Branch points of area-minimizing projective planes

TL;DR

This paper proves that area-minimizing minimal projective planes in three-dimensional manifolds are immersions without branch points, extending fundamental theorems to nonorientable surfaces and resolving a historical question on branch line directions.

Contribution

It extends the fundamental theorem of branched immersions to nonorientable surfaces and shows minimal projective planes are free of branch points.

Findings

Minimal area projective planes are immersions without branch points.

Extended the fundamental theorem of branched immersions to nonorientable surfaces.

Resolved Courant's 1950 question on branch line directions.

Abstract

Minimal surfaces in a Riemannian manifold $M^n$ are surfaces which are stationary for area: the first variation of area vanishes. In this paper we focus on surfaces of the topological type of the real projective plane $\R P^2$. We show that a minimal surface $f:\R P^2\to M^3$ which has the smallest area, among those mappings which are not homotopic to a constant mapping, is an immersion. That is, $f$ is free of branch points. As a major step toward treating minimal surfaces of the type of the projective plane, we extend the fundamental theorem of branched immersions to the nonorientable case. We also resolve a question on the directions of branch lines posed by Courant in 1950.