Finiteness of the number of ends of minimal submanifolds in euclidean space

TL;DR

This paper extends classical value distribution theory to properly immersed minimal submanifolds in Euclidean space, proving finiteness of their ends under certain geometric conditions.

Contribution

It establishes a finiteness result for the number of ends of minimal submanifolds with finite projective volume, generalizing the Denjoy-Ahlfors theorem.

Findings

Finiteness of the number of ends for minimal submanifolds with finite projective volume.

Bound on the number of ends based on intersection properties with planes.

Extension of value distribution concepts to higher codimension minimal submanifolds.

Abstract

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in R^N. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codimensionn meeting any n-plane passing through the origin in at most k points has no more c(n,N)k ends.