j. differential geometry 61 (2002) 335-340

ISOPERIMETRIC INEQUALITY FOR THE SECOND EIGENVALUE OF A SPHERE

NIKOLAI NADIRASHVILI

Abstract

We prove Hersch’s type isoperimetric inequality for the second positive eigenvalue on a two dimensional sphere.

1. Introduction

Let (S^{2}, g) be a Riemannian manifold diffeomorphic to the twodimensional sphere. Assume that the area of (S^{2}, g) is equal to the area of a unit sphere in ℝ^{3}:

Area(S^{2}, g) = 4π.

Denote by

0 = λ_{0} < λ_{1} ≤ λ_{2} ≤ …

the spectrum of the Laplacian on (S^{2}, g). The classical isoperimetric inequality of Hersch states that

λ_{1} ≤ 2,

and equality is attained only when (S^{2}, g) is a standard sphere in ℝ^{3} (see [1]).

The goal of this paper is to prove a similar inequality for λ_{2}.

Theorem. Let g be a metric on S^{2} such that Area(S^{2}, g) = 4π. Then

λ_{2} < 4.

Received 09/03/2002.

In the theorem, the inequality turns into equality if we break the sphere (S^{2}, g) into two round spheres, both of area 2π. The inequality is isoperimetric, i.e., there exists a sequence of metrics on S^{2} of areas 4π for which λ_{2} tends to 4. To get such metrics, one can simply connect two round spheres of area 2π by a thin passage.

It is an interesting question to get sharp upper bounds for all the eigenvalues λ_{n} in terms of Area(S^{2}, g). In [2], Korevaar proved that there exists a constant C such that

λ_{n} ≤ \frac{1}{Area(S^{2}, g)}C · n.

We expect that the last inequality holds for C = 8π, with equality when the sphere is broken into n equal round spheres.

2. Proof of the theorem

(1) Let (ds)^{2} be the standard metric on the round sphere (S^{2}, can). We may assume without loss of generality that the metric g is conformally equivalent to (ds)^{2}, i.e., that g = ω(ds)^{2}. There exists a unique point e ∈ ℝ^{3}, |e| < 1, so that, for the Möbius map

μ : S^{2} → S^{2}

given by the formula

μ(x) = μ_{e}(x) = \frac{(1 − |e|^{2})x − (1 − 2⟨e, x⟩ + |x|^{2})e}{1 − 2⟨e, x⟩ + |e|^{2}|x|^{2}}, x ∈ ℝ^{3},

we have the orthogonality relations

∫_{S^{2}} X_{i} ∘ μdg = 0, i = 1, 2, 3,

where the X_{i} are the coordinate functions in ℝ^{3} ([1], [3]). The same result is also true for any nonsingular measure ω on S^{2}.

For X ∈ ℝ^{3} and s ∈ S^{2}, put X_{s} = ⟨X, s⟩. Let s_{0} maximize the integral

∫_{S^{2}} X_{s}^{2} ∘ μdg

over all s ∈ S^{2}. Put

u = u_{μ} = X_{s_{0}} ∘ μ.

Then since X_{1}^{2} + X_{2}^{2} + X_{3}^{2} = 1 on S^{2}

∫_{S^{2}} u^{2}dg ≥ \frac{1}{3} ∫_{S^{2}} dg

and

∫_{S^{2}} |∇u|^{2}dg = ∫_{S^{2}} |∇u|^{2}ds = ∫_{S^{2}} |∇X_{i}|^{2}ds = \frac{8π}{3} .

Hence

λ := ∫_{S^{2}} |∇u|^{2}dg/ ∫_{S^{2}} u^{2}dg ≤ 2.

We call the function u a quasieigenfunction and λ a quasieigenvalue of the metric g. If there is only one choice of the point s_{0} which maximizes the above integral, then we say that the quasieigenfunction u is simple. The nodal set of u is a circle on S^{2}, which we denote by N = N_{u} = N_{g}. The center of N will be −e.

