Capillary Christoffel-Minkowski problem

Abstract

The result of Guan and Ma (Invent. Math. 151 (2003)) states that if $\ddot{I}{•}^{-1/k}:{\mathbb{S}}^n\to (0,\infty )$ is spherically convex, then $\ddot{I}•$ arises as the $\ddot{I}{ƒ}_k$ 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 $\hat{I}¸\in (0,\ddot{I}€/2)$: if $\ddot{I}{•}^{-1/k}:{\mathcal{C}}_{\hat{I}}¸\to (0,\infty )$ is a capillary function and spherically convex, then $\ddot{I}•$ is the $\ddot{I}{ƒ}_k$ curvature of a strictly convex capillary hypersurface.