A Brunn-minkowski Theory for Minimal Surfaces
نویسنده
چکیده
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 R3 and noticed that minimal hedgehogs of R3 constitute a real vector space [14]. In 1996, the author noticed that the square root of the area of minimal hedgehogs of R3 that are modelled on the closure of a connected open subset of S2 is a convex function of the support function [5]. In this paper, the author (i) gives new geometric inequalities for minimal surfaces of R3; (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 BrunnMinkowski theory to minimal surfaces. Two characterizations of the catenoid among minimal hedgehogs are given.
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