Value distribution of the Gauss map and the total curvature of complete minimal surface in R^m

Xiaokang Mo

Vol. 163 (1994), No. 1, 159–174

DOI: 10.2140/pjm.1994.163.159

Abstract

The aim of this paper is to prove the following Theorem 1 Let S be a complete non-degenerate minimal surface in R^m such that its generalized Gauss map f intersects only a finite number of times the hyperplanes A₁,…,Aq in CP^m−1 in general position. If q > m(m + 1)∕2, then S must have finite total curvature.

Mathematical Subject Classification 2000

Primary: 53A10

Secondary: 32H25, 32H30

Milestones

Received: 3 December 1990

Published: 1 March 1994