The Calabi-Yau problem for minimal surfaces with Cantor ends

TL;DR

This paper demonstrates that for various types of minimal and holomorphic immersions, one can construct complete conformal minimal surfaces with Cantor set ends in three-dimensional space and analogous results in higher-dimensional complex manifolds.

Contribution

It establishes the existence of minimal and holomorphic immersions with Cantor set ends in diverse complex and real manifolds, extending the Calabi-Yau problem to new settings.

Findings

Existence of complete minimal surfaces with Cantor ends in D space.

Extension of results to holomorphic and Legendrian immersions.

Applicability to self-dual and anti-self-dual Einstein four-manifolds.

Abstract

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in $\mathbb R^3$ with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least $2$, for holomorphic null immersions into $\mathbb C^n$ with $n\ge 3$, for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions in any self-dual or anti-self-dual Einstein four-manifold.