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

TL;DR

This paper proves Escobar's conjecture on a sharp lower bound for the first nonzero Steklov eigenvalue for manifolds with nonnegative sectional curvature, confirming the conjecture in this specific curvature setting.

Contribution

It confirms Escobar's conjecture for manifolds with nonnegative sectional curvature using a novel combination of weighted Reilly-type formulas and Pohozaev identities.

Findings

First nonzero Steklov eigenvalue is bounded below by boundary curvature c

Equality holds only for Euclidean balls of radius 1/c

Method involves weighted Reilly formula and Pohozaev identity

Abstract

It was conjectured by Escobar [J. Funct. Anal. 165 (1999), 101--116] that for an $n$-dimensional ($n\geq 3$) 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.