First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces

TL;DR

This paper establishes upper bounds for the first eigenvalues of the Jacobi operator in constant mean curvature hypersurfaces, leading to new rigidity results and insights into related spectral problems and higher-dimensional invariants.

Contribution

It provides novel geometric bounds for eigenvalues of the Jacobi operator and applies these to derive rigidity results for CMC hypersurfaces and boundary-related spectral problems.

Findings

Upper bounds for the first eigenvalue of the Jacobi operator for CMC hypersurfaces.

Rigidity results for the area of CMC hypersurfaces under spectral and curvature conditions.

Bounds for the Jacobi--Steklov eigenvalue and related boundary rigidity results.

Abstract

In this paper, we obtain geometric upper bounds for the first eigenvalue $\lambda_1(J)$ of the Jacobi operator for both closed and compact with boundary hypersurfaces having constant mean curvature (CMC). As an application, we derive new rigidity results for the area of CMC hypersurfaces under suitable conditions on $\lambda_1(J)$ and the curvature of the ambient space. We also address the Jacobi--Steklov problem, proving geometric upper bounds for its first eigenvalue $\sigma_1(J)$ and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.