Geometric Optimal Transportation: A PDE Approach to Minimal Curves and Surfaces

Abstract

This paper explores the connection between geometric optimal transportation and partial differential equations (PDEs), with a focus on minimal curves and surfaces. We develop a PDE approach to compute optimal transportation maps and distances, and apply this framework to study the geometry of minimal curves and surfaces. Our results establish a link between optimal transportation and geometric flows, and provide new insights into the geometry of minimal surfaces.