A Brunn-Minkowski theory for minimal surfaces

Abstract

The aim of this paper is to motivate the development of a Brunn-Minkowski theory for minimal surfaces. In 1988, H. Rosenberg and E. Toubiana studied a sum operation for finite total curvature complete minimal surfaces in $\mathbb{R}^{3}$ and noticed that minimal hedgehogs of $\mathbb{R}^{3} $ constitute a real vector space [14]. In 1996, the author noticed that the square root of the area of minimal hedgehogs of $\mathbb{R}^{3}$ that are modelled on the closure of a connected open subset of $\mathbb{S}^{2}$ is a convex function of the support function [5]. In this paper, the author (i) gives new geometric inequalities for minimal surfaces of $\mathbb{R}^{3}$; (ii) studies the relation between support functions and Enneper-Weierstrass representations; (iii) introduces and studies a new type of addition for minimal surfaces; (iv) extends notions and techniques from the classical Brunn-Minkowski theory to minimal surfaces. Two characterizations of the catenoid among minimal hedgehogs are given.