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Summer 2004 A Brunn-Minkowski theory for minimal surfaces
Yves Martinez-Maure
Illinois J. Math. 48(2): 589-607 (Summer 2004). DOI: 10.1215/ijm/1258138401


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.


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Yves Martinez-Maure. "A Brunn-Minkowski theory for minimal surfaces." Illinois J. Math. 48 (2) 589 - 607, Summer 2004.


Published: Summer 2004
First available in Project Euclid: 13 November 2009

zbMATH: 1076.53009
MathSciNet: MR2085429
Digital Object Identifier: 10.1215/ijm/1258138401

Primary: 53A10
Secondary: 52A40

Rights: Copyright © 2004 University of Illinois at Urbana-Champaign


Vol.48 • No. 2 • Summer 2004
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