Eigenvalues of the Laplacian on balls with spherically symmetric metrics

In this article we will explore Dirichlet Laplace eigenvalues of balls with spherically symmetric metrics. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radii. As a special case we shall show that the Dirichlet Laplace eigenvalues of balls with small radii on the sphere are smaller than the corresponding eigenvalues of the Euclidean balls with the same radii. The opposite correspondence is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.


Introduction
It is well known that the Dirichlet Laplace eigenvalues of the ball B r 0 (0) ⊂ R n with radius r 0 are on the form One of the tools used in the proof of (1.1) is comparison theorems for Sturm-Liouville equations on the radial part of the eigenfunction.
Recall that a spherically symmetric (also called a rotationally symmetric) metric on a ball in Euclidean space is of the form dr ⊗dr + f 2 (r )g S n−1 . Recently in [4,Lem. 3.1], the first eigenvalue of a ball with a spherically symmetric metric was compared to the first eigenvalue of a ball with the Euclidean metric. When applied to hyperbolic space and the sphere the result gives a generalization of (1.1) for the first eigenvalue found in [4,Thm. 3.3]. As a general consequence of this result, Borisov and Freitas showed in [4] that the first Dirichlet Laplace eigenvalue on balls on hyperbolic space is larger than the first eigenvalue on Euclidean space. Prior, a lower bound in the case of hyperbolic space for the first Dirichlet Laplace eigenvalue on balls was shown by Artamoshin, see [1]. In the same article, Artamoshin also computed all the eigenvalues corresponding to radial eigenfunctions in 3-dimensional hyperbolic space.
In this article, we will present two different proofs of inequalities similar to (1.1) for all eigenvalues of spheres and hyperbolic spaces, see Theorem 3.1. Both proofs will heavily depend on the existence of differentiable families of eigenfunctions and corresponding eigenvalues. The construction of these families for constant curvature spaces will be outlined in Sect. 3, and generalized to balls with spherically symmetric metrics in Sect. 5. This parametrized family of Dirichlet eigenvalues of the geodesic ball B r ( p) with radius r centered at p will be denoted by λ m,l (r ) and satisfies lim r →0 r 2 λ m,l (r ) = ( j l m+n/2−1 ) 2 .
We will show that λ m,l (r ) satisfies a lower and upper bound similar to (1.1). Every eigenvalue on balls with spherically symmetric metrics is of the form λ m,l (r ). As a consequence, we will demonstrate on hyperbolic space with curvature K that For the first eigenvalue corresponding to the case m = 0 and l = 1 this was previously shown by Randol, see [5, p. 46]. The outline of the paper is as follows: We will start by introducing some facts related to eigenvalues of manifolds with spherically symmetric metrics in Sect. 2. Of special importance for this paper is representing eigenfunctions of balls with spherically symmetric metrics using spherical harmonics. In Sect. 3 we will prove a version of (1.1) for all eigenvalues using Sturm-Liouville theory as outlined in [2,3]. The goal of Sect. 4 is to prove a comparison equality between eigenvalues of balls with different radii smaller than the injectivity radius for any Riemannian manifold. Denote by λ(t) a differentiable family of Dirichlet eigenvalues satisfying lim t→0 t 2 λ(t) = j l m+n/2−1 2 with corresponding differentiable family of eigenfunctions u t on the ball B t ( p) centered at p with radius t. As a corollary of the main result in Sect. 4, one gets that any differentiable family satisfies In Sect. 5, we will apply (1.2) to manifolds with spherically symmetric metrics. The symmetry will allow us to simplify the expression significantly, and we obtain where g is a radial function depending only on the geometry. For the eigenvalue λ 0,1 (r ) this result can be found in [4,Lem. 3.1]. As an application of formula (1.3) we give in Sect. 6 another proof of the theorem presented in Sect. 3 for eigenvalues of spheres and hyperbolic spaces.
The author would like to thank E. Malinnikova and the anonymous reviewer of the paper for several helpful remarks and corrections.

