Sharp bounds for the first non-zero Stekloff eigenvalues

Abstract

Let (M, , ) be an n( 2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problemswhere is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first nonzero eigenvalues of the above problems will be denoted by p 1 and q 1 , respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then p 1 √ λ 1 ( √ λ 1 + λ 1 − (n − 1)c 2 )/{(n − 1)c} with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius 1 c , here λ 1 denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q 1 nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius 1 c . Finally, we show that q 1 A/V and that if the equality holds and if there is a point x 0 ∈ ∂M such that the mean curvature of ∂M at x 0 is no less than A/{nV }, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively. ✩ Partially supported by CNPq.

Isoperimetric control of the Steklov spectrum

“…The series of papers by J. Escobar [10][11][12] is influential. For more recent results, see [2,30,14]. For higher eigenvalues in the planar situation, see [15,16].…”

“…See for instance [29, p. 38 and p. 453] and [27]. Recently, Wang and Xia [30] studied this question for the first non-zero eigenvalues of both operators. Under the assumption that Ricci curvature of M is non-negative and that the principal curvatures of ∂M are bounded below by a positive constant κ, they proved that…”

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

“…When ∂Ω has positive mean curvature a lower bounds for d 1 is available: Recently, Wang-Xia [33] have extended Theorem 4.5 to compact manifolds with boundary. Theorem 4.5 seems to say that the infimum of d 1 in the class of convex domains might be strictly positive.…”

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

“…Discussion and previous results. Quantitative estimates relating individual Steklov eigenvalues to eigenvalues of the tangential Laplacian ∆ Σ have been studied in [28,6,2,3,19,30,25,29]. They are relatively easy to obtain if the manifold M is isometric (or quasi-isometric with some control) to a product near its boundary.…”