(2) Denote by M the set of all spherical caps on S^{2}. Take a ∈ M. We denote by a^{∗} the adjacent cap to a, namely S^{2} \\overline{a}, and by B(a) ⊆ S^{2} the boundary circle of a. Recall that B(a) = B(a^{∗}). For each a ∈ M, there exists a unique conformal reflection C_{a} : S^{2} → S^{2} which changes the orientation of S^{2} and is the identity on B(a). We have that C_{a} = C_{a^{∗}} and C_{a}(a) = a^{∗}. Let g_{a} = ω_{a}(ds)^{2} be the image of the metric g under C_{a}. Put

G_{a} = \begin{cases}(ω + ω_{a})(ds)^{2} & on a, \\ 0 & on a^{∗}. \end{cases}

Then Area(S^{2}, G_{a}) = Area(a, G_{a}) = 4π.

(3) Take a ∈ M and let u = u_{a} be a quasieigenfunction of G_{a}. Define

U = \begin{cases}u & on a, \\ u ∘ C_{a} & on a^{∗}. \end{cases}

Then

∫_{S^{2}} Udg = 0,
∫_{S^{2}} U^{2}dg = ∫_{S^{2}} u^{2}dG_{a}

and

∫_{S^{2}} |∇U|^{2}ds = 2 ∫_{a} |∇u|^{2}ds.

Since

∫_{S^{2}} |∇u|^{2}ds/ ∫_{S^{2}} u^{2}dG_{a} ≤ 2,

we get

∫_{S^{2}} |∇U|^{2}ds/ ∫_{S^{2}} U^{2}dg < 4.

Thus, if the quasieigenfunction u is non-simple, then we get a twodimensional space of functions U on S^{2} for which the last inequality will hold. Hence, by the variational principle for eigenvalues, it follows that the second eigenvalue of (S^{2}, g) will be less than 4. Therefore, we may assume without loss of generality that, for all a ∈ M, the quasieigenfunction u_{a} of the metric G_{a} is simple. Then, for any a ∈ M, the function u_{a} is determined up to a sign. The circle n(a) := N_{u_{a}} depends continuously on a. Denote by Ω ⊆ M the set of all a such that n(a) ∩ B(a) = ∅.

(4) Since M ≅ S^{2} × I, we have π_{1}(M) = 0. Thus, by a suitable choice of sign of u_{a}, we may assume that the functions u_{a} are continuously dependent on the parameter a ∈ M. Hence, if we denote by s(a) ∈ S^{2} the point where u_{a} attains its supremum, then we get a continuous map

s : M → S^{2}.

Denote by p(a) ∈ S^{2} the center of the cap a. Denote by g(a) the orthogonal projection of the vector s(a) on the plane tangent to S^{2} at p(a), namely, T_{p(a)}S^{2}. Note that g(a) = 0 if and only if g(a^{∗}) = 0.

There exists a finite collection of sets E_{i} ⊆ M, i = 1, …, N, such that each E_{i} is diffeomorphic to a ball and ⋃ E_{i} = M. By Sard’s theorem for any ε > 0 there exists a smooth map φ_{1} : M → T S^{2} with φ_{1}(a) ∈ T_{p(a)}S^{2} for all a ∈ M such that |φ_{1}| < ε; and, if we define F_{1} = f +φ_{1}, then F_{1}^{(−1)}(0)∩E_{1} is a nonsingular one-dimensional set. We can define inductively a sequence of φ_{i} and F_{i} = F_{i−1} + φ_{i−1} such that |φ_{i}| < ε and F_{i}^{(−1)}(0) ∩ (E_{1} ∪ ⋯ ∪ E_{i}) is a nonsingular one-dimensional set. If we put F = F_{N}, then F^{(−1)}(0) is a union of nonsingular curves and |f − F | < Nε Since, for all a ∈ M, n(a) does not coincide with B(a), for sufficiently small ε > 0 we have F^{(−1)}(0) ⊆ Ω.

(5) Let Γ be the zero set of F. As discussed above, we assume that ε > 0 is so small that Γ ⊆ Ω. Then Γ is a finite union of smooth curves, without intersection.

