The equality case in the substatic Heintze-Karcher inequality (Q6642892)

A Riemannian manifold \((M,g)\) of dimension \(n\geq2\), equipped with a non-negative smooth function \(f\) satisfying\N\begin{align*}\Nf\text{Ric}-\nabla\nabla f+\Delta fg\geq0,\N\end{align*}\Nsuch that \(\partial M=\{f=0\}\) is a compact, minimal, regular level-set of \(f\) (i.e., \(\nabla f\neq0\) on \(\partial M\)), is known as a substatic Riemannian manifold with horizon boundary. In [\textit{J. Li} and \textit{C. Xia}, J. Differ. Geom. 113, No. 3, 493--518 (2019; Zbl 1433.53059)] and [\textit{M. Fogagnolo} and \textit{A. Pinamonti}, J. Math. Pures Appl. (9) 163, 299--317 (2022; Zbl 1496.53056)], the classical Heintze-Karcher inequality [\textit{E. Heintze} and \textit{H. Karcher}, Ann. Sci. Éc. Norm. Supér. (4) 11, No. 4, 451--470 (1978; Zbl 0416.53027)] is generalized to hold for domains \(\Omega\subset(M,g)\) with smooth boundary \(\partial\Omega=\Sigma\cup\partial M\), such that \(\Sigma\) is a smooth, connected, strictly mean convex hypersurface in \((M,g)\), homologous to \(\partial M\), and the equality case of the Heintze-Karcher-type inequality happens if and only if \(\Sigma\) is umbilical.\N\NThe authors refine the mentioned Heintze-Karcher-type inequality for substatic Riemannian manifolds with horizon boundary, by giving a more explicit characterization of the equality case. Moreover, if \((M,g)\) is in addition a warped product manifold, then the authors improve the Heintze-Karcher-type inequality in [\textit{S. Brendle}, Publ. Math., Inst. Hautes Étud. Sci. 117, 247--269 (2013; Zbl 1273.53052)], by removing assumption (H4) therein. In particular, this is used to provide an improved version of the Alexandrov-type theorem [\textit{S. Brendle}, Publ. Math., Inst. Hautes Étud. Sci. 117, 247--269 (2013; Zbl 1273.53052), Theorem 1].