue λ_{1}(h_{1}), showing that its multiplicity is n. This completes our proof.

Remarks. (1) The argument used in our proof above can be viewed as a way of translating order properties of the zeros of the u_{m}’s into order properties of the corresponding eigenvalues λ_{i}(h_{m}). In fact, our approach extends easily to an interlacing result for the zeros (and hence for the associated eigenvalues) of u_{m} and u_{m+1} for arbitrary m. Further ordering properties of the Dirichlet eigenvalues of spherical caps in 𝕊^{n}, at least for even n (and surely the case of odd n could be handled similarly), may be inferred from the papers of Baginski [12], [13]. We note, though, that all of Baginski’s results are presented in terms of the zeros of the associated Legendre functions P_{ν}^{m} in the variable ν (for m an integer). To make the connection to the present setting, one makes use of the formulas in Section 6 following equation (6.4) (or equations (3.17) and (3.18) in [8], with the correction that the upper index in both associated Legendre functions should be negated), which relate our functions u_{m} as defined above to associated Legendre functions (up to constant factors). Note, in this connection, that y_{1}(θ) = u_{0}(θ; λ_{1}) and y_{2}(θ) = u_{1}(θ; λ_{2}) up to constant factors (as proved in Lemma 3.1 above; throughout this section we take these factors to be 1). In general, one has u_{m}(θ; λ) proportional to (sin θ)^{1−n/2} P_{ν}^{−(n/2−1+m)}(cos θ), where λ and ν are related by λ = (ν − n/2 + 1)(ν + n/2). We thank the referee for calling Baginski’s papers to our attention.

(2) Another proof of Lemma 3.1 follows by mimicking our proof of Lemma 3.1 of [9]. With τ = λ_{2}(h_{0}) and v as the associated eigenfunction we can assume that for some a ∈ (0, θ_{1}) v > 0 on (0, a) and v < 0 on (a, θ_{1}), and hence that v(a) = 0, v′(a) < 0. Also we take λ = λ_{1}(h_{1}) and set g = u_{1}(θ; λ) and h = u_{0}(θ; λ). Since c_{1} = λ/n > 0 it is clear that g > 0 on (0, θ_{1}) and that g(0) = 0 = g(θ_{1}). It also follows that g = −h′ and h satisfies

−h′′ − (n − 1) cot θ h′ = λh

while v satisfies

−v′′ − (n − 1) cot θ v′ = τv.

From the last two equations we obtain

[(sin θ)^{n−1}(vh′ − v′h)]′ = −(λ − τ)(sin θ)^{n−1}vh

and integration from a to θ_{1} produces

(3.12) (sin θ_{1})^{n−1}v′(θ_{1})h(θ_{1}) − (sin a)^{n−1}v′(a)h(a) = (λ − τ) ∫_{a}^{θ_{1}} vh sin^{n−1} θ dθ.

We now argue by contradiction, so assume τ ≤ λ. Since v(a) = 0, v = u_{0}(θ; τ), h = u_{0}(θ; λ) and τ ≤ λ, by the fact that the positive zeros of u_{0}(θ; λ) are decreasing with increasing λ it follows that the first positive zero of h is less than or equal to a and, since h′ = −g < 0 on (0, θ_{1}), it therefore follows that h < 0 on (a, θ_{1}]. But now (3.12) gives a contradiction, since its right-hand side is greater than or equal to 0 while its left-hand side is negative (note that v′(a) < 0 and v′(θ_{1}) > 0 since a and θ_{1} must be successive zeros of v).

(3) Yet another proof of Lemma 3.1 would be via the level–ordering results of Baumgartner, Grosse, and Martin [18], [19]. Specifically, see our papers [3], [4] where proofs for a ball in ℝ^{n} occur and also the papers [16], [17] of Baumgartner, which give extensions to cases arising from separation of variables in spherical coordinates in spaces of constant curvature.

For future reference we note the following lemma.

Lemma 3.2. If 0 < θ_{1} < π, the first eigenfunction of (3.1) with m = 0, i.e., y_{1}(θ) ≡ u_{0}(θ; λ_{1}), is strictly decreasing on [0, θ_{1}] (y_{1} > 0 on [0, θ_{1}) is our convention for y_{1} here and throughout this paper; this follows from our choice c_{0} = 1).

Proof. u_{0}(θ; λ_{1}) satisfies

−(sin^{n−1} θ u′_{0})′ = λ_{1}(θ_{1}) sin^{n−1} θ u_{0} > 0

in [0, θ_{1}) (since λ_{1}(Ω) > 0 follows from the variational characterization of the eigenvalues of −Δ via the Rayleigh quotient ∫_{Ω} |∇φ|^{2}/ ∫_{Ω} φ^{2}), which implies that sin^{n−1} θ u′_{0} is decreasing in [0, θ_{1}). Hence sin^{n−1} θ u′_{0} < (sin^{n−1} θ u′_{0}) |_{θ=0}= 0, which proves the lemma.

Having identified λ_{1} and λ_{2} for −Δ on a spherical cap with Dirichlet boundary conditions we are now in position to investigate their behaviors and, in particular, that of λ_{2}/λ_{1}. Since our concern will be with how these functions vary with θ_{1}, the geodesic radius of the spherical cap, we shall denote λ_{1} and λ_{2} by λ_{1}(θ_{1}) and λ_{2}(θ_{1}) throughout the remainder of this section. Associated with equation (3.1) is the one–dimensional Schrödinger operator

H_{m}(θ_{1}) = −\frac{d^{2}}{dθ^{2}} + \frac{(2m + n − 1)(2m + n − 3)}{4 sin^{2} θ} − \frac{(n − 1)^{2}}{4} (3.13)

acting on L^{2}((0, θ_{1}), dθ) with Dirichlet boundary conditions imposed at 0 and θ_{1}. The operators H_{m}(θ_{1}) form a family of self–adjoint operators. It is clear by Lemma 3.1 that λ_{1}(θ_{1}) = λ_{1}(H_{0}(θ_{1})) and λ_{2}(θ_{1}) = λ_{1}(H_{1}(θ_{1})).

We now analyze how λ_{1} and λ_{2} vary with θ_{1} by using perturbation theory [34], [36], [45]. To be successful at this we need to work on a fixed interval (0, θ_{1}) and we do this by observing that the eigenvalue problem H_{m}(cθ_{1})v = λv on (0, cθ_{1}) can be rescaled to

[−\frac{1}{c^{2}} \frac{d^{2}}{dt^{2}} + \frac{(2m + n − 1)(2m + n − 3)}{4 sin^{2} ct} − \frac{(n − 1)^{2}}{4} ] v = λv for t ∈ (0, θ_{1}) (3.14)

which is equivalent to

[−\frac{d^{2}}{dθ^{2}} + \frac{(2m + n − 1)(2m + n − 3)c^{2}}{4 sin^{2} cθ} − \frac{(n − 1)^{2}c^{2}}{4} ] v = c^{2}λv for θ ∈ (0, θ_{1}). (3.15)

As was done above for H_{m}(θ_{1}), we define an operator H̃_{m}(c) on L^{2}(0, θ_{1}) via the differential expression appearing on the left-hand side of (3.15). It is clear from (3.15) that λ_{k}(H̃_{m}(c)) = c^{2}λ_{k}(H_{m}(cθ_{1})) and thus, in particular, that

λ_{1}(cθ_{1}) = c^{−2}λ_{1}(H̃_{0}(c)) (3.16)

and

λ_{2}(cθ_{1}) = c^{−2}λ_{1}(H̃_{1}(c)). (3.17)

What we intend to do now is to determine the derivatives λ′_{1}(θ_{1}) and λ′_{2}(θ_{1}) using perturbation theory and the fact that

λ′_{j}(θ_{1}) = \frac{1}{θ_{1}} \frac{dλ_{j}(cθ_{1})}{dc} |_{c=1} . (3.18)

Since H̃_{m}(c) is an analytic family in c for c near 1 we can apply regular Rayleigh–Schrödinger perturbation theory [34], [36], [45]. In fact

H̃_{m}(c) = H_{m}(θ_{1}) + \frac{(2m + n − 1)(2m + n − 3)}{4} (\frac{c^{2}}{sin^{2} cθ} − \frac{1}{sin^{2} θ}) − \frac{(n − 1)^{2}(c^{2} − 1)}{4}
(3.19)  = H_{m}(θ_{1}) + V_{m}(θ; c)

and since V_{m}(θ; c) is analytic in c for c near 1 we are assured that the operators H̃_{m}(c) form an analytic family of type (A) for c near 1 (see [45], p. 16 for the definition of analytic family of type (A), or see [34], p. 154; also see Chapter 7 of [36] for the definitive account of analytic perturbation theory). This allows us to compute the derivatives of the eigenvalues of H̃_{m}(c) using the first-order perturbation formula (cf. Kato [36], p. 391, eq. (3.18))

\frac{dλ_{1}(H̃_{m}(c))}{dc} |_{c=1}= \frac{∫_{0}^{θ_{1}} [\frac{∂V_{m}}{∂c} (θ; c) |_{c=1}] v_{m}^{2} dθ}{∫_{0}^{θ_{1}} v_{m}^{2} dθ} (3.20)

where the functions v_{m}(θ) denote first eigenfunctions of H_{m}(θ_{1}) = H̃_{m}(1).

By (3.18) we have

\frac{d}{dθ_{1}} [\frac{λ_{2}(θ_{1})}{λ_{1}(θ_{1})}] = \frac{1}{θ_{1}} [\frac{d}{dc} (\frac{λ_{2}(cθ_{1})}{λ_{1}(cθ_{1})})] |_{c=1}
= \frac{1}{θ_{1}} [λ_{1}(θ_{1}) (\frac{d}{dc}λ_{2}(cθ_{1}) |_{c=1}) − λ_{2}(θ_{1}) (\frac{d}{dc}λ_{1}(cθ_{1}) |_{c=1})] \frac{1}{λ_{1}(θ_{1})^{2}} .