Let Q ⊆ M be the set of all points q ∈ M such that B(q) is a big circle on S^{2}. Then Q ≅ S^{2} and p maps Q into S^{2}.

If we restrict the map F to Q, then we get a vector field on S^{2} which we denote by v. Put Z = Q ∩ Γ. Then the critical points of v are precisely the image of Z under p. By taking a small variation of F, we may assume without loss of generality that Γ intersects Q transversally at Z.

Let γ ⊆ Γ be a curve. Assume that the intersection γ ∩ Q has more than one point. Let z_{1} and z_{2} be subsequent points on γ of γ ∩ Q. Let ξ_{t} ⊆ M, t ∈ [1, 2], be a continuous family of small loops around γ such that ξ_{1} ⊆ Q, ξ_{2} ⊆ Q and p(ξ_{i}) is a loop around p(z_{i}), i = 1, 2. The orientations of ξ_{1} and ξ_{2} on Q are clearly opposite. Therefore, the orientations of p(ξ_{1}) and p(ξ_{2}) on S^{2} are also opposite. On the other hand, the rotation of the vector F (a) as the point a goes around the loop ξ_{t} is independent of t. Therefore, the indices of v at the points p(z_{1}) and p(z_{2}) are opposite.

Therefore, there exists a curve ζ ⊆ Γ which has odd points of intersection with Q. Such a curve ζ is necessarily unclosed, and hence the endpoints of ζ are on ∂M. Let us parametrize ζ by ζ = {ζ_{t}, t ∈ [0, 1]}, so that ζ_{0}, ζ_{1} ∈ ∂M. By the oddness of ζ ∩ Q, ζ_{t} tends to a point as t → 0 and to S^{2} as t → 1. Consequently,

Area(ζ_{t}, g) → 0 as t → 0,
Area(ζ_{t}^{∗}, g) → 0 as t → 1.

Denote u_{t} := u_{ζ_{t}}, N_{t} := N_{u_{t}}. Then

∫_{ζ_{t}} u_{t}^{2}dg → 0 as t → 0

and

∫_{ζ_{t}^{∗}} u_{t}^{2}dg → 0 as t → 1.

Thus there exists t_{0} ∈ (0, 1) so that

∫_{ζ_{t_{0}}} u_{t_{0}}^{2}dg = ∫_{ζ_{t_{0}}^{∗}} u_{t_{0}}^{2}dg.

Define

U = \begin{cases}u_{t_{0}} & on ζ_{t_{0}}, \\ u_{t_{0}} ∘ C_{ζ_{t_{0}}} & on ζ_{t_{0}}^{∗}. \end{cases}

Since ζ_{t_{0}} ∈ Ω, N_{t_{0}} ∩ N_{t_{0}}^{∗} = ∅. Hence the function U has three nodal domains: D_{1}, D_{2}, D_{3} such that ∂D_{1} = N_{t_{0}}, ∂D_{2} = N_{t_{0}}^{∗} and D_{3} = S^{2} \ (D_{1} ∪ D_{2}). Since

∫_{D_{1}} U^{2}dg = ∫_{D_{2}} U^{2}dg,

we have for i = 1, 2, 3 that

\frac{∫_{D_{i}} |∇U|^{2}dg}{∫_{D_{i}} U^{2}dg} = 2λ,

where λ is the quasieigenvalue of the quasieigenfunction u_{ζ_{t_{0}}} . Since λ < 2, the theorem follows from the variational principle for the second eigenvalue of (S^{2}, g).

References

[1] J. Hersch, Quatre propriétés isopérimétriques de membranes sphérique homogènes, C.R. Acad. Paris 270 (1970) 1645–1648, MR 45 #1444, Zbl 0224.73083.

[2] N. Korevaar, Upper bounds for eigenvalues of conformal metrics, J. Differential Geom. 37 (1993) 73–93, MR 94d:58153, Zbl 0794.58045.

[3] R. Schoen & S.-T. Yau, Lectures on Differential Geometry, International Press, 1994, MR 97d:53001, Zbl 0830.53001.

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