Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complete minimal surfaces with Cantor ends in minimally convex domains

Antonio Alarcon

Abstract

We survey the recent history of the conformal Calabi-Yau problem consisting in determining the complex structures admitted by complete bounded minimal surfaces in $\mathbb{R}^3$. Moreover, we prove that for any minimally convex domain $Ω$ in $\mathbb{R}^3$ and any compact Riemann surface $R$ there is a Cantor set $C$ in $R$ whose complement $R\setminus C$ is the complex structure of a complete proper minimal surface in $Ω$.

Complete minimal surfaces with Cantor ends in minimally convex domains

Abstract

We survey the recent history of the conformal Calabi-Yau problem consisting in determining the complex structures admitted by complete bounded minimal surfaces in . Moreover, we prove that for any minimally convex domain in and any compact Riemann surface there is a Cantor set in whose complement is the complex structure of a complete proper minimal surface in .

Paper Structure

This paper contains 2 sections, 7 theorems, 11 equations.

Table of Contents

  1. The conformal Calabi-Yau problem: a brief survey
  2. Complete minimal surfaces with Cantor ends in minimally convex domains

Key Result

Theorem 1.2

An open Riemann surface $M$ is the underlying complex structure of a complete nonflat minimal surface in $\mathbb{R}^3$ contained between two parallel planes if and only if $M$ admits a nonconstant bounded harmonic function.

Theorems & Definitions (10)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5: AlarconForstnericLopez2021Book
  • Theorem 1.6: AlarconForstnericLopez2021Book
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof : Proof of Theorem \ref{['th:']}