Thus, showing that λ_{2}(θ_{1})/λ_{1}(θ_{1}) increases with increasing θ_{1} comes down to showing that

0 < \frac{1}{λ_{2}(θ_{1})} (\frac{d}{dc}λ_{2}(cθ_{1}) |_{c=1}) − \frac{1}{λ_{1}(θ_{1})} (\frac{d}{dc}λ_{1}(cθ_{1}) |_{c=1}) , (3.21)

which, by equations (3.16) and (3.17), reduces to showing

0 < \frac{1}{λ_{2}(θ_{1})} (\frac{d}{dc}λ_{1}(H̃_{1}(c)) |_{c=1}) − \frac{1}{λ_{1}(θ_{1})} (\frac{d}{dc}λ_{1}(H̃_{0}(c)) |_{c=1}) . (3.22)

From (3.19) we obtain

\frac{∂V_{m}}{∂c} (θ; c) |_{c=1}= \frac{1}{2}(2m + n − 1)(2m + n − 3) csc^{2} θ(1 − θ cot θ) − \frac{1}{2}(n − 1)^{2}

so that, by (3.20),

\frac{d}{dc}λ_{1}(H̃_{1}(c)) |_{c=1}= \frac{1}{2}(n − 1)\frac{∫_{0}^{θ_{1}} [(n + 1) csc^{2} θ(1 − θ cot θ) − (n − 1)] v_{1}^{2} dθ}{∫_{0}^{θ_{1}} v_{1}^{2} dθ}, (3.23)

and

\frac{d}{dc}λ_{1}(H̃_{0}(c)) |_{c=1}= \frac{1}{2}(n − 1)\frac{∫_{0}^{θ_{1}} [(n − 3) csc^{2} θ(1 − θ cot θ) − (n − 1)] v_{0}^{2} dθ}{∫_{0}^{θ_{1}} v_{0}^{2} dθ}. (3.24)

The functions v_{0} and v_{1} are related to u_{0}(θ; λ_{1}) and u_{1}(θ; λ_{2}) by v_{0} = u_{0} sin^{(n−1)/2} θ and v_{1} = u_{1} sin^{(n−1)/2} θ respectively.

Introducing the functions ℓ(θ) = cot θ−θ csc^{2} θ and m(θ) = −ℓ′(θ)/2 = csc^{2} θ (1−θ cot θ) we can write (3.23) as

\frac{d}{dc}λ_{1}(H̃_{1}(c)) |_{c=1}= \frac{1}{2}(n − 1)\frac{∫_{0}^{θ_{1}} [(n + 1)m(θ) − (n − 1)] v_{1}^{2} dθ}{∫_{0}^{θ_{1}} v_{1}^{2} dθ}, (3.25)

and (3.24) as

\frac{d}{dc}λ_{1}(H̃_{0}(c)) |_{c=1}= \frac{1}{2}(n − 1)\frac{∫_{0}^{θ_{1}} [(n − 3)m(θ) − (n − 1)] v_{0}^{2} dθ}{∫_{0}^{θ_{1}} v_{0}^{2} dθ}, (3.26)

respectively. Using the relations m(θ) = −ℓ′(θ)/2, ℓ(θ) cot θ = m(θ)−1, and v_{0}^{2} = u_{0}^{2} sin^{n−1} θ, we can rewrite the numerator of the right-hand side of (3.26) as

∫_{0}^{θ_{1}}[(n − 1)(m(θ) − 1) − 2 m(θ)] v_{0}^{2} dθ = ∫_{0}^{θ_{1}}[(n − 1)ℓ(θ) cot θ + ℓ′(θ)] u_{0}^{2} sin^{n−1} θ dθ
= ∫_{0}^{θ_{1}}[ℓ(θ) sin^{n−1} θ]′ u_{0}^{2} dθ
(3.27)  = ∫_{0}^{θ_{1}}[−2ℓ(θ) u_{0}(θ) u′_{0}(θ)] sin^{n−1} θ dθ,

where the last equality follows by integrating by parts (both boundary terms vanish). Finally, from (3.26) and (3.27) we obtain

\frac{d}{dc}λ_{1}(H̃_{0}(c)) |_{c=1}= (n − 1)\frac{∫_{0}^{θ_{1}} [−ℓ(θ) u_{0}(θ) u′_{0}(θ)] sin^{n−1} θ dθ}{∫_{0}^{θ_{1}} v_{0}^{2} dθ}. (3.28)

At this point we need the following properties of the functions ℓ(θ) and m(θ).

Lemma 3.3. The function

ℓ(θ) ≡ cot θ − θ csc^{2} θ (3.29)

is negative, decreasing, and concave for 0 < θ < π. Moreover, the function

m(θ) ≡ −\frac{1}{2}ℓ′(θ) = csc^{2} θ (1 − θ cot θ) (3.30)

is positive, increasing, and convex for 0 < θ < π. Also, m(0) = 1/3, m(π/2) = 1, and m(π^{−}) = ∞.

Proof. Using the product representation of sin θ, i.e., sin θ = θ ∏_{k=1}^{∞}(1 − θ^{2}/(kπ)^{2}), one has

(3.31)  cot θ = ∑_{k=−∞}^{∞} \frac{1}{θ + kπ}  and  csc^{2} θ = ∑_{k=−∞}^{∞} \frac{1}{(θ + kπ)^{2}}

(convergence of the series for cot θ here is understood in the sense of symmetric partial sums). From (3.31) we obtain the following representation for ℓ(θ)

ℓ(θ) = − ∑_{k=1}^{∞} [\frac{kπ}{(kπ − θ)^{2}} − \frac{kπ}{(kπ + θ)^{2}} ] . (3.32)

It follows from (3.32) that ℓ(θ) < 0 for 0 < θ < π. Also from (3.32) we have

m(θ) = −\frac{ℓ′(θ)}{2} = ∑_{k=1}^{∞} [\frac{kπ}{(kπ + θ)^{3}} + \frac{kπ}{(kπ − θ)^{3}} ] , (3.33)

which is positive for 0 < θ < π. Thus, m(θ) is positive and ℓ(θ) is decreasing for 0 < θ < π. Taking derivatives again we find,

m′(θ) = 3 ∑_{k=1}^{∞} [\frac{kπ}{(kπ − θ)^{4}} − \frac{kπ}{(kπ + θ)^{4}} ] . (3.34)

The right-hand side of (3.34) is positive for θ ∈ (0, π). Hence, m(θ) is increasing and ℓ(θ) is concave in (0, π). It also follows from (3.34) that m′(θ) is increasing, and therefore m(θ) is convex in (0, θ_{1}). Lastly, the values of m(θ) at θ = 0, π/2, and π^{−} are found by explicit evaluation.

Remark. In fact the function ℓ and all its derivatives are negative for 0 < θ < π.

With all these preliminary results in hand we are ready to prove Theorems 1.2 and 1.3.

Proof of Theorem 1.2. Since λ̃_{1}(c) ≡ λ_{1}(H̃_{0}(c)) = c^{2}λ_{1}(cθ_{1}) (see (3.16) above), we just need to prove that

\frac{d}{dc}λ̃_{1}(c) |_{c=1}= \frac{1}{θ_{1}} \frac{d}{dθ_{1}} (θ_{1}^{2}λ_{1}(θ_{1})) < 0. (3.35)

This inequality follows from (3.28) and Lemmas 3.2 and 3.3. Note that it holds for all θ_{1} ∈ (0, π).

Proof of Theorem 1.3. Since λ_{2}(θ_{1}) > λ_{1}(θ_{1}) and dλ̃_{1}/dc |_{c=1}< 0, to prove (3.22) and therefore

\frac{d}{dθ_{1}} (\frac{λ_{2}(θ_{1})}{λ_{1}(θ_{1})}) > 0

reduces to showing (since λ̃_{1}λ̃′_{2} − λ̃_{2}λ̃′_{1} can be grouped as λ̃_{1}(λ̃′_{2} − λ̃′_{1}) − (λ̃_{2} − λ̃_{1})λ̃′_{1})

\frac{dλ̃_{2}}{dc} |_{c=1} −\frac{dλ̃_{1}}{dc} |_{c=1}> 0, (3.36)

where λ̃_{2}(c) ≡ λ_{1}(H̃_{1}(c)). From equations (3.25) and (3.26) this is equivalent to proving

(n + 1)\frac{∫_{0}^{θ_{1}} m(θ)v_{1}^{2} dθ}{∫_{0}^{θ_{1}} v_{1}^{2} dθ} > (n − 3)\frac{∫_{0}^{θ_{1}} m(θ)v_{0}^{2} dθ}{∫_{0}^{θ_{1}} v_{0}^{2} dθ},

which is obviously true for n ≤ 3 since m(θ) is positive. For n ≥ 4 it suffices to show

\frac{∫_{0}^{θ_{1}} m(θ)v_{1}^{2} dθ}{∫_{0}^{θ_{1}} v_{1}^{2} dθ} > \frac{∫_{0}^{θ_{1}} m(θ)v_{0}^{2} dθ}{∫_{0}^{θ_{1}} v_{0}^{2} dθ}.

The fact that g = v_{1}/v_{0} = y_{2}/y_{1} is an increasing function of θ on [0, θ_{1}] for 0 < θ_{1} ≤ π/2 (see Section 4 below) implies that v̂_{1} ≡ v_{1}/(∫_{0}^{θ_{1}} v_{1}^{2} dθ)^{1/2} and v̂_{0} ≡ v_{0}/(∫_{0}^{θ_{1}} v_{0}^{2} dθ)^{1/2} must have exactly one crossing in (0, θ_{1}). Since m(θ) is positive and increasing in [0, π) the desired inequality (3.36) follows by applying Lemma 2.7 of [14], p. 69; this inequality is also known as Bank’s inequality [15] (see also [27] or [6], p. 607).

Remark. For n = 2 we can obtain a stronger version of Theorem 1.3 which holds for all θ_{1} ∈ (0, π). Since m(θ) is increasing and m(0) = 1/3 we have (n + 1)m(θ) − (n − 1) ≥ (n + 1)/3 − (n − 1) = 2(2 − n)/3 = 0 if n = 2. Thus, from (3.25) we have

\frac{dλ̃_{2}}{dc} |_{c=1}> 0 if n = 2.

