Compact complete proper minimal immersions in strictly convex bounded regular domains of R^3

TL;DR

This paper constructs conformal complete proper minimal immersions of finite topological type into strictly convex bounded domains in R^3, extending the theory of minimal surfaces with boundary.

Contribution

It introduces a method to produce compact complete proper minimal immersions of arbitrary finite topological type into convex domains in R^3.

Findings

Existence of minimal immersions for any finite topological type

Extensions to continuous maps on the boundary

Construction within strictly convex bounded domains

Abstract

Consider a strictly convex bounded regular domain $C$ of $\R^3$. For any arbitrary finite topological type we find a compact Riemann surface $\mathcal{M}$, an open domain $M\subset \mathcal{M}$ with the fixed topological type, and a conformal complete proper minimal immersion $X:M\to C$ which can be extended to a continuous map $X:\overline{M}\to \overline{C}$.