The Minkowski problem, new constant curvature surfaces in R^3, and some applications

TL;DR

This paper constructs smooth convex bodies in R^3 with boundary surfaces of constant curvature, using the Minkowski problem, and explores applications to geometric analysis, including harmonic diffeomorphisms and capillary surfaces.

Contribution

It introduces a complete family of convex bodies with boundary surfaces of constant curvature, extending the Minkowski problem and deriving multiple geometric applications.

Findings

Existence of convex bodies with prescribed curvature properties.

Applications to harmonic diffeomorphisms between spherical domains.

Solutions to capillary surface problems in R^3.

Abstract

Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies $K$ in $\mathbb{R}^3$ whose boundary surface consists of an open surface $S$ with constant Gauss curvature (respectively, constant mean curvature) and $m$ planar compact discs $\bar{D_1},...,\bar{D_m},$ such that the Gauss map of $S$ is a homeomorphism onto $\mathbb{S}^2-\{p_j\}_{j=1}^m$ and $D_j\bot p_j,$ for all $j.$ We derive applications to the generalized Minkowski problem, existence of harmonic diffeomorphisms between domains of $\mathbb{S}^2,$ existence of capillary surfaces in $\mathbb{R}^3,$ and a Hessian equation of Monge-Ampere type.