This inequality together with (3.35) implies (3.36) (or, even more directly, (3.22)) and therefore Theorem 1.3 for n = 2 and 0 < θ_{1} < π. In fact, this argument shows that for all θ_{1} ∈ (0, π) θ_{1}^{2}λ_{1}(θ_{1}) is decreasing and θ_{1}^{2}λ_{2}(θ_{1}) is increasing. These lead immediately to the fact that λ_{2}(θ_{1})/λ_{1}(θ_{1}) and θ_{1}^{2}[λ_{2}(θ_{1}) − λ_{1}(θ_{1})] are both increasing for 0 < θ_{1} < π. For n = 3 a related argument allows us to show that θ_{1}^{2}[λ_{2}(θ_{1}) − λ_{1}(θ_{1})] is increasing for 0 < θ_{1} < π, and hence that Theorem 1.3 extends to 0 < θ_{1} < π in that case as well. For n ≥ 4 we only have a proof of Theorem 1.3 for 0 < θ_{1} ≤ π/2.

Next we prove two inequalities between the first two Dirichlet eigenvalues of a geodesic ball which are needed in Section 4.

Theorem 3.1. Let λ_{1} and λ_{2} be the first two eigenvalues of the Dirichlet Laplacian on a geodesic ball contained in a hemisphere of 𝕊^{n}. Denote its (geodesic) radius by θ_{1}. Then

\frac{λ_{2} − n}{λ_{1}} ≥ \frac{n + 2}{n} for 0 < θ_{1} ≤ π/2 (3.37)

with equality if and only if θ_{1} = π/2 (i.e., for the hemisphere).

Remarks. (1) If we consider π/2 < θ_{1} < π then inequality (3.37) is reversed (and is a strict inequality, i.e., (λ_{2} − n)/λ_{1} < (n + 2)/n for π/2 < θ_{1} < π). This inequality is actually quite interesting since it allows us to control the way λ_{2} goes to n (from above) as θ_{1} → π^{−} in terms of λ_{1} (since λ_{1} goes to 0 as θ_{1} → π^{−}). The proof of this reversed inequality will not be given, since it follows by making suitable modifications to the proof of Theorem 3.1, as given below.

(2) Theorem 3.1 is the analog for 𝕊^{n} of our Lemma 2.2 of [7] for the Euclidean case. Indeed, the inequality of Lemma 2.2 follows (except that we do not get a strict inequality) if we consider (3.37) in the limit θ_{1} → 0^{+} (the “Euclidean limit”). Since λ_{1} ≈ α^{2}/θ_{1}^{2} and λ_{2} ≈ β^{2}/θ_{1}^{2} for θ_{1} near 0 where α and β are the Bessel function zeros j_{n/2−1,1} and j_{n/2,1} (see [1] for the notation here), the inequality β^{2}/α^{2} ≥ (n + 2)/n follows.

Proof. The proof is similar to that of Lemma 2.2 of [7]. We assume that θ_{1} < π/2 through most of the proof, returning to the case θ_{1} = π/2 (which can be treated explicitly in terms of elementary functions) only at the end. We shall use a suitable trial function (based on y_{2}) in the Rayleigh quotient for λ_{1} = λ_{1}(θ_{1}):

(3.38) λ_{1} ≤ \frac{∫_{0}^{θ_{1}}(u′)^{2} sin^{n−1} θ dθ}{∫_{0}^{θ_{1}} u^{2} sin^{n−1} θ dθ} = −\frac{∫_{0}^{θ_{1}} u(sin^{n−1} θ u′)′ dθ}{∫_{0}^{θ_{1}} u^{2} sin^{n−1} θ dθ}.

It suffices that the trial function u be real and continuous on [0, θ_{1}], have u(0) finite and u(θ_{1}) = 0, and be such that all the integrals occurring above exist as finite real numbers. This includes, in particular, the case u = y_{2}/ sin θ, which we now adopt. With this choice we find

− (sin^{n−1} θ u′)′ = −\frac{d}{dθ} [sin^{n−1} θ (\frac{y_{2}}{sin θ})′]
(3.39)  = − sin^{n−2} θ y′′_{2} − (n − 3) sin^{n−2} θ cot θ y′_{2}
+ (n − 3) sin^{n−4} θ y_{2} − (n − 2) sin^{n−2} θ y_{2},

and, upon using the differential equation satisfied by y_{2} (i.e., (3.1) with m = 1), we have

(3.40) − (sin^{n−1} θ u′)′ = 2 sin^{n−2} θ cot θ y′_{2} + (λ_{2} − n + 2) sin^{n−2} θ y_{2} − 2 sin^{n−4} θ y_{2}.

That the expression on the right here has a finite limit as θ → 0^{+} follows from the fact that y_{2}(θ) can be taken as u_{1}(θ) and u_{1}(θ) = c_{1}θ + O(θ^{3}) as θ → 0^{+} (see (3.1) and (3.2)). It now follows from (3.38) and (3.40) (since for θ_{1} < π/2, u = y_{2}/ sin θ does not satisfy the equation that y_{1} does, we can write a strict inequality here) that

(3.41) λ_{1} ∫_{0}^{θ_{1}} uy_{2} sin^{n−2} θ dθ < ∫_{0}^{θ_{1}} u [2 cot θ y′_{2} + (λ_{2} − n + 2)y_{2} − 2 csc^{2} θ y_{2}] sin^{n−2} θ dθ.

Since we would like to show that λ_{1} < n(λ_{2} − n)/(n + 2), it will be enough to show that the right-hand side of (3.41) is less than or equal to [n(λ_{2} − n)/(n + 2)] ∫_{0}^{θ_{1}} uy_{2} sin^{n−2} θ dθ, or, equivalently,

(n+2) ∫_{0}^{θ_{1}} u [2 cot θ y′_{2} + (λ_{2} − n + 2)y_{2} − 2 csc^{2} θ y_{2}] sin^{n−2} θdθ ≤ n(λ_{2}−n) ∫_{0}^{θ_{1}} uy_{2} sin^{n−2} θdθ or

(3.42) 2 ∫_{0}^{θ_{1}} u [(n + 2) cot θ y′_{2} + (λ_{2} + 2)y_{2} − (n + 2) csc^{2} θ y_{2}] sin^{n−2} θdθ ≤ 0.

We now rewrite this integral using y_{2} = u_{1}(θ; λ_{2}) (where λ_{2} = λ_{2}(θ_{1})) and proceed to simplify the integrand using the relations (3.3), (3.6) developed for the u_{m}’s. First, using (3.3) with m = 1 the expression in square brackets becomes

(n + 2) cot θ[cot θ u_{1} − u_{2}] + (λ_{2} + 2)u_{1} − (n + 2) csc^{2} θ u_{1}

or, since cot^{2} θ − csc^{2} θ = −1,

−(n + 2) cot θ u_{2} + (λ_{2} − n)u_{1}.

Finally, we use the recursion relation (3.6) with m = 2 to see that

u_{3} = (n + 2) cot θ u_{2} − (λ − n)u_{1}

and hence, since we are taking λ = λ_{2}, inequality (3.42) may be rewritten simply as

−2 ∫_{0}^{θ_{1}} uu_{3} sin^{n−2} θ dθ ≤ 0,

which clearly holds if u_{3} > 0 for 0 < θ < θ_{1} since u = y_{2}/ sin θ and y_{2} > 0 on (0, θ_{1}). To see that u_{3} > 0 for 0 < θ < θ_{1} we use the relation (3.7), first for m = 1 and then for m = 2. With m = 1 we have

sin^{n} θ u_{2}(θ) = (λ_{2} − n) ∫_{0}^{θ}(sin t)^{n}u_{1}(t) dt,

showing that u_{2}(θ; λ_{2}) > 0 for 0 < θ ≤ θ_{1} since u_{1}(θ; λ_{2}) = y_{2}(θ) > 0 on (0, θ_{1}) and λ_{2} = λ_{2}(θ_{1}) > n for 0 < θ_{1} ≤ π/2 by domain monotonicity of Dirichlet eigenvalues (and the fact that λ_{2}(π/2) = 2(n + 1) > n). This in turn yields u_{3} > 0 for 0 < θ ≤ θ_{1} if 0 < θ_{1} < π/2 since by (3.7) with m = 2 we have

sin^{n+1} θ u_{3}(θ) = (λ_{2} − 2(n + 1)) ∫_{0}^{θ}(sin t)^{n+1}u_{2}(t) dt,

and we know by domain monotonicity that λ_{2}(θ_{1}) > λ_{2}(π/2) = 2(n + 1) for 0 < θ_{1} < π/2.

Thus we have proved inequality (3.37) of our theorem for 0 < θ_{1} < π/2 (with strict inequality in (3.37)). It only remains to show that equality holds when θ_{1} = π/2 and this is elementary since in this case we can solve explicitly for all eigenvalues and eigenfunctions without even the need of special functions. One finds that λ_{1} = n with eigenfunction x_{n+1} = cos θ and that λ_{2} = 2(n + 1) with exactly n linearly independent eigenfunctions x_{i}x_{n+1} for i = 1, 2, …, n. Clearly (λ_{2} − n)/λ_{1} = (n + 2)/n and our proof is complete.

Remark. To fill in the picture for the hemisphere in 𝕊^{n}, we note that y_{1} = cos θ and y_{2} = sin θ cos θ, and thus u_{1} = sin θ cos θ, u_{2} = sin^{2} θ, and u_{3} ≡ 0. In this case u = y_{2}/ sin θ ≡ y_{1} and (3.38) becomes an equality.

Lemma 3.4. With notation as above,

λ_{2} − λ_{1} > \frac{n − 1}{sin^{2} θ_{1}} (3.43)

for 0 < θ_{1} ≤ π/2.

