Branch points of area-minimizing projective planes and a question of Courant

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 treat two topics on branch points of minimal surfaces. In the first, we show that a minimal surface \(f:\mathbb RP^2\rightarrow M^3\) which has the smallest area, among those mappings from the projective plane which are not homotopic to a constant mapping, is an immersion. That is, \(f\) is free of branch points, including especially false 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. In the second topic, we resolve, in the negative, a question on the directions of curves of self-intersection at a true branch point, which was posed by Courant (Dirichlet’s principle, conformal mapping and minimal surfaces. Wiley, New York, 1950).