Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant

TL;DR

This paper establishes area bounds for stable capillary constant mean curvature hypersurfaces with nonpositive Yamabe invariant in Riemannian manifolds, and proves local splitting and rigidity results under certain conditions.

Contribution

It provides new area estimates and a local rigidity theorem for stable capillary hypersurfaces with nonpositive Yamabe invariant, including a splitting characterization of the ambient manifold.

Findings

Area estimates for stable capillary CMC hypersurfaces with nonpositive Yamabe invariant.

A local splitting theorem for manifolds along embedded, energy-minimizing hypersurfaces.

Conditions under which the manifold is isometric to a product with Einstein or Ricci-flat factors.

Abstract

We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $\Sigma$ with nonpositive Yamabe invariant that are properly immersed in a Riemannian $n$-dimensional manifold $M$ with scalar curvature $R^M$ and mean curvature of the boundary $H^{\partial M}$ bounded from below. We also prove a local rigidity result in the case $\Sigma$ is embedded and $\mathcal{J}$-energy-minimizing. In this case, we show that $M$ locally splits along $\Sigma$ and is isometric to $(-\varepsilon,\varepsilon)\times \Sigma, dt^2 + e^{-2Ht}g)$, where $g$ is Einstein, or Ricci flat, $H\geq 0$ and $\partial\Sigma$ is totally geodesic.