Proof. The proof is similar to that of Lemma 2.1 of [7]. We use u = y_{2} in the Rayleigh quotient for λ_{1} (i.e., in (3.38) above). Since

− (sin^{n−1} θ y′_{2})′ = sin^{n−1} θ (λ_{2} − \frac{n − 1}{sin^{2} θ}) y_{2},

we get from (3.38)

(3.44) λ_{1} < λ_{2} − \frac{∫_{0}^{θ_{1}} u^{2}((n − 1)/ sin^{2} θ) sin^{n−1} θ dθ}{∫_{0}^{θ_{1}} u^{2} sin^{n−1} θ dθ}

(again, y_{2} does not satisfy the equation that y_{1} does, so (3.44) is strict). Now the lemma follows by the monotonicity of sin θ in (0, θ_{1}) for 0 < θ_{1} ≤ π/2.

To conclude this section we present the following results which are needed in Section 4.

Theorem 3.2. Let p(θ) = −u′_{0}(θ, λ_{1})/u_{0}(θ, λ_{1}) = u_{1}(θ, λ_{1})/u_{0}(θ, λ_{1}). Then, p(θ) is positive, strictly increasing, and strictly convex on (0, θ_{1}) for 0 < θ_{1} ≤ π/2. Moreover, p(0) = 0 and p(θ) → ∞ as θ → θ_{1}^{−}.

Proof. The analog of this result in the Euclidean case was proved in Lemma 2.3 of [7]. That p(0) = 0 and p(θ) → ∞ as θ → θ_{1}^{−} follow from the boundary behavior of u_{0}. Using the raising and lowering identities (3.3) and (3.4) with m = 0 and m = 1 respectively and with λ fixed at λ_{1}(θ_{1}), i.e.,

u_{1} = −u′_{0}

(3.45)

λ_{1}u_{0} = u′_{1} + (n − 1) cot θ u_{1},

one obtains

u_{0}^{2} p′ = λ_{1}u_{0}^{2} + u_{1}^{2} − (n − 1) cot θ u_{0}u_{1} ≡ σ(θ), (3.46)

and in similar fashion

(sin^{n−1} θ σ(θ))′ = (n − 1) sin^{n−3} θ u_{0}u_{1}. (3.47)

The function σ(θ) is finite at θ = 0, and thus σ(θ) sin^{n−1} θ = 0 at θ = 0. Since u_{0} is positive and decreasing in (0, θ_{1}), u_{0}u_{1} = −u_{0}u′_{0} > 0 there. Hence, σ(θ) > 0 in (0, θ_{1}] and therefore by (3.46) p is increasing there. Clearly p(θ) > 0 in (0, θ_{1}), since p(0) = 0 and p is increasing there. Moreover, from (3.46) we find

u_{0}^{2}p′′ = (n − 1)\frac{u_{0}^{2}p}{sin^{2} θ} − (n − 1) cot θ σ + 2pσ ≡ s(θ). (3.48)

From (3.45), (3.46), (3.47), and (3.48) we obtain

(3.49) (sin^{n+1} θ s(θ))′ = 2 sin^{n} θ σ(θ) \{\frac{σ}{u_{0}^{2}} sin θ + (n − 1) sin θ + 2p cos θ\} > 0

in (0, θ_{1}) for 0 < θ_{1} ≤ π/2. Since sin^{n+1} θ s(θ) = 0 at θ = 0, this implies that s(θ) > 0 in (0, θ_{1}) and therefore, by (3.48), p′′ > 0 there and the proof is complete.

Remark. The function p satisfies the Riccati equation

p′ = λ_{1} + p^{2} − (n − 1) cot θ p. (3.50)

An alternative proof of Theorem 3.2 can be given directly from (3.50). In particular, the fact that p is increasing follows from (3.50) and the convexity of cot θ in (0, π/2) using arguments similar to the ones used to prove Theorem 4.1 below.

Lemma 3.5. The function p(θ) cot θ is strictly increasing on (0, θ_{1}) for 0 < θ_{1} < π/2. If θ_{1} = π/2, p(θ) = tan θ, so p(θ) cot θ ≡ 1 in that case.

Proof. Consider the function r(θ) ≡ p(θ) cot θ − λ_{1}/n. Using (3.47), (3.48), and (3.50) one can show that r(θ) satisfies the equation

(sin^{n−1} θ u_{0}^{2}r′)′ = 2n sin^{n−3} θ u_{0}^{2}r. (3.51)

Since the function p(θ) is odd and analytic in θ for θ near 0, we can expand it in odd powers of θ. Inserting a series expansion in the Riccati equation (3.50) we find

p(θ) = \frac{λ_{1}}{n} θ + \frac{λ_{1}}{3n^{2}(n + 2)} (3λ_{1} + n(n − 1)) θ^{3} + O(θ^{5})

for θ near 0. Substituting this into the definition of r(θ) we find

r(θ) ≈ \frac{λ_{1}}{n^{2}(n + 2)}(λ_{1} − n)θ^{2}

as θ → 0^{+}. Since λ_{1}(θ_{1}) > n for θ_{1} < π/2 (which follows from the fact that λ_{1}(π/2) = n and λ_{1}(θ_{1}) is decreasing in θ_{1}), r(θ) is positive in a neighborhood of 0. Now, r(θ) is a continuously differentiable function in [0, θ_{1}), it is positive in a neighborhood of 0, and it goes to infinity as θ → θ_{1}^{−}. This implies that either r(θ) is an increasing function in [0, θ_{1}) or that it has a positive local maximum. However, this latter possibility is ruled out by (3.51) (any possible positive critical point of r(θ) must be a local minimum). Thus, r(θ) is increasing on (0, θ_{1}) if 0 < θ_{1} < π/2. If θ_{1} = π/2, u_{0} = cos θ, hence u′_{0} = − sin θ, p(θ) = tan θ, and finally r(θ) ≡ 0 since λ_{1} = n.

Remark. In the Euclidean case, the analog of Lemma 3.5 (i.e., Lemma 2.4 of [7]) was an immediate consequence of the convexity of the function analogous to the function p(θ) used here. For 𝕊^{n}, as we have seen, the proof is somewhat more involved.

4 Monotonicity properties of g and B

In this section we prove the key properties of the functions that occur in our rearrangement procedure in Section 6. These functions are

q(θ) = sin θ\frac{g′(θ)}{g(θ)}, (4.1)

and

B(θ) = g′(θ)^{2} + \frac{n − 1}{sin^{2} θ}g(θ)^{2} = [q^{2} + (n − 1)] (\frac{g}{sin θ})^{2} , (4.2)

where

g(θ) = \frac{y_{2}(θ)}{y_{1}(θ)} = \frac{u_{1}(θ; λ_{2}(θ_{1}))}{u_{0}(θ; λ_{1}(θ_{1}))}. (4.3)

Our objectives in this section are to prove that g(θ) is increasing and B(θ) is decreasing in [0, θ_{1}] for 0 < θ_{1} ≤ π/2. For the hemisphere (i.e., for θ_{1} = π/2) we can explicitly compute g(θ) = sin θ and B(θ) = (n − 1) + cos^{2} θ (see the remark following the proof of Theorem 3.1). It is obvious that g is increasing and B is decreasing in this case. Thus, we can assume in the rest of this section that θ_{1} < π/2.

Since g and sin θ are positive, that g is increasing will be a simple consequence of showing q ≥ 0. On the other hand, from (4.1) and (4.2) it follows that

(4.4) B′(θ) = 2 [qq′ − (cos θ − q)(q^{2} + n − 1)\frac{1}{sin θ}] (\frac{g(θ)}{sin θ})^{2} .

Hence, that B is decreasing will be a consequence of showing that q′ ≤ 0 and 0 ≤ q ≤ cos θ. Thus, in order to prove the desired properties of g and B we only need to show that 0 ≤ q(θ) ≤ cos θ and q′(θ) ≤ 0 for 0 ≤ θ ≤ θ_{1}.

The strategy we use to prove these results for q follows the same general method used in [7] and [9]. Since we shall need the boundary behavior of q at the two endpoints θ = 0 and θ = θ_{1} (this is necessitated by the fact that the coefficients in the right-hand side of the differential equation for q, equation (4.7) below, become singular at the two endpoints), we give these now. By Taylor–Frobenius expansion, we find

q(0) = 1,  q′(0) = 0,  q′′(0) = 2 (\frac{λ_{1}}{n} − \frac{λ_{2}}{n + 2} − \frac{2 − n}{2(n + 2)}) , (4.5)

q(θ_{1}) = 0, q′(θ_{1}) = −\frac{1}{3} (λ_{2} − λ_{1} − \frac{n − 1}{sin^{2} θ_{1}} ) sin θ_{1}. (4.6)

Throughout this section we will use λ_{1} and λ_{2} to denote λ_{1}(θ_{1}) and λ_{2}(θ_{1}), respectively. From (4.5) and Theorem 3.1 we have that q′′(0) < −1 for θ_{1} < π/2. From (4.6) and Lemma 3.4 we have that q′(θ_{1}) < 0. Therefore, q < cos θ on an interval just to the right of 0 and q > 0 on an interval just to the left of θ_{1}.

In order to prove that q ≥ 0, q′ ≤ 0, and q ≤ cos θ for 0 ≤ θ ≤ θ_{1} we analyze the ordinary differential equation satisfied by q. To obtain the differential equation for q first we differentiate (4.1) with respect to θ, and use our choice of g = y_{2}/y_{1}. Then we use the equation (3.1), with m = 0 and λ = λ_{1}(θ_{1}), satisfied by y_{1}, and the same equation, but this time with m = 1 and λ = λ_{2}(θ_{1}), satisfied by y_{2}. Thus, we obtain

(4.7) q′ = 2pq − (n − 2)q cot θ − \frac{q^{2} + 1 − n}{sin θ} − (λ_{2} − λ_{1}) sin θ.

Here, the function p ≡ −y′_{1}/y_{1} obeys the Riccati equation

p′ − p^{2} + (n − 1) cot θ p − λ_{1} = 0 (4.8)

associated to equation (3.1) with m = 0 and λ = λ_{1}(θ_{1}) (note that (4.7) is also a Riccati equation). We proved in Section 3 that p is positive, strictly increasing, and strictly convex in (0, θ_{1}), for 0 < θ_{1} ≤ π/2, p(0) = 0, and p → ∞ as θ → θ_{1}^{−} (see Theorem 3.2).

