Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface

TL;DR

This paper establishes sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces with boundary, relating geometric curvature conditions to eigenvalue estimates, with equality characterizations.

Contribution

It provides new lower bounds for Steklov eigenvalues under curvature constraints and characterizes the cases of equality, extending spectral geometry results.

Findings

Derived a lower bound for the first eigenvalue involving boundary curvature and Gaussian curvature.

Characterized the equality case as a Euclidean disk with zero Gaussian curvature.

Established a sharp lower bound for the fourth-order Steklov eigenvalue.

Abstract

Let $\Omega$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial \Omega$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -\alpha$ for a constant $\alpha \ge 0$, then the first eigenvalue $\sigma_1$ of the Steklov-type eigenvalue problem satisfies \[ \sigma_1 + \frac{\alpha}{\sigma_1} \ge c. \] Moreover, equality holds if and only if $\Omega$ is a Euclidean disk of radius $\frac{1}{c}$ and $\alpha = 0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $\Omega$.