The structure and rigidiy of FBCMC hypersurfaces

TL;DR

This paper establishes rigidity results for free boundary minimal hypersurfaces under certain curvature conditions in 5-manifolds, extending previous results and methods to the free boundary setting.

Contribution

It extends rigidity theorems to 5-dimensional manifolds and characterizes one-endedness of free boundary constant mean curvature hypersurfaces.

Findings

Rigidity of stable free boundary minimal hypersurfaces under curvature conditions

Extension of Wu's result to 5-dimensions

Characterization of one-endedness for free boundary CMC hypersurfaces

Abstract

We prove that the combination of strict positivity of $k$-tri-Ricci curvature with non-negative $3$-intermediate Ricci curvature forces rigidity of two-sided stable free boundary minimal hypersurface in a 5-manifold with bounded geometry and weakly convex boundary. This improves the result of Wu \cite{Wuyujie2023} to 5-dimensions and also extends the method of Hong-Yan \cite{Hong-cmc nonexis} to the free boundary case. We give a characterization of one-endedness for weakly stable free boundary constant mean curvature(FBCMC) hypersurfaces, which is an extension of Cheng-Cheung-Zhou \cite{Cheng-Cheung-Zhou2008}.