Escobar's Conjecture on a sharp lower bound for the first nonzero Steklov eigenvalue

Abstract

It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101--116] that for an $n$-dimensional ($n\ge 3$) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by $c>0$, the first nonzero Steklov eigenvalue is greater than or equal to $c$ with equality holding only on isometrically Euclidean balls with radius $1/c$. In this paper, we confirm this conjecture in the case of nonnegative sectional curvature. The proof is based on a combination of Qiu--Xia's weighted Reilly-type formula with a special choice of the weight function depending on the distance function to the boundary, as well as a generalized Pohozaev-type identity.

Peking Math. J. (to appear)

References in corpus (2)

• The Obata equation with Robin boundary condition
• Spectral geometry of the Steklov problem

Cited by in corpus (3)

• An upper bound for the first nonzero Steklov eigenvalue
• Wentzel-Laplace eigenvalues comparison
• Sharp eigenvalue estimates and related rigidity theorems