A Heintze-Karcher inequality with free boundaries and applications to capillarity theory (Q6592080)

Given a bounded open connected set \(\Omega \subset \mathbb{R}^{n+1}\) with smooth boundary \(\partial \Omega\) and positive scalar mean curvature \(H_{\Omega}\), the Heintze-Karcher inequality states that\N\[\N(n+1)\mathrm{Vol}(\Omega) \leq \int_{\partial \Omega} \frac{n}{H_{\Omega}} \\N\]\Nwith equality if and only if \(\Omega\) is a Euclidean ball.\N\NThis interesting connection has motivated many authors to investigate the Heintze-Karcher inequality for proving rigidity theorems. In the paper under review, the authors obtain a new form of the Heintze-Karcher inequality for mean convex hypersurfaces with boundary lying on curved substrates and apply it to characterize the shape of a droplet inside a smooth container in the capillarity regime.