Value of first eigenvalue of some minimal hypersurfaces embedded in the unit sphere

: We prove that the first nonzero eigenvalue of the Laplace-Beltrami operator of equator-like minimal submanifold embedded in the sphere S n + 1 is equal to n . The proof uses the spectral properties of the heat kernel operator corresponding to the submanifold


Introduction
In the theory of minimal submanifolds in a sphere, an interesting question asks about the value of first nonzero eigenvalue of the Laplacian for a minimal hypersurface Σ embedded in (n + 2)-unit sphere S n+1 in R n+2 .In its list of famous problems, the following question has been raised by S. T. Yau (problem 100, [17]).Conjecture: [17] Let Σ be a minimal hypersurface embedded in the n + 1-unit sphere S n+1 .Then, λ 1 (Σ) = n.
The upper bound λ 1 (Σ) ≤ n is not obvious, and was obtained before the statement of the conjecture due to Takahashi [15].Just after the conjecture was published, Choi and Wang proved that λ 1 (Σ) ≥ n/2.In fact, they proved a more general statement based on Reilly's formula, see [9].Until this day, it was the best known lower bound in the general case.Many important steps towards this conjecture has been done by proving the conjecture for some minimal homogenous hypersurfaces due to Muto-Ohnita-Urakawa [12], Kotani [10] and Solomon [13,14].Recently, Z. Tang and W. Yan proved that the conjecture is valid for closed minimal isoparametric hypersurfaces [16].In a recent work, S Deshmukh has proved some results related to the conjecture in [6].For the case when λ 1 (Σ) < n, it is shown that one has the following alternative, either λ 1 (Σ) ≤ (1 In the opposite case, when λ 1 (Σ) = n, either Σ is isometric to the unit sphere S n or otherwise k 0 ≤ n − 1/n.A generalization of this work for pseudo-umbilical hypersurface in the unit sphere has been proved by M. A. Choudhary in [2].
The method we are going to use in the paper are very different to the previous works, which have studied in this topic.Indeed, we are going to focus on the spectral properties of the Laplacian of a special type of immersed minimal submanifolds in the unit sphere.One of the most important objects in spectral geometry is the heat kernel operator associated with a given Riemmanian manifold, which corresponds to the solution of the heat equation on the manifold.The first nonzero eigenvalue controls the rate of growth of the heat kernel when time tends to infinity.

The main result
Let Σ be the hypersurface given by the locus of vanishing of some smooth function ψ on the unit sphere S n+1 i.e., We assume that Σ is embedded in S n+1 , which amounts to say that the gradient of ψ never vanishes on Σ.Thus, Σ is Riemannian submanifold on S n+1 , in particular it has an induced metric which gives rise to the corresponding Laplace-Beltrami operator ∆ Σ which is also self-adjoint.The spectrum of ∆ Σ which is discrete, has a least nonzero eigenvalue, λ 1 (Σ).Let us consider the polar coordinates parametrization, (σ, θ) coming from the stereographic projection.Suppose ψ is chosen so that Σ is a minimal embedded hypersurface in the n + 1-unit sphere S n+1 .Then, one has the following positive answer to the conjecture for minimal submanifolds satisfying some tranversality conditions.Theorem 1.1.Let Σ a minimal embedded hypersurface in the n + 1-unit sphere S n+1 and assume that the normal bundle of Σ is a one-dimensional subspace of T (S n+1 ) generated by the vector field ∂ θ coming from the stereographic projection.Then,

