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Complete proper minimal surfaces in convex bodies of $\mathbb{R}^3$ (II): The behavior of the limit set

Francisco Martin, Santiago Morales

TL;DR

The paper addresses the asymptotic behavior and existence of complete proper minimal disks inside convex bodies in $\mathbb{R}^3$. It develops a constructive, density-based approach built on Meeks' trick, a Completeness Lemma, and a Properness Lemma to approximate any given minimal disk by a complete one whose boundary limit set is arbitrarily close to a prescribed curve on the boundary of a strictly convex domain. The main achievement is proving that for any regular strictly convex bounded domain $C$ and any regular Jordan curve $\Gamma\subset\partial C$, there exists a complete proper minimal immersion $\psi_{(\Gamma,\varepsilon)}:\mathbb{D}\to C$ with $\delta^H(\psi_{(\Gamma,\varepsilon)}(\partial\mathbb{D}),\Gamma)<\varepsilon$, which yields that every bounded regular domain admits a complete properly immersed minimal disk and allows approximation of limit sets by small boundary pieces. The results provide sharp control over the limit sets and establish universality-type consequences, including compact-set approximations of boundary data and a robust framework for constructing complete minimal surfaces with prescribed asymptotic behavior. The methods significantly advance the explicit, controlled construction of complete minimal surfaces in convex domains and enhance understanding of their global geometry and boundary interaction.

Abstract

Let $D$ be a regular strictly convex bounded domain of $\mathbb{R}^3$, and consider a regular Jordan curve $Γ\subset \partial D$. Then, for each $ε>0$, we obtain the existence of a complete proper minimal immersion $ψ_ε:\mathbb{D} \to D$ satisfying that the Hausdorff distance $δ^H(ψ_ε(\partial \mathbb{D}), Γ) < ε,$ where $ψ_ε(\partial \mathbb{D})$ represents the limit set of the minimal disk $ψ_ε(\mathbb{D}).$ This result has some interesting consequences. Among other things, we can prove that any bounded regular domain $R$ in $\mathbb{R}^3$ admits a complete proper minimal immersion $ψ: \mathbb{D} \longrightarrow R$.

Complete proper minimal surfaces in convex bodies of $\mathbb{R}^3$ (II): The behavior of the limit set

TL;DR

The paper addresses the asymptotic behavior and existence of complete proper minimal disks inside convex bodies in . It develops a constructive, density-based approach built on Meeks' trick, a Completeness Lemma, and a Properness Lemma to approximate any given minimal disk by a complete one whose boundary limit set is arbitrarily close to a prescribed curve on the boundary of a strictly convex domain. The main achievement is proving that for any regular strictly convex bounded domain and any regular Jordan curve , there exists a complete proper minimal immersion with , which yields that every bounded regular domain admits a complete properly immersed minimal disk and allows approximation of limit sets by small boundary pieces. The results provide sharp control over the limit sets and establish universality-type consequences, including compact-set approximations of boundary data and a robust framework for constructing complete minimal surfaces with prescribed asymptotic behavior. The methods significantly advance the explicit, controlled construction of complete minimal surfaces in convex domains and enhance understanding of their global geometry and boundary interaction.

Abstract

Let be a regular strictly convex bounded domain of , and consider a regular Jordan curve . Then, for each , we obtain the existence of a complete proper minimal immersion satisfying that the Hausdorff distance where represents the limit set of the minimal disk This result has some interesting consequences. Among other things, we can prove that any bounded regular domain in admits a complete proper minimal immersion .

Paper Structure

This paper contains 8 sections, 11 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction and background
  2. Minimal surface background
  3. Background on convex bodies and Hausdorff distance
  4. Preliminary Lemmas
  5. Proof of Lemma \ref{['lem:complete']}
  6. Proof of Theorem \ref{['th:meeks']}
  7. Proof of Lemma \ref{['lem:nadi']}
  8. Proof of the main Theorems

Key Result

Lemma 1

Consider $\Gamma$ a closed analytic curve. Let $P$ be a polygon, $X:\overline{\operatorname{Int} P} \rightarrow \mathbb{R}^3$ a conformal minimal immersion, and $r,\epsilon$ positive constants satisfying: Then, for all $s>0$ there exist a polygon $\widetilde{P}$ and a conformal minimal immersion $\widetilde{X} : \overline{\operatorname{Int} \widetilde{P}} \rightarrow \mathbb{R}^3$ verifying:

Figures (5)

  • Figure 1: Theorem C states that every bounded domain with regular boundary admits a complete properly immersed minimal surface. Besides, the boundary limit set is close to a small simple closed curve $\Gamma$ on the positively curved part of the boundary of the domain.
  • Figure 2: The minimal disk $X(P)$ and the tube $T(\Gamma,\mu).$
  • Figure 3: The set $\varpi_i$ and its image in $\mathbb{R}^3.$
  • Figure 4: The domains $R$ and $R'$ and the curve $\Gamma.$
  • Figure 5: We translate $U$ in the direction of the interior of $D$ until it does not touch $\psi(\mathbb{D})$.

Theorems & Definitions (27)

  • Lemma 1: Completeness Lemma
  • Theorem 1: Meeks' trick
  • Lemma 2: Properness Lemma, convex-I
  • Lemma 3
  • Remark 1
  • Claim 2.1
  • Claim 2.2
  • Proposition 1
  • proof
  • Theorem 2
  • ...and 17 more