Eigenvalues of balls with spherically symmetric metrics
Let (M, g) be a Riemannian manifold of dimension n and fix a point p ∈ M. We will use the notations r p (x) = dist(x, p) to denote the radial distance, B r 0 ( p) = r −1 p ([0, r 0 )) to be the geodesic ball, and S r 0 ( p) = r −1 p (r 0 ) to denote the geodesic sphere. Since we work primarily with geodesic spherical coordinates, we let S n−1 denote the (n − 1)-sphere with radius 1 equipped with standard round metric g S n−1 . Then we can denote the geodesic spherical coordinates (r , θ) ∈ [0, ρ) × S n−1 with respect to the point p and the radius ρ, see [6, Sect.III. 1.]. Furthermore, we will assume that (M, g) has a ball B ρ ( p) with a spherically symmetric metric centered at p with radius ρ > 0, i.e. the metric g on the ball B ρ ( p) can be written in the form By [12,Prop. 1.4.7] the function f : [0, ρ) → [0, ∞) satisfies All smooth surfaces of revolution where the surface of revolution intersects the axis at a point p have balls centered at p with spherically symmetric metrics. Writing the Laplacian in geodesic spherical coordinates gives We refer the reader to [12,Sect.4.2.3] for more information about spherically symmetric metrics.
We will study the Dirichlet problem In the case u(r , θ) = R t (r ) (θ ), we refer to the parametrized family R t (r ) as the radial part of the solution and (θ) as the spherical part. The parametrized family R t will be chosen to be continuous in t, and we will show in Sect. 5 that it can also be taken to be differentiable in t. In the case of the sphere this was shown in [5, Chap. II.5] with a similar approach. It is well known that the spherical part is always an eigenfunction of the S n−1 with eigenvalue m(m + n − 2) for some m ∈ N ∪ {0}. The radial part R t solves the equation with R t (t) = 0 and where R t (r ) is bounded for r near 0. Introduce the norm The operator is unbounded and symmetric on L 2 f = L 2 ((0, t), f n−1 (r )dr) defined on the subspace Recall that H 2 loc (0, t) is the space where the second weak derivatives are locally in L 2 (0, t). The eigenvalues of L m are simple, since if u, v are two eigenfunctions with the same eigenvalue λ, we have This means that f n−1 (r )(uv − vu ) is constant. Using u(t) = v(t) = 0 we get that the Wronskian uv − vu is zero. Therefore, u and v are linearly dependent.
We will let λ m,l (t) be the l'th eigenvalue of the problem (2.2) with the corresponding eigenfunction R t m,l (r ) which is positive for small values of r and L 2 f -normalized.
and R t m,l is the eigenfunction corresponding to the l'th eigenvalue of (2.2).
Proof Let u(r , θ) = R(r ) (θ ) be an eigenfunction written in geodesic spherical coordinates with eigenvalue λ(t). Using a standard separation of variables argument, we get This leads to being a spherical harmonic function with corresponding eigenvalue m(m + n − 2) for some m ∈ N ∪ {0}, and R satisfying (2.2). Since u is a solution to (2.1) and hence smooth, we need that R(t) = 0 and that R(r ) is bounded for r near 0. The only thing left to show is that each eigenfunction can be written as a sum of eigenfunctions on the form R(r ) (θ ). Since the spherical harmonics are the eigenfunctions of S n−1 , we know that the spherical harmonics form an orthonormal basis for L 2 (S n−1 ). This means that an arbitrary eigenfunction u(r , θ) with eigenvalue λ(t) can be written as where m,k is the k'th spherical harmonic function with eigenvalue m(m + n − 2) and E m is the eigenspace with eigenvalue m(m + n − 2). Using that the spherical Laplacian is symmetric together with the Laplacian in geodesic spherical coordinates gives In particular, the function a m,k (r ) m,k (θ ) is an eigenfunction for all m and k. By orthogonality of eigenfunctions with different eigenvalues we get that u can be written in the form where a m,k (r ) m,k (θ ) has eigenvalue λ(t).

