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Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces

Jinyu Gao, Guanghan Li

TL;DR

The paper develops a generalized, weighted anisotropic Hsiung-Minkowski integral formula for $\omega_0$-capillary hypersurfaces in the half-space, unifying closed and capillary cases via the Cahn-Hoffman framework and the capillary support function $\bar u$. Leveraging this formula alongside the anisotropic Heintze-Karcher inequality and Newton-Maclaurin relations, it proves Alexandrov-type rigidity results, establishing that certain curvature-sum identities force the hypersurface to be an $\omega_0$-capillary Wulff shape, and it derives uniqueness results for the anisotropic Orlicz-Christoffel-Minkowski problem as well as a new proof of the Euclidean capillary $L_p$-Minkowski problem for $p\ge 1$. The work also yields two principal Alexandrov-type theorems for strictly convex anisotropic capillary hypersurfaces, illustrating that rigidity persists under generalized curvature conditions in anisotropic capillary geometry. Overall, the results advance the understanding of rigidity and Minkowski-type problems in anisotropic capillary settings with boundary.

Abstract

In this paper, we obtain a new Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in the half-space, which includes the weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity for closed hypersurfaces as special cases. As applications, we prove some anisotropic Alexandrov-type theorems and rigidity results for anisotropic capillary hypersurfaces. Specially, the uniqueness of the solution to the anisotropic Orlicz-Christoffel-Minkowski problem is obtained, and thus a new proof is provided for the uniqueness of the solution to $L_p$-Minkowski problem with $p\geq 1$ in the Euclidean capillary convex bodies geometry.

Generalized Minkowski formulas and rigidity results for anisotropic capillary hypersurfaces

TL;DR

The paper develops a generalized, weighted anisotropic Hsiung-Minkowski integral formula for -capillary hypersurfaces in the half-space, unifying closed and capillary cases via the Cahn-Hoffman framework and the capillary support function . Leveraging this formula alongside the anisotropic Heintze-Karcher inequality and Newton-Maclaurin relations, it proves Alexandrov-type rigidity results, establishing that certain curvature-sum identities force the hypersurface to be an -capillary Wulff shape, and it derives uniqueness results for the anisotropic Orlicz-Christoffel-Minkowski problem as well as a new proof of the Euclidean capillary -Minkowski problem for . The work also yields two principal Alexandrov-type theorems for strictly convex anisotropic capillary hypersurfaces, illustrating that rigidity persists under generalized curvature conditions in anisotropic capillary geometry. Overall, the results advance the understanding of rigidity and Minkowski-type problems in anisotropic capillary settings with boundary.

Abstract

In this paper, we obtain a new Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces in the half-space, which includes the weighted Hsiung-Minkowski formula and classical anisotropic Minkowski identity for closed hypersurfaces as special cases. As applications, we prove some anisotropic Alexandrov-type theorems and rigidity results for anisotropic capillary hypersurfaces. Specially, the uniqueness of the solution to the anisotropic Orlicz-Christoffel-Minkowski problem is obtained, and thus a new proof is provided for the uniqueness of the solution to -Minkowski problem with in the Euclidean capillary convex bodies geometry.
Paper Structure (6 sections, 19 theorems, 103 equations)

This paper contains 6 sections, 19 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of Theorem \ref{['lemma:hminkow-w0']} and Corollary \ref{['cor:minkowski neq']}
  4. Proof of theorem \ref{['Thm-Alexan-a=b-w0']} and uniqueness of the $L_p$-Minkowski problem
  5. Proof of Alexandrov-type theorems \ref{['thm1.9']}$\sim$\ref{['thm1.10']}
  6. Reference

Key Result

Theorem 1.1

Suppose $\left(N^{n+1}, g^N\right)$ has constant curvature and $(\Sigma,g)$ is a closed oriented hypersurface. Assume $X \in \Gamma\left(\phi^*(T N)\right)$ is a conformal vector field along $\Sigma$, and $f$ is a smooth function on $\Sigma$. Then for $0 \leqslant k \leqslant n-1$, we have Here $H_k=\sigma_k(\kappa)/\binom{n}{k}$ is the normalized $k$-th mean curvature, Lie derivative of $g^N$ sa

Theorems & Definitions (33)

  • Theorem 1.1: Kwong, 2016, KKK2016*Theorem 1.1
  • Theorem 1.2: Jia-Wang-Xia-Zhang, 2023, Jia-Wang-Xia-Zhang2023*Theorem 1.3
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 23 more