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Willmore-type inequality in unbounded convex sets

Xiaohan Jia, Guofang Wang, Chao Xia, Xuwen Zhang

TL;DR

The paper extends the classical Willmore inequality to unbounded convex sets in Euclidean space and to anisotropic settings. It develops a Minkowski-norm framework with Wulff geometry, defines an anisotropic parallel-flow mechanism, and uses volume estimates of tubular neighborhoods to derive a Willmore-type bound tied to the asymptotic volume ratio AVR_F(K). Equality is rigid, forcing Σ to be a Wulff sphere segment and K503Ω to be the corresponding cone, or, in the anisotropic capillary/half-space cases, Σ to be a Wulff shape with cone-like complement. The work also connects to half-space capillary problems by recasting them as anisotropic free-boundary problems via modified Minkowski norms, yielding sharp inequalities and characterizations of equality in both isotropic and anisotropic capillary settings. These results unify extrinsic Willmore-type estimates with anisotropic capillary geometry and have potential implications for relative isoperimetric problems in unbounded domains.

Abstract

In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set $K\subset\mathbb{R}^{n+1}$ $(n\ge 2)$, for any embedded hypersurface $Σ\subset K$ with boundary $\partialΣ\subset \partial K$ satisfying a certain contact angle condition, there holds $$\frac1{n+1}\int_Σ\vert{H}\vert^n{\rm d}A\ge{\rm AVR}(K)\vert\mathbb{B}^{n+1}\vert.$$ Moreover, equality holds if and only if $Σ$ is a part of a sphere and $K\setminusΩ$ is a part of the solid cone determined by $Σ$. Here $Ω$ is the bounded domain enclosed by $Σ$ and $\partial K$, $H$ is the normalized mean curvature of $Σ$, and ${\rm AVR}(K)$ is the asymptotic volume ratio of $K$. We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space.

Willmore-type inequality in unbounded convex sets

TL;DR

The paper extends the classical Willmore inequality to unbounded convex sets in Euclidean space and to anisotropic settings. It develops a Minkowski-norm framework with Wulff geometry, defines an anisotropic parallel-flow mechanism, and uses volume estimates of tubular neighborhoods to derive a Willmore-type bound tied to the asymptotic volume ratio AVR_F(K). Equality is rigid, forcing Σ to be a Wulff sphere segment and K503Ω to be the corresponding cone, or, in the anisotropic capillary/half-space cases, Σ to be a Wulff shape with cone-like complement. The work also connects to half-space capillary problems by recasting them as anisotropic free-boundary problems via modified Minkowski norms, yielding sharp inequalities and characterizations of equality in both isotropic and anisotropic capillary settings. These results unify extrinsic Willmore-type estimates with anisotropic capillary geometry and have potential implications for relative isoperimetric problems in unbounded domains.

Abstract

In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set , for any embedded hypersurface with boundary satisfying a certain contact angle condition, there holds Moreover, equality holds if and only if is a part of a sphere and is a part of the solid cone determined by . Here is the bounded domain enclosed by and , is the normalized mean curvature of , and is the asymptotic volume ratio of . We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space.
Paper Structure (9 sections, 22 theorems, 112 equations, 1 figure)

This paper contains 9 sections, 22 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minkowski norm and anisotropic geometry
  4. Anisotropic capillary hypersurfaces in a half-space
  5. Properties of unbounded convex sets
  6. Willmore inequality in unbounded convex sets
  7. Willmore inequalities in a half-space
  8. An alternative approach in the half-space case
  9. Capillary hypersurfaces in a half-space

Key Result

Theorem 1

Let $(M^{n+1},g)$$(n\geq2)$ be a complete Riemannian manifold with nonnegative Ricci curvature and $\Omega\subset M$ a bounded open set with smooth boundary. Then where ${\rm AVR}(g)$ is the asymptotic volume ratio of M (i.e. the limit $\lim_{r\rightarrow\infty}\frac{(n+1)\lvert B_r(p)\rvert}{r^{n+1}\lvert\mathbb{S}^n\rvert}$, which exists thanks to the Bishop-Gromov volume comparison theorem). M

Figures (1)

  • Figure 1: $\widetilde{K}$ and $\mathcal{C}_{x_0}$.

Theorems & Definitions (35)

  • Theorem 1: AFM20*Theorem 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2: JWXZ23*Proposition 3.1
  • Definition 2.3
  • Proposition 2.4: JWXZ23*Remark 2.1
  • Proposition 2.5: JWXZ23*Proposition 3.2
  • ...and 25 more