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Stable $(r+1)$-th capillary hypersurfaces

Jinyu Guo, Haizhong Li, Chao Xia

TL;DR

The paper develops a variational framework for higher-order capillary hypersurfaces by introducing the $(r+1)$-th energy functional $\mathcal{E}_{r+1}$ and restricting to volume-preserving and angle-preserving variations. It derives the first and second variation formulas, establishing a stability notion on a constrained function space and expressing the second variation via the Jacobi-type operator $L_r$ and Newton tensors $P_r$. Central to the results are higher-order Minkowski-type formulas in both Euclidean half-spaces and hyperbolic horoballs, which yield admissible test functions and enable rigidity classifications. Consequently, stable $(r+1)$-th capillary hypersurfaces in the Euclidean half-space are spherical caps, while in hyperbolic space the stability analysis on horospheres implies total umbilicity (not totally geodesic under elliptic-point assumptions), extending classical capillary stability to higher order with precise geometric-analytic tools.

Abstract

In this paper, we propose a new definition of stable $(r+1)$-th capillary hypersurfaces from variational perspective for any $1\leq r\leq n-1$. More precisely, we define stable $(r+1)$-th capillary hypersurfaces to be smooth local minimizers of a new energy functional under volume-preserving and contact angle-preserving variations. Using the new concept of the stable $(r+1)$-th capillary hypersurfaces, we generalize the stability results of Souam \cite{Souam} in a Euclidean half-space and Guo-Wang-Xia \cite{GWX} in a horoball in hyperbolic space for capillary hypersurface to $(r+1)$-th capillary hypersurface case.

Stable $(r+1)$-th capillary hypersurfaces

TL;DR

The paper develops a variational framework for higher-order capillary hypersurfaces by introducing the -th energy functional and restricting to volume-preserving and angle-preserving variations. It derives the first and second variation formulas, establishing a stability notion on a constrained function space and expressing the second variation via the Jacobi-type operator and Newton tensors . Central to the results are higher-order Minkowski-type formulas in both Euclidean half-spaces and hyperbolic horoballs, which yield admissible test functions and enable rigidity classifications. Consequently, stable -th capillary hypersurfaces in the Euclidean half-space are spherical caps, while in hyperbolic space the stability analysis on horospheres implies total umbilicity (not totally geodesic under elliptic-point assumptions), extending classical capillary stability to higher order with precise geometric-analytic tools.

Abstract

In this paper, we propose a new definition of stable -th capillary hypersurfaces from variational perspective for any . More precisely, we define stable -th capillary hypersurfaces to be smooth local minimizers of a new energy functional under volume-preserving and contact angle-preserving variations. Using the new concept of the stable -th capillary hypersurfaces, we generalize the stability results of Souam \cite{Souam} in a Euclidean half-space and Guo-Wang-Xia \cite{GWX} in a horoball in hyperbolic space for capillary hypersurface to -th capillary hypersurface case.
Paper Structure (8 sections, 35 theorems, 180 equations, 2 figures)

This paper contains 8 sections, 35 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $(r+1)$-th capillary hypersurfaces and stabilities
  4. Rigidity for stable $(r+1)$-th capillary hypersurfaces in a half-space
  5. $(r+1)$-th capillary hypersurfaces supported on a horosphere
  6. Key formulae for $(r+1)$-th capillary hypersurfaces supported on a horosphere
  7. Rigidity for stable $(r+1)$-th capillary hypersurfaces supported on a horosphere
  8. Proof of the first variational formula

Key Result

Theorem 1.1

Let $0\leq r\leq n-1$. An immersed n-dimensional closed constant $(r+1)$-th mean curvature hypersurface in space formsWe regard an open hemi-sphere as a spherical space form in this paper. is stable if and only if it is a geodesic sphere.

Figures (2)

  • Figure 1: Hypersurface $M$ with contact angle $\theta$ in the half-space $\mathbb{R}^{n+1}_{+}$.
  • Figure 2: Hypersurface $M$ supported on horosphere $\mathcal{H}$.

Theorems & Definitions (56)

  • Theorem 1.1: AdCACRBCo
  • Theorem 1.2: Souam3
  • Theorem 1.3
  • Theorem 1.4: GWX3
  • Theorem 1.5
  • Corollary 1.1
  • Lemma 2.1: Guan
  • Lemma 2.2: Guan
  • Proposition 2.1: BCo
  • Proposition 2.2: WX
  • ...and 46 more