Having derived the equation for q, we are ready to prove the necessary facts about q. We start by showing q ≥ 0 in [0, θ_{1}], which we prove by contradiction. Assume q is negative somewhere in [0, θ_{1}]. Since q(0) = 1 and q is positive to the left of θ_{1} (and q is continuous) this implies that there are two points α, β, say, with 0 < α < β < θ_{1} such that q(α) = q(β) = 0 and q′(α) ≤ 0, q′(β) ≥ 0. At points in (0, θ_{1}) where q = 0 it follows from (4.7) that

q′ = \frac{n − 1}{sin θ} − (λ_{2} − λ_{1}) sin θ. (4.9)

Since λ_{2} > λ_{1} and sin θ is increasing in θ for 0 < θ < π/2, the right-hand side of (4.9) is strictly decreasing in θ, and hence it is not possible to have α < β with q(α) = q(β) = 0 and q′(α) ≤ 0, q′(β) ≥ 0. Therefore q ≥ 0 in [0, θ_{1}].

The proof that q ≤ cos θ follows the same ideas. Define the function ψ = cos θ − q. From (4.7) we get

(4.10) ψ′ = (λ_{2} − λ_{1} − n) sin θ + (2 p + \frac{ψ}{sin θ})(ψ − cos θ) − (n − 1)ψ cot θ.

We have already shown that ψ(θ) = cos θ −q(θ) is positive in a neighborhood of θ = 0. Also, if θ_{1} < π/2, ψ(θ_{1}) = cos θ_{1} > 0. Now assume that ψ is negative somewhere in [0, θ_{1}]. This implies that there are two points r, s, say, with 0 < r < s < θ_{1} such that ψ(r) = ψ(s) = 0 and ψ′(r) ≤ 0, ψ′(s) ≥ 0. At points in (0, θ_{1}) where ψ = 0 we have from (4.10) that

\frac{1}{sin θ}ψ′ = (λ_{2} − λ_{1} − n) − 2 p cot θ. (4.11)

From Lemma 3.5 it follows that the right-hand side of (4.11) is strictly decreasing in θ, hence it is not possible to have r < s with ψ(r) = ψ(s) = 0 and ψ′(r) ≤ 0, ψ′(s) ≥ 0. Thus ψ ≥ 0, and hence q ≤ cos θ in [0, θ_{1}].

Finally, we show that q(θ) is decreasing in [0, θ_{1}]. It is convenient now to write (4.7) in an alternative form and use convexity arguments. Specifically we consider

(4.12) q′ = 2 p(θ) q + (n − 2)(1 − q) cot θ + \frac{1 − q^{2}}{sin θ} + (n − 2) tan(θ/2) − (λ_{2} − λ_{1}) sin θ

and, since λ_{2} − λ_{1} > 0, q ≤ 1, and n ≥ 2, the right-hand side of (4.12), F (θ, q), say, is convex in θ for fixed q because each of the functions cot θ, csc θ, tan(θ/2), and − sin θ is individually convex on the interval [0, θ_{1}] for 0 < θ_{1} ≤ π/2. Also the function p(θ) is convex on the same interval (see Theorem 3.2 above). To see how these facts imply that q′ ≤ 0 observe that if not we could find three points α_{1}, α_{2}, α_{3} in (0, θ_{1}), where q(α_{1}) = q(α_{2}) = q(α_{3}), q′(α_{1}) < 0, q′(α_{2}) > 0, and q′(α_{3}) < 0. But then we would have (we use q for the common value of q(α_{i}) here)

0 < q′(α_{2}) = F (α_{2}, q) = F (μ α_{1} + (1 − μ)α_{3}, q)
(4.13)  < μF (α_{1}, q) + (1 − μ)F (α_{3}, q) = μq′(α_{1}) + (1 − μ)q′(α_{3}) < 0,

a contradiction. We have used the strict convexity of F (θ, q) in θ here. The parameter μ is strictly between 0 and 1 and determines α_{2} as a convex combination of α_{1} and α_{3}, that is, α_{2} = μ α_{1} + (1 − μ)α_{3}. To summarize this section we state the results above as a theorem.

Theorem 4.1. With q(θ), B(θ), and g(θ) as defined in (4.1), (4.2), and (4.3), respectively, the inequalities 0 ≤ q ≤ cos θ and q′ ≤ 0 hold on [0, θ_{1}] for 0 < θ_{1} ≤ π/2. It follows that g(θ) is increasing and B(θ) is decreasing there as well.

5 Chiti’s comparison argument in 𝕊^{n}

Here we need an extension of Chiti’s comparison result [24], [25], [26], [27] (see also Appendix A of [6]), given originally for domains in ℝ^{n}, to the case of domains in 𝕊^{n}.

We let 𝕊^{n}(ρ) denote the n–dimensional sphere of radius ρ (hence of constant sectional curvature κ = 1/ρ^{2}). 𝕊^{n} will always denote 𝕊^{n}(1). Define the function

S_{ρ}(r) = ρ sin(r/ρ).

It is well known that, in geodesic polar coordinates, the metric on 𝕊^{n}(ρ) is

ds^{2} = dr^{2} + S_{ρ}(r)^{2}|dω|^{2},

where r represents geodesic distance from a point and |dω|^{2} is the canonical metric for 𝕊^{n−1}. One should think of r as ρ times the angle θ from the north pole. Thus r runs from 0 to ρπ.

We will always assume that r = 0 (the north pole) is the center for our spherical rearrangements (this is not a restriction since the metric is identical in geodesic polar coordinates about any point). Then for a bounded domain Ω ⊂ 𝕊^{n}(ρ) Ω^{⋆}, the spherical rearrangement of Ω, will denote the geodesic ball about r = 0 having the same volume as Ω, i.e., |Ω^{⋆}| = |Ω|.

For a nonnegative function f defined on Ω we define two rearranged functions f^{#} and f^{⋆}. The decreasing rearrangement f^{#} of f is a function from [0, |Ω|] to ℝ which is equimeasurable with f and nonincreasing. We will use s as the argument of f^{#} in most instances. The symmetric decreasing (or spherical decreasing) rearrangement f^{⋆} of f is a function defined on Ω^{⋆} which is invariant under rotations about r = 0, equimeasurable with f, and nonincreasing with respect to r. f^{⋆} is a function of x ∈ Ω^{⋆} but because of its symmetry we will abuse notation and write f^{⋆}(r) where r is the geodesic distance from the center of Ω^{⋆}. With this understanding we have

f^{⋆}(r) = f^{#}(A(r))

where

A(r) ≡ s = nC_{n} ∫_{0}^{r} S_{ρ}(τ)^{n−1} dτ (5.1)

is the n-volume of the geodesic ball of radius r in 𝕊^{n}(ρ). Here C_{n} = π^{n/2}/Γ(\frac{n}{2} + 1) denotes the volume of the unit ball in ℝ^{n} (and nC_{n} is its “surface area”, i.e., |𝕊^{n−1}|). We shall also have occasion to use increasing rearrangements. These will be denoted f_{#} (the increasing rearrangement of f) and f_{⋆} (the symmetric or spherical increasing rearrangement of f) and their definitions are analogous to those of f^{#} and f^{⋆}, respectively. For further information on rearrangements the reader is referred to the book of Hardy, Littlewood, and Pólya [32] and the many other references given in [6],[51].

Finally, we also need the classical isoperimetric inequality extended to 𝕊^{n}(ρ) in its sharp form. We let L(t) be the function giving the (n − 1)–dimensional volume of the geodesic ball of radius r, i.e.,

L(r) = nC_{n}S_{ρ}(r)^{n−1} = A′(r). (5.2)

Then, for example, for any domain in 𝕊^{2} one has

L^{2} ≥ 4πA − (A/ρ)^{2} = 4πA − κA^{2} (5.3)

(see, e.g., [41]). Here A = |Ω| and L is the length of the boundary of Ω. Equality occurs in (5.3) if and only if Ω is a geodesic ball. In 𝕊^{n}, n > 2, the sharp classical isoperimetric inequality cannot be given in as explicit a form as (5.3). For a bounded domain Ω we define H_{n−1}(∂Ω) to be the (n − 1)–dimensional volume of ∂Ω. The isoperimetric inequality on 𝕊^{n}(ρ) then reads

H_{n−1}(∂Ω) ≥ H_{n−1}(∂Ω^{⋆}) (5.4)

with equality if and only if Ω is a geodesic ball (see Burago and Zalgaller [21], p. 86, Theorem 10.2.1). In terms of the function L(r) defined above (5.4) may be written

H_{n−1}(∂Ω) ≥ L(θ(|Ω|)) = nC_{n}S_{ρ}(θ(|Ω|))^{n−1}, (5.5)

where θ(s) is the inverse function to the function A defined in (5.1).

With all these ingredients we state our extension of Chiti’s comparison result (throughout the rest of this section we set ρ = 1, the extension to arbitrary ρ being straightforward).

Theorem 5.1. Let Ω be a bounded domain in 𝕊^{n} and let λ_{1} and u_{1} denote the first Dirichlet eigenvalue and eigenfunction of the Laplacian on Ω. Let B_{λ_{1}} be the geodesic ball of such a radius that λ_{1} is also the first Dirichlet eigenvalue of the Laplacian on B_{λ_{1}}. Let v_{1} > 0 be the first Dirichlet eigenfunction on B_{λ_{1}} and fix its normalization so that ∫_{Ω} u_{1}^{2} dσ = ∫_{B_{λ_{1}}} v_{1}^{2} dσ.

Then there is a value r_{1} ∈ (0, θ(|B_{λ_{1}}|)) such that

v_{1}(r) ≥ u_{1}^{⋆}(r) for r ∈ [0, r_{1}], (5.6)

and

v_{1}(r) ≤ u_{1}^{⋆}(r) for r ∈ [r_{1}, θ(|B_{λ_{1}}|)], (5.7)

where θ(s) is the function defined following (5.5) above.

