geometric analysis

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

arXiv:2405.18233

summary

The paper derives geometric upper bounds for the first eigenvalue of the Jacobi operator on constant mean curvature hypersurfaces (both closed and with boundary) and uses these bounds to obtain rigidity results for area and boundary length, also addressing the Jacobi‑Steklov problem and connections to Yamabe invariants in higher dimensions.

Abstract

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

21 pages. Final version, to appear in J. Geom. Anal

Topics & keywords

#constant mean curvature#jacobi operator#eigenvalue estimates#rigidity theorems#hypersurfaces#jacobi-steklov problemfirst eigenvalueJacobi operatorJacobi-SteklovCMC hypersurfacesrigidityYamabe invariant
First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces · wovepaper