A note on the lower bounds of the first nonzero Steklov eigenvalue on compact manifolds

TL;DR

This paper develops new lower bounds for the first nonzero Steklov eigenvalue on compact manifolds with convex boundary, generalizing previous results and exploring curvature and conformal transformation effects.

Contribution

The authors introduce a novel weight function under specific sectional curvature conditions to improve lower bound estimates for the Steklov eigenvalue.

Findings

Constructed a new weight function for curvature assumptions.

Provided generalized lower bounds for the Steklov eigenvalue.

Analyzed eigenvalue bounds under conformal transformations.

Abstract

Let $(\Omega^{n+1}, g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$, satisfying that ${\text{Ric}_{\Omega}}\ge 0$ and $\Sigma$ is strictly convex, more precisely, its second fundamental form $h\ge cg_{\Sigma}$ for some positive constant $c$. Escobar {\cite{escobar1997geometry}} considered the first nonzero Steklov eigenvalue $\sigma_1$ of $(\Omega^{n+1}, g)$ and proved that $\sigma_1\geq c$ when $n=1$ and $\sigma_1>{\frac{c}{2}}$ when $n \geq 2$. He then conjectured {\cite{escobar1999isoperimetric}} that the first nonzero Steklov eigenvalue $\sigma_1\ge c$. Very recently, Xia and Xiong {\cite{xia2023escobar}} confirmed Escobar's conjecture in the case that $\Omega$ has nonnegative sectional curvature, by constructing a weight function and using appropriate integral identities. In this paper, we construct a new weight function under certain sectional curvature assumptions and provide some new lower bounds for the first nonzero Steklov eigenvalue, which can be considered as generalizations of the results of Escobar and Xia-Xiong. As an application of the weight function, we also consider lower bound estimate of the first nonzero Steklov eigenvalue under conformal transformations.