Proof. Let u_{1}, respectively λ_{1}, be the lowest eigenfunction, respectively eigenvalue, of the Dirichlet problem on Ω ⊂ 𝕊^{n}, i.e.,

−Δu_{1} = λ_{1}u_{1} in Ω, u_{1} = 0 on ∂Ω. (5.8)

Define Ω_{t} = {x | u_{1}(x) > t} and ∂Ω_{t} = {x | u_{1}(x) = t}. Let μ_{1}(t) = |Ω_{t}|, and |∂Ω_{t}| ≡ H_{n−1}(∂Ω_{t}), where H_{n−1}(dσ) denotes (n−1)–dimensional measure on 𝕊^{n}. Then we have (see, e.g., Talenti [51], p. 709, eq. (32))

−μ′_{1}(t) = ∫_{Ω_{t}} \frac{1}{|∇u_{1}|}H_{n−1}(dσ), (5.9)

for almost every t > 0. Applying Gauss’s theorem to (5.8), we have

λ_{1} ∫_{Ω_{t}} u_{1} dσ = ∫_{∂Ω_{t}} |∇u_{1}|H_{n−1}(dσ), (5.10)

since the outward normal to Ω_{t} is given by −∇u_{1}/|∇u_{1}|. Using the Cauchy–Schwarz inequality and equations (5.9) and (5.10) we obtain

|∂Ω_{t}|^{2} = (∫_{∂Ω_{t}} H_{n−1}(dσ))^{2} ≤ (−μ′_{1}(t))λ_{1} ∫_{Ω_{t}} u_{1} dσ. (5.11)

As discussed above, if Ω is a domain in 𝕊^{n}, the classical isoperimetric inequality is given by

H_{n−1}(∂Ω) ≥ H_{n−1}(∂Ω^{⋆}) (5.12)

where Ω^{⋆} is a geodesic ball having the same volume as Ω. The (n − 1)–dimensional measure of ∂Ω^{⋆}, H_{n−1}(∂Ω^{⋆}), is given in terms of θ_{1}(Ω^{⋆}), the geodesic radius of Ω^{⋆}, by

H_{n−1}(∂Ω^{⋆}) = nC_{n}(sin θ_{1}(Ω^{⋆}))^{n−1}, (5.13)

where C_{n} = π^{n/2}/Γ(\frac{n}{2} + 1) is the volume of the unit ball in ℝ^{n} (and nC_{n} is the volume of 𝕊^{n−1}). Therefore, from (5.12) and (5.13) it follows that

|∂Ω_{t}| = H_{n−1}(∂Ω_{t}) ≥ nC_{n}(sin θ_{1}(Ω_{t}^{⋆}))^{n−1}. (5.14)

Hence, from (5.11) we have

λ_{1} ∫_{Ω_{t}} u_{1} dσ ≥ n^{2}C_{n}^{2}[sin θ_{1}(Ω_{t}^{⋆})]^{2n−2} (−\frac{1}{μ′_{1}(t)}) . (5.15)

Finally one uses the fact that

∫_{Ω_{t}} u_{1} dσ = ∫_{0}^{μ_{1}(t)} u_{1}^{#}(s) ds, (5.16)

which follows directly from the definition of u_{1}^{#}, the decreasing rearrangement of u_{1} on the interval [0, |Ω|] (here s is a variable denoting volume and is related to the geodesic radial variable θ via s = nC_{n} ∫_{0}^{θ}(sin r)^{n−1} dr, i.e., ds/dθ = nC_{n}(sin θ)^{n−1}).

Since u_{1}^{#}(s) is the inverse function to μ_{1}(t) we have

−\frac{du_{1}^{#}}{ds} = −\frac{1}{μ′_{1}(t)},

which, combined with (5.15) and (5.16), yields

−\frac{du_{1}^{#}}{ds} ≤ λ_{1}n^{−2}C_{n}^{−2}(sin θ(s))^{2−2n} ∫_{0}^{s} u_{1}^{#}(s′) ds′. (5.17)

Now, if we view v_{1} as a function of the volume s (here we will abuse notation and continue to call it v_{1}) rather than as a function of θ (or r), with s = nC_{n} ∫_{0}^{θ}(sin r)^{n−1} dr, it satisfies (5.17) with equality, i.e.,

−\frac{dv_{1}}{ds} = λ_{1}n^{−2}C_{n}^{−2}(sin θ(s))^{2−2n} ∫_{0}^{s} v_{1}(s′) ds′. (5.18)

In fact, (5.18) is an integrated version of equation (3.1) with m = 0 and λ = λ_{1} in the variable s. Having obtained the relations (5.17) and (5.18) for u_{1}^{#}(s) and v_{1}(s) respectively, we will prove that under the normalization condition imposed on them, they are either identical or they cross only once in the interval (0, |B_{λ_{1}}|) (in the sense specified by Theorem 5.1). All the arguments we give below depend on the continuity of u_{1}^{#} and v_{1}. The function v_{1} is in fact real analytic in [0, |B_{λ_{1}}|] and, furthermore, it is decreasing there as can be seen from Lemma 3.2 above (or from (5.18) and the fact that v_{1} > 0). The absolute continuity of u_{1}^{#} on [0, |Ω|] follows from arguments in [51]. Since u_{1}^{#} and v_{1} are normalized to have the same L^{2}–norm, they either are identical or they cross. If they coincide, then B_{λ_{1}} = Ω^{⋆}, and Theorem 5.1 is proved since any r ∈ (0, θ(|B_{λ_{1}}|)) will serve as r_{1}. Next, following Chiti [26] we conclude that u_{1}^{#}(0) cannot exceed v_{1}(0). In fact, if u_{1}^{#}(0) ≥ v_{1}(0) it follows by mimicking the proof of the main theorem in [26] that v_{1}(s) ≤ u_{1}^{#}(s) for all s ∈ [0, |B_{λ_{1}}|) which, in turn, implies v_{1}(s) ≡ u_{1}^{#}(s) and we are back in the previous case. Thus, if v_{1}(s) ≢ u_{1}^{#}(s), v_{1}(s) > u_{1}^{#}(s) in a neighborhood of 0, and both functions being of the same norm they must cross at least once. Let s_{1} be the largest s ∈ (0, |B_{λ_{1}}|) such that u_{1}^{#}(s′) ≤ v_{1}(s′) for all s′ ≤ s. By the definition of s_{1}, there is an interval immediately to the right of s_{1} on which u_{1}^{#}(s) > v_{1}(s). Indeed, by continuity and the definition of s_{1}

∫_{0}^{s}[u_{1}^{#}(s′) − v_{1}(s′)] ds′ < 0 for 0 < s ≤ s_{1} + ϵ (5.19)

for some ϵ > 0. It now follows that u_{1}^{#}(s) > v_{1}(s) at least on the interval from s_{1} to s_{1} + ϵ since by the absolute continuity of u_{1}^{#}

(5.20) v_{1}(s) − u_{1}^{#}(s) = ∫_{s_{1}}^{s} [\frac{d}{ds}(v_{1} − u_{1}^{#})] ds
≤ λ_{1}n^{−2}C_{n}^{−2} ∫_{s_{1}}^{s} (sin θ(s′))^{2−2n} ∫_{0}^{s′}[u_{1}^{#}(s′′) − v_{1}(s′′)] ds′′ ds′
< 0 for s ∈ (s_{1}, s_{1} + ϵ]

by virtue of (5.19).

We will now show that u_{1}^{#}(s) > v_{1}(s) for all s ∈ (s_{1}, |B_{λ_{1}}|], which will prove the theorem. If not, then the point s_{2} defined to be the largest s ∈ (s_{1}, |B_{λ_{1}}|] for which u_{1}^{#}(s′) > v_{1}(s′) for all s_{1} < s′ < s, would be less than |B_{λ_{1}}|, and we would have

u_{1}^{#}(s) > v_{1}(s) for s ∈ (s_{1}, s_{2}) (5.21)

with u_{1}^{#}(s_{1}) = v_{1}(s_{1}) and u_{1}^{#}(s_{2}) = v_{1}(s_{2}). In this case, we can define the function

w(s) = \begin{cases}v_{1}(s) & for s ∈ [0, s_{1}] ∪ [s_{2}, |B_{λ_{1}}|] \\ u_{1}^{#}(s) & for s ∈ (s_{1}, s_{2}). \end{cases}

It follows from (5.17) and (5.18) that w satisfies

−\frac{dw}{ds}(s) ≤ λ_{1}n^{−2}C_{n}^{−2}(sin θ(s))^{2−2n} ∫_{0}^{s} w(s′) ds′, (5.22)

with strict inequality for all s > s_{1}. From w(s) define the function

g(x) = w(s(θ))

for x ∈ B_{λ_{1}} where θ is the polar angle (angle from the north pole) corresponding to x. Thus g is a radial function on B_{λ_{1}} (assumed centered at the north pole). Because of (5.22) (or (5.20) or (5.21)), g cannot be the groundstate of the Laplacian with Dirichlet boundary conditions on B_{λ_{1}} (but it is certainly an admissible trial function for λ_{1}). Therefore,

λ_{1} < \frac{∫_{B_{λ_{1}}} |∇g|^{2} dσ}{∫_{B_{λ_{1}}} g^{2} dσ} . (5.23)

By standard change of variables,

∫_{B_{λ_{1}}} g^{2} dσ = ∫_{0}^{|B_{λ_{1}}|} w^{2}(s) ds (5.24) and

∫_{B_{λ_{1}}} |∇g|^{2} dσ = n^{2}C_{n}^{2} ∫_{0}^{|B_{λ_{1}}|}(sin θ(s))^{2n−2}w′(s)^{2} ds. (5.25)

Using (5.22) (substitute for one of the w′(s)’s in (5.25), using the fact that −w′(s) > 0) and integration by parts we get

∫_{B_{λ_{1}}} |∇g|^{2} dσ ≤ λ_{1} ∫_{0}^{|B_{λ_{1}}|} w(s)^{2} ds. (5.26)

From (5.23), (5.24), and (5.26) we get a contradiction, and the theorem follows.

