Bumpy Riemannian metrics and closed parametrized minimal surfaces in Riemannian manifolds

TL;DR

This paper demonstrates that for generic metrics on high-dimensional manifolds, all prime parametrized minimal surfaces are smooth, nondegenerate, and correspond to critical points with symmetries, advancing understanding of minimal surface regularity.

Contribution

It establishes generic conditions ensuring minimal surfaces are free of branch points and form nondegenerate critical submanifolds, linking geometry and symmetry in high dimensions.

Findings

Prime parametrized minimal surfaces are free of branch points.

Such surfaces form nondegenerate critical submanifolds.

Critical submanifolds have dimension equal to the automorphism group of the domain.

Abstract

This article proves that if M is a smooth manifold of dimension at least four, then for generic choice of metric on M, all prime parametrized minimal surfaces in M are free of branch points and lie on nondegenerate critical submanifolds for the two-variable energy function which have the same dimension as the group of complex automorphisms of the domain Riemann surface.