Optimal transport and the Gauss curvature equation

TL;DR

This paper links the problem of prescribing Gauss curvature and the Gauss map image for graphs to an optimal transport problem on the sphere, providing existence, regularity, and gradient blowup results.

Contribution

It introduces a novel formulation of the Gauss curvature prescription as an optimal transport problem on the sphere, enabling new analytical insights.

Findings

Existence and regularity of solutions under mild curvature assumptions

Quantitative gradient blowup analysis within the optimal transport framework

Connection between geometric PDEs and optimal transport theory

Abstract

In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second boundary value problem. Our main observation is that this problem can be posed as an optimal transport problem where the target is a subset of the lower hemisphere of $\mathbb{S}^n$. As a result we obtain existence and regularity of solutions under mild assumptions on the curvature, as well as a quantitative version of a gradient blowup result due to Urbas, which turns out to fall within the optimal transport framework.