A BRUNN-MINKOWSKI THEORY FOR MINIMAL SURFACES

摘要

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 R~3 and noticed that minimal hedgehogs of R~3 constitute a real vector space. In 1996, the author noticed that the square root of the area of minimal hedgehogs of R~3 that are modelled on the closure of a connected open subset of S~2 is a convex function of the support function. In this paper, the author (ⅰ) gives new geometric inequalities for minimal surfaces of 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.