Capillary Christoffel-Minkowski problem

TL;DR

This work extends the classical Christoffel–Minkowski problem to capillary geometry in the half-space by prescribing the $k$-th curvature $\sigma_k$ of strictly convex capillary hypersurfaces with a fixed contact angle $\theta$. The authors adapt Guan–Ma's framework, constructing a continuity path $\phi_t$ and proving a Full Rank Theorem to ensure openness, together with robust a priori estimates to achieve closedness, thereby proving existence and uniqueness (modulo horizontal translations) of smooth capillary hypersurfaces with $\sigma_k(\tau^{\sharp}[s])=\phi$ under suitable convexity and boundary conditions. A key corollary shows uniqueness when $\phi=\ell^{−k}$, illustrating the capillary analogue of the classical result. The results provide a comprehensive capillary-extension of the Minkowski problem, including $C^0$–$C^2$ bounds and higher regularity, and lay a foundation for further capillary curvature problems.

Abstract

The result of Guan and Ma (Invent. Math. 151 (2003)) states that if $φ^{-1/k} : \mathbb{S}^n \to (0,\infty)$ is spherically convex, then $φ$ arises as the $σ_k$ curvature (the $k$-th elementary symmetric function of the principal radii of curvature) of a strictly convex hypersurface. In this paper, we establish an analogous result in the capillary setting in the half-space for $θ\in(0,π/2)$: if $φ^{-1/k} : \mathcal{C}_θ \to (0,\infty)$ is a capillary function and spherically convex, then $φ$ is the $σ_k$ curvature of a strictly convex capillary hypersurface.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Theorem 3.1: Full Rank Theorem