Capillary hypersurfaces, Heintze-Karcher's inequality and Zermelo's navigation (Q6619976)

The classical Heintze-Karcher inequality, see [\textit{E. Heintze} and \textit{H. Karcher}, Ann. Sci. Éc. Norm. Supér. (4) 11, No. 4, 451--470 (1978; Zbl 0416.53027)], states that, given an open bounded set \(\Omega\subset\mathbb{R}^{n+1}\) with \(C^2\), mean convex boundary \(\Sigma\), there holds \N\[\N\int_{\Sigma}\frac1H\mathrm{d}\mathcal{H}^{n} \geq\frac{n+1}n\vert{\Omega}\vert,\N\]\Nwith equality if and only if \(\Sigma\) is a round sphere. This provides an alternative approach to prove the Alexandrov's soap bubble theorem: the only closed, embedded hypersurfaces in \(\mathbb{R}^{n+1}\) with constant (\(k\)-th order) mean curvature are round spheres.\N\NThe focus of the paper is to establish a Heintze-Karcher-type inequality for capillary hypersurfaces in Euclidean ball hypersurfaces that meet \(\partial\mathbb{B}^{n+1}\) transversally from the interior with constant contact angle. As an application, the authors prove the corresponding Alexandrov's theorem for capillary hypersurfaces in the Euclidean ball.\N\NTo prove the Heintze-Karcher-type inequality, the authors combine the idea of Heintze-Karcher's original paper [loc. cit.] investigating the geodesic normal flow, with \textit{S. Brendle}'s monotonicity approach [Publ. Math., Inst. Hautes Étud. Sci. 117, 247--269 (2013; Zbl 1273.53052)]. The advances of this paper consist largely in the fact that, through the introduction of a Finsler metric, the authors transform the (Riemannian) capillary problem in \((\mathbb{B}^{n+1},g_{\mathrm{euc}})\) to a (Finsler) free boundary problem in \(\mathbb{B}^{n+1}\) with respect to the Finsler metric, by virtue of which studying geodesic normal flow makes sense. This idea is closely related to \textit{E. Zermelo}'s navigation problem [Z. Angew. Math. Mech. 11, 114--124 (1931; Zbl 0001.34101)], which asks: ``Suppose a ship is sailing on a calm sea with a mild breeze. How can the captain steer the ship so as to reach a destination in the shortest possible time?''