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The Calabi-Yau problem for minimal surfaces with Cantor ends

Franc Forstneric

TL;DR

This work resolves a Calabi--Yau-type problem for minimal surfaces by showing that every compact connected or bordered open Riemann surface $M$ contains a Cantor set $C$ whose complement $M\setminus C$ admits a complete conformal minimal immersion into $\mathbb{R}^3$ with a bounded image (and, in higher dimension, a bounded embedding into $\mathbb{R}^5$). The authors develop and combine tools from the Enneper--Weierstrass representation, Oka theory via the holomorphic null quadric $\mathbf{A}_*$, Runge--Mergelyan approximation on admissible sets, and an intrinsic-radius enlargement lemma to iteratively extend, approximate, and control the surface while driving completeness. The construction yields a Cantor-type end for the minimal surface, and the method extends to holomorphic immersions into complex manifolds ($\dim \ge 2$), holomorphic null immersions ($n\ge 3$), holomorphic Legendrian immersions, and oriented superminimal surfaces in (anti-)self-dual Einstein 4-manifolds through corresponding geometric correspondences. These results provide new, first-of-their-kind examples of complete bounded minimal surfaces with Cantor ends and reveal a broad axiomatic framework for Calabi--Yau-type properties across multiple geometric settings.

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.

The Calabi-Yau problem for minimal surfaces with Cantor ends

TL;DR

This work resolves a Calabi--Yau-type problem for minimal surfaces by showing that every compact connected or bordered open Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion into with a bounded image (and, in higher dimension, a bounded embedding into ). The authors develop and combine tools from the Enneper--Weierstrass representation, Oka theory via the holomorphic null quadric , Runge--Mergelyan approximation on admissible sets, and an intrinsic-radius enlargement lemma to iteratively extend, approximate, and control the surface while driving completeness. The construction yields a Cantor-type end for the minimal surface, and the method extends to holomorphic immersions into complex manifolds (), holomorphic null immersions (), holomorphic Legendrian immersions, and oriented superminimal surfaces in (anti-)self-dual Einstein 4-manifolds through corresponding geometric correspondences. These results provide new, first-of-their-kind examples of complete bounded minimal surfaces with Cantor ends and reveal a broad axiomatic framework for Calabi--Yau-type properties across multiple geometric settings.

Abstract

We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least , for holomorphic null immersions into with , 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.
Paper Structure (4 sections, 6 theorems, 12 equations, 1 figure)

This paper contains 4 sections, 6 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tools used in the proof
  3. Proof of Theorem \ref{['th:main']}
  4. Generalization to other geometries

Key Result

Theorem 1.1

In every compact connected Riemann surface, $M$, there is a Cantor set $C$ whose complement admits a complete conformal minimal immersion $M\setminus C \to\mathbb{R}^3$ with bounded image. There also exist a Cantor set $C$ in $M$ and a complete conformal minimal embedding $M\setminus C \hookrightarr

Figures (1)

  • Figure 3.1: A central cross removed from a rectangle $P$

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: Definition 1.12.9 in AlarconForstnericLopez2021
  • Definition 2.2: Definition 3.1.2 in AlarconForstnericLopez2021
  • Definition 2.3: Definition 3.1.3 in AlarconForstnericLopez2021
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7
  • Remark 2.8
  • ...and 5 more