Dirichlet Laplace eigenvalues of model spaces
In this section we will work on the model space (M K , g K ) with constant sectional curvature K and dimension n. These spaces are special examples of spherically symmetric spaces. Denote by sin K (r ) the function and cos K (r ) = sin K (r ) . Then the metric centered at a point p can be written as where the radius of the ball is given by In Sect. 2 we showed that the solutions to the Dirichlet Laplace eigenvalue problem has a basis on the form u r 0 (r , θ) = R r 0 m,l (r ) m (θ ) on L 2 (B r 0 ( p)). In this case, the radial part R r 0 m,l solves the equation with the condition R r 0 m,l (r 0 ) = 0 and R r 0 m,l (r ) is bounded for r near 0. It is known, see e.g. [5, p. 318], that where j l m+n/2−1 is the l'th positive zero of the Bessel function J m+n/2−1 . We are now ready to state the main theorem.
Theorem 3.1 Let λ m,l (r 0 ) be as above. Furthermore, when K > 0 we will assume that r 0 < π/ √ K . In the case m > 0 or n > 2 we have that When m = 0 and n = 2 the result simplifies to

Remark 3.2 •
In the case of the sphere S 2 Theorem 3.1 is known from [2,3]. Additionally, for the first eigenvalue the theorem is known from [4, Thm. 3.3]. • When K < 0 we have that For any K we have the limit • For n = 3 and m = 0 we have that Theorem 3.1 simplifies to For hyperbolic space, this equality was shown in [1].
To compare the eigenfunctions of (3.1) to the Bessel equation we are going to use the Sturm-Picone comparison theorem.
in the interval [a, b]. Assume that 0 < p 2 ≤ p 1 and q 1 ≤ q 2 , and let z 1 and z 2 be two consecutive zeros of y 1 . Then either y 2 has a zero in the interval (z 1 , z 2 ), or y 1 = cy 2 for some non-zero constant c.

Proof of Theorem 3.1
In the proof we will assume that for K > 0 we have r 0 < π Let Jm be a Bessel function of orderm. Then u C (r ) := √ r Jm(Cr) is a solution to Our goal is to apply the Sturm-Picone comparison theorem to (3.2) and (3.3). We will use Sturm-Picone comparison twice with the C in (3.3) being two different constants which we will denote by C 1 and C 2 .
Notice that 1 sin 2 K (r ) − 1 r 2 is increasing with the lower limit We use the notation am := 4m 2 − 1 4 For the next part we will need a lower bound for the eigenvalue. We will show that for K > 0 and m > 0 we have the lower bound .

(3.4)
When m > 0 the first zero r 0 of R r 0 m,l occurs after the first extremal point r max . Without loss of generality, we can assume that the first extremal point is a maximum. At the point r max we have (R r 0 m,l ) (r max ) = 0 and (R r 0 m,l ) (r max ) < 0. Using this together with (3.1) we obtain Hence (3.4) follows. For negatively curved spaces we will use that which can be found in e.g. [5, p. 46]. When n > 2 or m > 0 we have that 1 − 4m 2 ≤ 0. We set In this case

Using Theorem 3.3 for solutions to the Eqs. (3.2) and (3.3) leads to the estimate of r 0 by
Solving for λ m,l (r 0 ) gives When m = 0 and n = 2 we have that Hence we obtain Corollary 3.4 Assume that K < 0. Then in the notation of Theorem 3.1 we have Proof By [5, p. 46] we know that the smallest eigenvalue λ 0,1 (r 0 ) satisfies Using this lower bound together with taking the limit as r 0 goes towards infinity of the upper bound in Theorem 3.1 gives the result for the case when m > 0 or n > 2.
Hence we are only left with the case when n = 2 and m = 0. Consider the operators .
Then both L 0 and L 1 are negative symmetric operators on the L 2 -space with the inner-product For φ ∈ X r 0 one has the inequality The Rayleigh min-max principle for negative operators gives that where the supremum is taken over all l-dimensional subspaces X l of X r 0 . This implies that the eigenvalues satisfy λ 0,l (r 0 ) ≤ λ 1,l (r 0 ).
Taking the limit as r 0 → ∞ gives proving the claim.

