Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

TL;DR

This paper proves that any map from a finite set on a compact Riemann surface can be extended to a complete minimal surface in three-dimensional space with finite total curvature, and explores optimal hitting problems with concrete curvature bounds.

Contribution

It establishes the extension of boundary maps to complete minimal immersions with finite total curvature and introduces new results on hitting sets and curvature bounds.

Findings

Any boundary map extends to a complete minimal surface with finite total curvature.

Constructs sets of points in a plane that impose curvature bounds on minimal surfaces.

Provides explicit relationships between hitting sets and total curvature.

Abstract

We prove that, given a compact Riemann surface $\Sigma$ and disjoint finite sets $\varnothing\neq E\subset\Sigma$ and $\Lambda\subset\Sigma$, every map $\Lambda \to \mathbb{R}^3$ extends to a complete conformal minimal immersion $\Sigma\setminus E\to \mathbb{R}^3$ with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in $\mathbb{R}^3$ with finite total curvature. To this respect we provide, for each integer $r\ge 1$, a set $A\subset\mathbb{R}^3$ consisting of $12r+3$ points in an affine plane such that if $A$ is contained in a complete nonflat orientable immersed minimal surface $X\colon M\to\mathbb{R}^3$, then the absolute value of the total curvature of $X$ is greater than $4\pi r$.