Doubly connected minimal surfaces and extremal harmonic mappings

TL;DR

This paper investigates the interplay between quasiconformal and harmonic mappings on doubly connected domains, deriving sharp estimates and applying them to minimal surface evolution problems.

Contribution

It combines quasiconformal and harmonic mapping theories to establish new sharp estimates for doubly connected domains and applies these results to the minimal surface Cauchy problem.

Findings

Derived sharp estimates for quasiconformal harmonic mappings.

Applied estimates to the Bjorling problem for minimal surfaces.

Provided bounds on the modulus of evolving doubly connected minimal surfaces.

Abstract

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.