6 The main result

After all the preliminaries developed in Sections 2 through 5 we are ready to prove our main result, i.e., Theorem 1.1 from which the PPW result for domains contained in a hemisphere of 𝕊^{n} follows as indicated in the introduction. As in our proof of the PPW conjecture for domains in ℝ^{n}, the starting point here is the use of the gap inequality, which is a variational estimate for the difference between the first two eigenvalues of the Laplacian. The gap inequality states that

λ_{2}(Ω) − λ_{1}(Ω) ≤ \frac{∫_{Ω} |∇P |^{2}u_{1}^{2} dσ}{∫_{Ω} P^{2}u_{1}^{2} dσ} (6.1)

provided ∫_{Ω} P u_{1}^{2} dσ = 0 and P ≢ 0. Here Ω is a domain in 𝕊^{n} and dσ is the standard volume element in 𝕊^{n}. The gap inequality follows from the Rayleigh–Ritz inequality for λ_{2} using P u_{1} as the trial function (hence the side condition P u_{1} ⊥ u_{1}) after a suitable integration by parts. To get the desired isoperimetric result out of this one must make very special choices of the function P: in particular, choices such that (6.1) is an equality if Ω is the appropriate geodesic ball.

A key element needed to guarantee the orthogonality of the trial functions P_{i}u_{1} to u_{1} which we will use in the sequel is the center of mass argument embodied in Theorem 2.1 above.

Concerning the choice of trial functions P_{i} we proceed as follows. Thinking of 𝕊^{n} as the unit sphere in ℝ^{n+1} and with the center of mass point for Ω fixed at the north pole we take

P_{i} = g(θ)\frac{x_{i}}{sin θ}, i = 1, 2, …, n, (6.2)

where θ represents the angle of a point from the positive x_{n+1}–axis (the direction of the north pole). In 𝕊^{n} the variable θ is the geodesic radial variable with respect to the north pole. Division by sin θ = \sqrt{1 − x_{n+1}^{2}} normalizes the n–vector (x_{1}, x_{2}, …, x_{n}). The choice of g(θ), as in the Euclidean case, is determined by the fact that we must have equality in (6.1) when Ω is a geodesic ball. For a geodesic ball of radius θ_{1}, u_{1} = c_{1}y_{1} where y_{1} satisfies

y_{1}′′ + (n − 1) cot θ y′_{1} + λ_{1}y_{1} = 0 (6.3)

with boundary conditions y_{1}(0) finite and y_{1}(θ_{1}) = 0. This is just equation (3.1) with m = 0 and λ = λ_{1}. Also, u_{2} = c_{2}(x_{i}/ sin θ)y_{2} (for any i = 1, 2, …, n) where y_{2} satisfies

y′′_{2} + (n − 1) cot θ y′_{2} + (λ_{2} − \frac{n − 1}{sin^{2} θ})y_{2} = 0 (6.4)

with boundary conditions y_{2}(0) = y_{2}(θ_{1}) = 0 (this is just equation (3.1) with m = 1 and λ = λ_{2}). The values λ_{1} and λ_{2} are to be taken as the least eigenvalues of these onedimensional radial problems. By Lemma 3.1 these are the correct identifications of λ_{1} and λ_{2} for our geodesic ball. One can express y_{1} and y_{2} in terms of associated Legendre functions. In fact,

y_{1}(θ) = (sin θ)^{1−n/2} P_{ν_{1}}^{−(n/2−1)}(cos θ)

and

y_{2}(θ) = (sin θ)^{1−n/2} P_{ν_{2}}^{−n/2}(cos θ)

up to unimportant constant factors, where the parameters ν_{1} and ν_{2} are related to the eigenvalues λ_{1} and λ_{2} respectively by

ν(ν + 1) = λ + \frac{n(n − 2)}{4}.

We follow Abramowitz and Stegun [1] in our notation here; note that their convention for associated Legendre functions makes (sin θ)^{1−n/2} P_{ν}^{−μ}, where μ = n/2 − 1 + m and m is a nonnegative integer, and not (sin θ)^{1−n/2} P_{ν}^{μ}, the “right” n-dimensional generalization of the familiar associated Legendre functions P_{ν}^{m}, m = 0, 1, …, from 𝕊^{2} (or ℝ^{3}). This means, in particular, that when n is even P_{ν}^{μ} can be substituted for P_{ν}^{−μ} (they are then proportional), while if n is odd Q_{ν}^{μ} can be used (the distinction here is between μ being an integer or half an odd integer). Using P_{ν}^{−μ} with μ as above circumvents this “even-odd effect”. Since we want equality in (6.1) when Ω is a geodesic ball, using the form of u_{1} and u_{2} for a geodesic ball we see that g(θ) must be essentially the quotient of y_{2} by y_{1}.

Let Ω ⊂ 𝕊^{n} be contained in a hemisphere. Let B_{λ_{1}} denote the geodesic ball in 𝕊^{n} having the same value of λ_{1} as Ω. By Sperner’s inequality [49] (see also [31]) λ_{1}(B_{λ_{1}}) = λ_{1}(Ω) ≥ λ_{1}(Ω^{⋆}) and by the properties of λ_{1} for geodesic balls (in particular, λ_{1} decreases as the radius of the ball increases), we see that θ_{1} ≤ π/2, where θ_{1} denotes the geodesic radius of B_{λ_{1}}. We now set g(θ) = y_{2}(θ)/y_{1}(θ) for 0 ≤ θ ≤ θ_{1}, g(θ) = g(θ_{1}) for θ_{1} ≤ θ ≤ π/2 and we extend g(θ) to a function on [0, π] by reflection about θ = π/2. (Note that this definition makes B(θ) = g′(θ)^{2} + (n − 1)g(θ)^{2}/ sin^{2} θ a decreasing function on [θ_{1}, π/2], since B(θ) = (n − 1)g(θ_{1})^{2}/ sin^{2} θ there.) We then apply the center of mass result (Theorem 2.1) to Ω and G̃(θ) = g(θ)/ sin θ obtaining a choice of Cartesian coordinate axes such that

∫_{Ω} g(θ)\frac{x_{i}}{sin θ}u_{1}^{2} dσ = 0, for i = 1, 2, …, n.

Using the functions P_{i} = g(θ)x_{i}/ sin θ in (6.1) we get

(6.5) (λ_{2} − λ_{1}) ∫_{Ω} g(θ)^{2}(\frac{x_{i}}{sin θ})^{2}u_{1}^{2} dσ ≤ ∫_{Ω} |∇ (g(θ)\frac{x_{i}}{sin θ})|^{2}u_{1}^{2} dσ

for 1 ≤ i ≤ n, and summing on i from 1 to n we obtain

λ_{2} − λ_{1} ≤ \frac{∫_{Ω} B(θ)u_{1}^{2} dσ}{∫_{Ω} g(θ)^{2}u_{1}^{2} dσ} , (6.6)

where

B(θ) = g′(θ)^{2} + \frac{n − 1}{sin^{2} θ}g(θ)^{2} (6.7)

as defined previously (see equation (4.2)). We observe that, in spite of the fact that Ω is contained in a hemisphere, our use of the center of mass result may imply that Ω does not lie in the northern hemisphere (i.e., θ would not be in (0, π/2) for all points in Ω). To remedy this situation, we observe that since Ω is contained in a hemisphere, −Ω ⊂ 𝕊^{n} \ Ω. Thus, if we let Ω_{±} = {x⃗ ∈ Ω | ±x_{n+1} > 0} we have Ω_{+} ∩ (−Ω_{−}) = ∅. Since g(θ) and B(θ) are both symmetric with respect to θ = π/2 it follows that the integrals over Ω can be replaced by integrals over Ω̃ = Ω_{+} ∪ (−Ω_{−}) with no change in their values if we agree to transplant u_{1} along with Ω_{−} to −Ω_{−}. This follows since g and B were defined to be symmetric about θ = π/2 and therefore they transplant into themselves. After moving the whole problem to the northern hemisphere (where g is increasing and B is decreasing) we can carry out all the further rearrangements in exact parallel with the Euclidean case, encountering no further difficulties.

To conclude the proof of Theorem 1.1 we need the following two chains of inequalities. We have

∫_{Ω} B(θ)u_{1}^{2} dσ = ∫Ω̃ B(θ)ũ_{1}^{2} dσ ≤ ∫_{Ω^{⋆}} B(θ)^{⋆}u_{1}^{⋆2} dσ
(6.8)  ≤ ∫_{Ω^{⋆}} B(θ)u_{1}^{⋆2} dσ ≤ ∫_{B_{λ_{1}}} B(θ)v_{1}^{2} dσ,

and

∫_{Ω} g(θ)^{2}u_{1}^{2} dσ = ∫Ω̃ g(θ)^{2}ũ_{1}^{2} dσ ≥ ∫_{Ω^{⋆}} g(θ)_{⋆}^{2}u_{1}^{⋆2} dσ
(6.9)  ≥ ∫_{Ω^{⋆}} g(θ)^{2}u_{1}^{⋆2} dσ ≥ ∫_{B_{λ_{1}}} g(θ)^{2}v_{1}^{2} dσ.

Here ũ_{1} represents u_{1} as transplanted to Ω̃ and v_{1} is the first eigenfunction of −Δ on B_{λ_{1}} with Dirichlet boundary conditions and normalized so that ∫_{Ω} u_{1}^{2} dσ = ∫_{B_{λ_{1}}} v_{1}^{2} dσ. The functions g and B are likewise based on the eigenfunctions of the ball B_{λ_{1}} (so that the θ_{1} that goes into the boundary value problems (6.3) and (6.4) that define them is the radius of the ball B_{λ_{1}}). In each of (6.8) and (6.9), the equality is trivial, the first inequality follows simply from rearrangement (see Section 5 for our notation), the second inequality is by virtue of the monotonicity properties of g(θ) and B(θ), and the last inequality follows from our 𝕊^{n} analog of Chiti’s comparison result (see Section 5 for details) and also uses the monotonicity properties of g and B again. Finally, from (6.6), (6.8), and (6.9) we obtain

λ_{2}(Ω) − λ_{1}(Ω) ≤ \frac{∫_{B_{λ_{1}}} B(θ)v_{1}^{2} dσ}{∫_{B_{λ_{1}}} g(θ)^{2}v_{1}^{2} dσ} = λ_{2}(B_{λ_{1}}) − λ_{1}(B_{λ_{1}}). (6.10)

Hence (since λ_{1}(B_{λ_{1}}) = λ_{1}(Ω))

λ_{2}(Ω) ≤ λ_{2}(B_{λ_{1}}) (6.11)

which concludes the proof of Theorem 1.1, it being clear from any of a number of our previous inequalities that equality obtains in (6.11) if and only if Ω is itself a ball.

