Estimate for the first fourth Steklov eigenvalue of a minimal hypersurface with free boundary

Rondinelle Batista, Barnabé Lima, Paulo Sousa and Bruno Vieira

Vol. 325 (2023), No. 1, 1–10

DOI: 10.2140/pjm.2023.325.1

Abstract

We explore the fourth-order Steklov problem of a compact embedded hypersurface ${\mathrm{\Sigma }}^n$ with free boundary in a $(n+1)$-dimensional compact manifold $M^{n+1}$ which has nonnegative Ricci curvature and strictly convex boundary. If $\mathrm{\Sigma }$ is minimal we establish a lower bound for the first eigenvalue of this problem. When $M=B^{n+1}$ is the unit ball in ${ℝ}^{n+1}$, if $\mathrm{\Sigma }$ has constant mean curvature $H^{\mathrm{\Sigma }}$ we prove that the first eigenvalue satisfies ${\sigma }_1\le n+\left|H^{\mathrm{\Sigma }}\right|$. In the minimal case ($H^{\mathrm{\Sigma }}=0$), we prove that ${\sigma }_1=n$.

Keywords

fourth Steklov eigenvalue, hypersurface with free boundary

Mathematical Subject Classification

Primary: 53C20, 53C42

Secondary: 35J40, 35P15

Milestones

Received: 25 October 2022

Revised: 15 March 2023

Accepted: 2 July 2023

Published: 3 September 2023