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Capillary $L_p$-Christoffel-Minkowski problem

Yingxiang Hu, Mohammad N. Ivaki

TL;DR

This work solves the capillary $L_p$-Christoffel--Minkowski problem in the half-space for $1<p<k+1$ within the class of even hypersurfaces by formulating a fully nonlinear capillary equation $\sigma_k(\tau^#[s]) = s^{p-1}\phi$ on $\mathcal{C}_\theta$ with the boundary condition $\nabla_\mu s = \cot\theta\,s$. The authors develop a non-collapsing height estimate, derive detailed regularity and curvature bounds, and establish a capillary constant rank theorem to ensure strict convexity. Existence and uniqueness are achieved via a degree-theoretic approach in the even class, complemented by a continuation/implicit-function argument and a convexity-based uniqueness proof using Alexandrov--Fenchel inequalities. Overall, the paper extends the capillary theory for the $L_p$-Minkowski problem to the intermediate range $1<p<k+1$ and provides a robust framework for existence, regularity, and rigidity in capillary convex geometry.

Abstract

We solve the capillary $L_p$-Christoffel--Minkowski problem in the half-space for $1<p<k+1$ in the class of even hypersurfaces. A crucial ingredient is a non-collapsing estimate that yields lower bounds for both the height and the capillary support function. Our result extends the capillary Christoffel--Minkowski existence result of \cite{HIS25}.

Capillary $L_p$-Christoffel-Minkowski problem

TL;DR

This work solves the capillary -Christoffel--Minkowski problem in the half-space for within the class of even hypersurfaces by formulating a fully nonlinear capillary equation on with the boundary condition . The authors develop a non-collapsing height estimate, derive detailed regularity and curvature bounds, and establish a capillary constant rank theorem to ensure strict convexity. Existence and uniqueness are achieved via a degree-theoretic approach in the even class, complemented by a continuation/implicit-function argument and a convexity-based uniqueness proof using Alexandrov--Fenchel inequalities. Overall, the paper extends the capillary theory for the -Minkowski problem to the intermediate range and provides a robust framework for existence, regularity, and rigidity in capillary convex geometry.

Abstract

We solve the capillary -Christoffel--Minkowski problem in the half-space for in the class of even hypersurfaces. A crucial ingredient is a non-collapsing estimate that yields lower bounds for both the height and the capillary support function. Our result extends the capillary Christoffel--Minkowski existence result of \cite{HIS25}.

Paper Structure

This paper contains 8 sections, 18 theorems, 237 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-collapsing Estimates
  4. Rotationally symmetric hypersurfaces
  5. Even hypersurfaces
  6. Regularity Estimates
  7. Strict Convexity
  8. Existence and uniqueness

Key Result

Theorem 1.1

Let $1 < p < k+1$, $\theta \in (0, \pi/2)$, and $\phi \in C^\infty(\mathcal{C}_\theta)$ be a positive function satisfying and the boundary condition Then there exists a unique even, strictly convex, capillary hypersurface $\Sigma \subset \overline{\mathbb{R}^{n+1}_+}$ with contact angle $\theta$ whose capillary support function $s$ solves

Figures (1)

  • Figure 1: Capillary vs. classical outer parallel hypersurfaces

Theorems & Definitions (39)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 29 more