Remarks. (1) Theorems 1.1 (i.e., inequality (6.11)) and 1.4 also hold under somewhat more general circumstances than for Ω contained in a hemisphere of 𝕊^{n}. In particular, they continue to hold if Ω ∩ (−Ω) = ∅, or, more generally, if Ω satisfies the “excess less than or equal to deficit property” with respect to the center of mass as north (or south!) pole as stated in (2.13) above for all k ∈ [0, 1]. We note, however, that even under these conditions Ω is constrained to have volume no larger than that of a hemisphere. See also Remark 4 in Section 2 following the proof of Theorem 2.1.

(2) In fact, it is enough that Ω satisfy θ_{1}(B_{λ_{1}}) ≤ π/2 (or, equivalently, λ_{1}(Ω) ≥ n), where θ_{1}(B_{λ_{1}}) is the geodesic radius of the ball B_{λ_{1}}, together with the “excess less than or equal to deficit property” (2.13) for all k ∈ [cos θ_{1}(B_{λ_{1}}), 1]. This condition allows us to prove that λ_{2}(Ω) ≤ λ_{2}(B_{λ_{1}}) even for certain domains which have volume larger than that of a hemisphere (as well as covering all previous cases). A variant of this condition also applies in the case of the Neumann problem (the “μ_{1} problem”) for domains in 𝕊^{n} (see [9] and certain of our remarks in Section 2 above). Then θ_{1}(Ω^{⋆}), the geodesic radius of Ω^{⋆}, should replace θ_{1}(B_{λ_{1}}) in the foregoing (and in this case we are still limited by |Ω| ≤ \frac{1}{2}|𝕊^{n}|). The reason for these values of θ_{1} is that these are the radii we end with in the respective problems, when all is said and done.

Acknowledgements

M.S.A. is grateful for the hospitality of Thomas Hoffmann–Ostenhof and the Erwin Schrödinger Institute (ESI) in Vienna, where some of this work was carried out. We would also like to thank the referee for several useful remarks.

References

[1] M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, D.C., 1964.

[2] M. A. Armstrong, Basic Topology, Springer–Verlag, New York, 1983.

[3] M. S. Ashbaugh and R. D. Benguria, Log–concavity of the ground state of Schrödinger operators: a new proof of the Baumgartner–Grosse–Martin inequality, Phys. Lett. A 131 (1988), 273–276.

[4] M. S. Ashbaugh and R. D. Benguria, Optimal lower bounds for eigenvalue gaps for Schrödinger operators with symmetric single–well potentials and related results, Maximum Principles and Eigenvalue Problems in Partial Differential Equations, P. W. Schaefer, editor, Pitman Research Notes in Mathematics Series, vol. 175, Longman Scientific and Technical, Harlow, Essex, United Kingdom, 1988, pp. 134-145.

[5] M. S. Ashbaugh and R. D. Benguria, Proof of the Payne–Pólya–Weinberger conjecture, Bull. Amer. Math. Soc. 25 (1991), 19–29.

[6] M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Annals of Math. 135 (1992), 601– 628.

[7] M. S. Ashbaugh and R. D. Benguria, A second proof of the Payne–Pólya–Weinberger conjecture, Commun. Math. Phys. 147 (1992), 181–190.

[8] M. S. Ashbaugh and R. D. Benguria, Isoperimetric inequalities for eigenvalue ratios, Partial Differential Equations of Elliptic Type, Cortona, 1992, A. Alvino, E. Fabes, and G. Talenti, editors, Symposia Mathematica, vol. 35, Cambridge University Press, Cambridge, 1994, pp. 1–36.

[9] M. S. Ashbaugh and R. D. Benguria, Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature, J. London Math. Soc.

(2) 52 (1995), 402–416.

[10] M. S. Ashbaugh and R. D. Benguria, On the Payne–Pólya–Weinberger conjecture on the n-dimensional sphere, General Inequalities 7 (Oberwolfach, 1995), C. Bandle, W. N. Everitt, L. Losonczi, and W. Walter, editors, International Series of Numerical Mathematics, vol. 123, Birkhäuser, Basel, 1997, pp. 111-128.

[11] M. S. Ashbaugh and H. A. Levine, Inequalities for the Dirichlet and Neumann eigenvalues of the Laplacian for domains on spheres, Journées “Équations aux Dérivées Partielles” (Saint–Jean–de–Monts, 1997), Exp. No. 1, 15 pp., École Polytechnique, Palaiseau, 1997.

[12] F. E. Baginski, Ordering the zeroes of Legendre functions P_{ν}^{m}(z_{0}) when considered as a function of ν, J. Math. Anal. Appl. 147 (1990), 296–308.

[13] F. E. Baginski, Comparison theorems for the ν-zeroes of Legendre functions P_{ν}^{m}(z_{0}) when −1 < z_{0} < 1, Proc. Amer. Math. Soc. 111 (1991), 395-402.

[14] C. Bandle, Isoperimetric Inequalities and Applications, Pitman Monographs and Studies in Mathematics, vol. 7, Pitman, Boston, 1980.

[15] C. Bandle and M. Flucher, Table of inequalities in elliptic boundary value problems, Recent Progress in Inequalities (Niš, 1996), G. V. Milovanovic, editor, Mathematics and its Applications, vol. 430, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998, pp. 97–125.

[16] B. Baumgartner, Level comparison theorems, Annals of Physics 168 (1986), 484–526.

[17] B. Baumgartner, Relative concavity of ground state energies as functions of a coupling constant, Phys. Lett. A 170 (1992), 1–4.

[18] B. Baumgartner, H. Grosse, and A. Martin, The Laplacian of the potential and the order of energy levels, Phys. Lett. B 146 (1984), 363–366.

[19] B. Baumgartner, H. Grosse, and A. Martin, Order of levels in potential models, Nucl. Phys. B 254 (1985), 528–542.

[20] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, fourth edition, Wiley, New York, 1989.

[21] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundlehren der mathematischen Wissenschaften 285, Springer–Verlag, Berlin, 1988.

[22] I. Chavel, Lowest-eigenvalue inequalities, Proc. Symp. Pure Math., vol. 36, Geometry of the Laplace Operator, R. Osserman and A. Weinstein, editors, Amer. Math. Soc., Providence, Rhode Island, 1980, pp. 79–89.

[23] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.

[24] G. Chiti, Norme di Orlicz delle soluzioni di una classe di equazioni ellitiche, Boll. Un. Mat. Ital. (5) 16–A (1979), 178-185.

[25] G. Chiti, A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators, J. Appl. Math. and Phys. (ZAMP) 33 (1982), 143–148.

[26] G. Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital. (6) 1–A (1982), 145–151.

[27] G. Chiti, A bound for the ratio of the first two eigenvalues of a membrane, SIAM J. Math. Anal. 14 (1983), 1163–1167.

[28] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience Publishers, New York, 1953.

[29] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.

[30] Faber, G., Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsber. Bayr. Akad. Wiss. München, Math.-Phys. Kl. 1923, 169-172.

[31] S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helvetici 51 (1976), 133–161.

[32] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, second edition, Cambridge University Press, Cambridge, 1952.

[33] E. M. Harrell II and P. L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Commun. Partial Diff. Eqs. 19 (1994), 2037–2055.

[34] P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, With Applications to Schrödinger Operators, Applied Mathematical Sciences, vol. 113, Springer–Verlag, New York, 1996.

[35] J. G. Hocking and G. S. Young, Topology, Addison–Wesley, Reading, Massachusetts, 1961.

[36] T. Kato, Perturbation Theory for Linear Operators, second edition, Grundlehren der mathematischen Wissenschaften 132, Springer–Verlag, Berlin, 1976.

[37] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), 97–100.

[38] E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9 (1926), 1–44. [English translation: Minimal properties of the sphere in three and more dimensions, Edgar Krahn 1894–1961: A Centenary Volume, Ü. Lumiste and J. Peetre, editors, IOS Press, Amsterdam, The Netherlands, 1994, pp. 139–174.]

[39] J. R. Munkres, Topology, A First Course, Prentice–Hall, Englewood Cliffs, New Jersey, 1975.

[40] J. R. Munkres, Elements of Algebraic Topology, Addison–Wesley, Menlo Park, California, 1984.

[41] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182– 1238.

[42] L. E. Payne, G. Pólya, and H. F. Weinberger, Sur le quotient de deux fréquences propres consécutives, Comptes Rendus Acad. Sci. Paris 241 (1955), 917–919.

[43] L. E. Payne, G. Pólya, and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289–298.

[44] J. W. S. Rayleigh, The Theory of Sound, second edition revised and enlarged (in two volumes), Dover Publications, New York, 1945 (republication of the 1894/1896 edition).

[45] M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV: Analysis of Operators, Academic Press, New York, 1978.

[46] R. D. Richtmyer, Principles of Advanced Mathematical Physics, vol. II, Springer–Verlag, New York, 1981.

[47] K. T. Smith, Primer of Modern Analysis, Bogden and Quigley, Tarrytown–on– Hudson, New York, 1971.

[48] E. H. Spanier, Algebraic Topology, McGraw–Hill, New York, 1966.

[49] E. Sperner, Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317–327.

[50] G. Szegő, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343–356.

[51] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 697–718.

[52] H. F. Weinberger, An isoperimetric inequality for the n-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633–636.

[53] E. F. Whittlesey, Fixed points and antipodal points, Amer. Math. Monthly 70 (1963), 807–821.