Spectral expansion of the Laplacian on Riemannian manifolds
We begin by some classical results that we are going to need.These can be found in many places (see e.g., [1,8]).Given an n-dimensional Riemannian manifold (M, g), one can define the Laplace-Beltrami operator which acts on smooth functions over M. In the local coordinate around the point x = (x 1 , . . ., x n ) with associated frame (∂ 1 , . . ., ∂ n ) which forms a basis of the tangent space T x (M), the Laplace-Beltrami operator takes the following form where g = det(g i j ) and (g i j ) = (g i j ) −1 .Assuming that M is a compact makes the operator −∆ M being a self-adjoint operators in L 2 (M).In particular, it has a discrete spectrum given by The spectral decompositon of the Hilbert space L 2 (M) with respect to ∆ M allows to write any where (Φ k ) k≥0 is a basis of eigenfunctions of L 2 (M).Associated to this, one can strongly continous operator in L 2 (M), P t = e −t∆ M satisfying the property P s+t = P s • P t such that ∥P t ∥ ≤ 1.It can be proved see e.g., [3] that the operator P t has a kernel K t : M × M → R for all t ≥ 0. This means that, for any function L 2 (M) The heat kernel characterizes the heat operator, and it can be obtained to perform the following evaluation The latter evaluation is allowed since one can identify the regular distribution with the function whenever it is continous, which is the case for the kernel operator.In can be shown that the function u(t, x) := P t f (x) satisfies the following equation with initial Dirichlet boundary condition The spectral decomposition of the heat kernel is given by for every x, y ∈ M. The exponential growth of K t (x, y) is controlled by the first eigenvalue λ 1 (M), which is nonzero for compact Riemannian manifolds with Dirichlet initial value condition.A fundamental result for long time behavior of the heat kernel it that for every x, y ∈ M one has (see e.g., [11]) The non-nullity of the first eigenvalue is granted by the fact that we consider the heat equation on a bounded domain of the sphere with Dirichlet boundary conditions.In fact, we can say much more about it since we are going to work with embedded closed minimal hypersurfaces in the unit sphere S n+1 .For this class of domains, the first eigenvalue is quite large in a certain sense, since it was proved by Choi and Wang that λ 1 (M) ≥ n 2 [9].In particular, the first eigenvalue is not zero.Yau's conjecture predicts that this value is maximal, in that λ 1 (M) = n for such hypersurfaces.Thus, the minimality condition for an hypersurface on the unit sphere implies maximality of the first eigenvalue.
For the sphere M = S n+1 , the eigenvalues of the operator (−∆ S n+1 ) acting on L 2 (S n+1 ) are given by in particular the first eigenvalue is given by In particular, using 2.4 and 2.6 for the unit sphere one has asymptotic estimate as t tends to infinity (2.7)
By now on, we use the parametrization of the unit sphere minus the north pole using the change of coordinates (x 1 , . . ., x n+1 ) ↔ (σ 1 , . . ., σ n+1 ) where σ n+1 = θ.The length element is given by The metric g |S n+1 in the local coordinates (σ 1 , . . ., σ n+1 , θ) is given by the diagonal matrix The metric g |S n+1 gives rise to the Christoffel symbols given by We can use this coefficients to define a connection on the tangent bundle of S n+1 .Let x ∈ S n+1 − N with local coordinates x = (σ 1 , . . ., σ n , θ) and corresponding orthonormal frame {∂ 1 , . . ., ∂ n , ∂ θ } with respect to the metric g |S n+1 that is, for every i, j = 1, . . ., n + 1 Using the basis {∂ 1 , . . ., ∂ n , ∂ θ } of T x (S n+1 ) we are able to define a bilinear map by assigning the values taken by this form at the elements of the basis of T (S n+1 ) by introducing the coefficients, where we have denoted ∂ n+1 = ∂ θ .The operator ∇ therefore defines a connection on the tangent bundle T (S n−1 ).Basic computations show that the Christoffel symbols relative to the metric g are symmetric in the sense and by uniqueness, ∇ is the Levi-Civita connection on T (S n+1 ).

The hypersurface Σ in S n+1 in the polar coordinate system
We consider the unit sphere equipped with the polar coordinates system (σ, θ) introduced in the previous paragraph.Thus, the hypersurface Σ in the coordinate system (σ, θ) is defined as follows Since Σ is embedded in S n+1 , the chain rule implies that gradient of ψ satisfies ∇ψ(x) 0 for any x ∈ Σ in the polar coordinates.The hypersurface Σ inherits a structure of Riemannian manifold given by a metric g Σ which the one induced by g |S n+1 with associated volume Riemmanian form dvol Σ = √ g Σ (σ, θ) dσ ∧ dθ which we do not need to explicit.In the local coordinates, we can assume that The vector fields ( ∂ ∂σ i ) are simply denoted ∂ i for 1 ⩽ i ⩽ n and sometimes we will denote either . With these notations, we have the orthonormal frame for T (S n+1 ) = {∂ 1 , . . ., ∂ n , ∂ θ } which extends the tangent bundle T (Σ) = {∂ 1 , . . ., ∂ n }.The tangent space of T (S n+1 ) can be splited as folllows: Since Σ is a smooth hypersurface, namely of codimension one, N(Σ) is a line bundle over Σ which is generated by the normal vector field ξ = ∂ θ .One can give an explicit expression for ξ, the metric g Σ and the mean curvature of Σ in function of the derivatives of u with respect to the frame {∂ 1 , . . ., ∂ n }, but we will not need it.Instead, we will use the general expression of the mean curvature in terms of the connections on Σ and S n+1 .Let us denote ∇ S n+1 (resp.∇ Σ ) the Levi-Civita connection of S n+1 (resp.Σ) relative to the metric g and the induced metric g |Σ given in polar coordinates.The second fundamental form II Σ of Σ in S n+1 is defined by for any two vectors fields in X, Y ∈ T (S n+1 ).In particular taking {∂ 1 , . . ., ∂ n } as an orthonormal basis for T (Σ), the previous relation applied to X = Y = ∂ i gives us Taking the sum we obtain the fundamental relation where 3. The proof of Theorem 1.1 By translating Σ using a rotation k ∈ SO(n + 1), one can sufficiently rotate the hypersurface Σ so that N Σ, that is, Σ ⊂ S n+1 − {N}.The hypothesis of Theorem 1.1 tells us that T (Σ) ⊥ ∂ θ and Σ is the graph of a smooth real valued function u : S n → [0, π], Since Σ is embedded in S n+1 , the gradient of ψ does not vanish, the implicit function theorem shows that (σ, θ) ∈ Σ − C if and only if θ = u(σ), in the coordinates (σ, θ).In particular, for any x = (σ, θ) ∈ V ∩Σ−C, one has ψ(σ, u(σ)) = 0 and at such point x, the hypersurface Σ only depends on the coordinates σ 1 , . . ., σ n .Thus, T x (Σ) is a hyperplane in T x (S n+1 ) with orthonormal basis {∂ 1 , . . ., ∂ n }.This basis extends to a local frame {∂ 1 , . . ., ∂ n , ∂ θ } of T x (S n+1 ) where ∂ n = ∂ θ .Thus, the normal direction is given by the line N x (Σ) generated by ∂ θ .
We give an explicit expression for the Laplacian of Σ in our setting which can also be found in [4,5].