## Illinois Journal of Mathematics

### A Brunn-Minkowski theory for minimal surfaces

Yves Martinez-Maure

#### 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.

#### Article information

Source
Illinois J. Math., Volume 48, Number 2 (2004), 589-607.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138401

Digital Object Identifier
doi:10.1215/ijm/1258138401

Mathematical Reviews number (MathSciNet)
MR2085429

Zentralblatt MATH identifier
1076.53009

#### Citation

Martinez-Maure, Yves. A Brunn-Minkowski theory for minimal surfaces. Illinois J. Math. 48 (2004), no. 2, 589--607. doi:10.1215/ijm/1258138401. https://projecteuclid.org/euclid.ijm/1258138401