Eigenvalues of balls for general Riemannian manifolds
Let (M, g) be a Riemannian manifold and consider the ball B t ( p) ⊂ M for p ∈ M and t less than the injectivity radius Inj( p) at p. Again, we will look at the Dirichlet problem We will assume that B t ( p) (u t ) 2 dvol = 1, in which case By using the Hadamard formula presented in [8, Cor. 2.1] one has, assuming that λ(t) and u t are differentiable with respect to t, that The notation u t n denotes the derivative with respect to the outward unit normal. We will use the following proposition to compare the eigenvalue of balls with spherical metric with the Dirichlet eigenvalue of balls in Euclidean space. (4.1) that are normalized in L 2 (B t ( p)). Denote by r the radial distance from the point p and let r 1 < Inj( p). Assume that both λ(t) and u t are differentiable in t for r 0 ≤ t ≤ r 1 . Then

Proposition 4.1 Let u t be a parametrized family of solutions to
In particular, if the conditions above are satisfied for r 0 = 0, then for some integers m and l. Hence The above proposition was proved for the first eigenvalue on B ρ ( p) with spherically symmetric metrics in [4,Lem. 3.1]. Since the first eigenfunction is radial when working on spherically symmetric metrics, the result simplifies to The proof of [4, Lem. 3.1] was based on variational methods. We will give another proof for general eigenvalues. We first develop the following lemma of independent interest.

Lemma 4.2 Let u be a solution to u + λu = 0 on the ball B t ( p) where t ≤ Inj( p).
Let ϕ : (0, ∞) → (0, ∞) be such that grad r ϕ(r (x)) is a smooth vector field. Then Proof We will use the notation X = grad r φ(r (x)) and Taking the divergence of V , one obtains Expanding the term Using the divergence theorem with div(V ) together with gives the result since grad r is the normal vector to S t ( p).

Remark 4.3
There are many examples of functions φ : (0, ∞) → (0, ∞) that satisfy the assumption in Lemma 4.2, such as φ(r ) = 1 r . In this case, Lemma 4.2 becomes Proof of Prop. 4. 1 We will write u instead of u t to simplify the notation. The boundary condition u = 0 on S t ( p) gives grad S t u = 0. Using Lemma 4.2 with ϕ(r (x)) = 1 implies that |grad u| 2 (r r + 1) Applying the divergence theorem to the last term of (4.2) gives Hence we get that By simplifying the expression we obtain Finally, integrating the identity above with respect to t gives the result. For the special case when r 0 = 0, we apply a result found in [5, p. 318], which states that

A Hadamard formula for spherically symmetric metrics
In this section we will assume that the n-dimensional Riemannian manifold (M, g) contains a ball B ρ ( p) with a spherically symmetric metric. We will use the notation introduced in Sect. 2. The goal is to show the following Hadamard formula.

Proof of Theorem 5.1
The proof of differentiability for both the eigenvalue and eigenfunction is quite standard, and a good exposition of perturbation theory of operators are given in [10]. Recall that R t m,l satisfies To get uniqueness, we will assume that the solution satisfies the normalization t 0 (R t m,l (r )) 2 f n−1 (r ) dr = 1, together with R t m,l (r ) being non-negative for small r . For simplicity, we will denote the operator where v is a twice differentiable function on the interval [0, 1]. Notice that R t m,l ( r t ) and λ m,l (t) are the eigenfunctions and eigenvalues of the operator A(t). We have already discussed the fact that the eigenvalue equation for the radial function only have simple eigenvalues in Sect. 4. It is well known by elliptic regularity that all the eigenvalues of the Laplacian are smooth, which implies that R t m,l (r ) is smooth for all r ∈ (0, t). Additionally, the compact embedding from H 2 to L 2 implies that the Laplacian, and hence also A(t), has compact resolvent. For the eigenvalue problem for the Laplace operator we can turn it into a bounded operator by restricting the operator to the Sobolev space H 2 (B t ( p)). Fix a t 0 ∈ (0, ρ). Then since we are considering smooth perturbations we have where lim t→t 0 S(t) goes to zero as O((t −t 0 ) 2 ) in the radial part of the H 2 -norm. By  Using Proposition 4.1 we get the following corollary.
Using the upper bound on F we get For the lower bound on F we have ≤ λ 0,l (r 1 ).
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Conflicts of interest
The author declares that there is no conflict of interest.

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