A note on the lower bounds of the first nonzero Steklov eigenvalue on compact manifolds
Yiwei Liu, Yi-Hu Yang
TL;DR
The paper develops curvature-aware lower bounds for the first nonzero Steklov eigenvalue on compact manifolds with convex boundary by introducing a novel weight function $V$ built from a curvature-driven ODE $\theta''+F\theta=0$ and its Greene–Wu approximation. Using weighted Reilly and Pohozaev identities, the authors prove a general bound $\sigma_1 \ge \sigma(c,K)$ that depends on the boundary convexity $c$ and the infimum of the radial sectional curvature $K$, with equality characterized by Euclidean ball isometry; these results generalize Escobar's conjecture and Xia–Xiong's case. The work also extends to conformal deformations, showing $\sigma_1(\widehat{g}) \ge c$ under a Hessian bound on the conformal factor, with rigidity when equality holds. The combination of the weight-function approach, curvature bounds, and conformal analysis provides a versatile framework for Steklov spectra beyond flat or nonnegative-curvature settings.
Abstract
Let $(Ω^{n+1}, g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $Σ$, satisfying that ${\text{Ric}_Ω}\ge 0$ and $Σ$ is strictly convex, more precisely, its second fundamental form $h\ge cg_Σ$ for some positive constant $c$. Escobar {\cite{escobar1997geometry}} considered the first nonzero Steklov eigenvalue $σ_1$ of $(Ω^{n+1}, g)$ and proved that $σ_1\geq c$ when $n=1$ and $σ_1>{\frac{c}{2}}$ when $n \geq 2$. He then conjectured {\cite{escobar1999isoperimetric}} that the first nonzero Steklov eigenvalue $σ_1\ge c$. Very recently, Xia and Xiong {\cite{xia2023escobar}} confirmed Escobar's conjecture in the case that $Ω$ 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.