Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant
Leandro F. Pessoa, Erisvaldo Véras, Bruno Vieira
TL;DR
This work provides sharp area estimates for stable capillary cmc hypersurfaces $\Sigma^{n-1}$ embedded or properly immersed in an $n$-manifold $M$ with boundary, under nonpositive Yamabe invariants and curvature bounds $R^M$, $H^{\partial M}$, together with a prescribed contact angle $\theta$. The authors leverage the $\mathcal{J}$-energy stability framework and Yamabe theory with boundary to derive lower bounds $A(\Sigma)^{\tfrac{2}{n-1}} \ge \dfrac{Q^{1,0}_g(\Sigma,\partial\Sigma)}{\inf R^M+\tfrac{n}{n-1}H^2} \ge \dfrac{\sigma^{1,0}(\Sigma,\partial\Sigma)}{\inf R^M+\tfrac{n}{n-1}H^2}$ and a boundary-area bound $A(\partial\Sigma)^{\tfrac{1}{n-2}} \ge \dfrac{\sin\theta}{2}\dfrac{Q^{0,1}_g(\Sigma,\partial\Sigma)}{\inf H^{\partial M}+H\cos\theta} \ge \dfrac{\sin\theta}{2}\dfrac{\sigma^{0,1}(\Sigma,\partial\Sigma)}{\inf H^{\partial M}+H\cos\theta}$, linking geometry to the Yamabe invariants. When equality occurs together with $\mathcal{J}$-minimization, the ambient space locally splits along $\Sigma$ into a product or warped product with $g$ Einstein or Ricci flat, revealing a strong rigidity mechanism for capillary cmc hypersurfaces. The results extend prior work of Barros and Longa to higher dimensions and the capillary setting, highlighting the interplay between ambient curvature, boundary geometry, and Yamabe-type invariants in governing area bounds and local rigidity.
Abstract
We prove area estimates for stable capillary $cmc$ (minimal) hypersurfaces $Σ$ 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 $Σ$ is embedded and $\mathcal{J}$-energy-minimizing. In this case, we show that $M$ locally splits along $Σ$ and is isometric to $(-\varepsilon,\varepsilon)\times Σ, dt^2 + e^{-2Ht}g)$, where $g$ is Einstein, or Ricci flat, $H\geq 0$ and $\partialΣ$ is totally geodesic.