Minimal immersions of compact bordered Riemann surfaces with free boundary

TL;DR

This paper proves the existence of branched minimal immersions with free boundary conditions for certain maps from bordered surfaces into Riemannian manifolds, and establishes bounds on their topological and geometric properties under curvature and convexity assumptions.

Contribution

It introduces a new existence theorem for free boundary minimal immersions with prescribed fundamental group action and provides bounds on their topology and area under specific geometric conditions.

Findings

Existence of branched minimal immersions with free boundary conditions.

Bounds on genus, boundary components, and area for low index minimal surfaces.

Applicability under curvature and convexity assumptions.

Abstract

Let N be a complete, homogeneously regular Riemannian manifold of dimension greater than 2 and let M be a compact submanifold of N. Let $\Sigma$ be a compact orientable surface with boundary. We show that for any continuous $f: (\Sigma, \partial \Sigma) \rightarrow (N, M)$ for which the induced homomorphism on certain fundamental groups is injective, there exists a branched minimal immersion of $\Sigma$ solving the free boundary problem $(\Sigma, \partial \Sigma) \rightarrow (N, M)$, and minimizing area among all maps which induce the same action on the fundamental groups as f. Furthermore, under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on M which is the boundary of N, we derive bounds on the genus, number of boundary components and area of any compact two-sided minimal surface solving the free boundary problem with low index.