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Perturbations of a minimal surface with triple junctions in $\mathbb{R}^2 \times \mathbb{S}^1$

Chen-Kuan Lee

TL;DR

This work addresses constructing stationary perturbations of a minimal surface with a circle of triple junctions in $\mathbb{R}^2\times \mathbb{S}^1$ that match prescribed boundary data. It adopts a non-parametric graph representation over the model surface $(\mathbf{Y}\times \mathbb{S}^1)\cap \mathbf{B}$, formulates the stationary condition as a quasilinear PDE system, and analyzes its linearization by splitting into Dirichlet and mixed boundary problems. Schauder estimates and a contraction-mapping argument yield existence and regularity of $\mathcal{C}^{2,\alpha}$-stationary perturbations for small boundary data, with the spine remaining a circle of triple junctions. The results provide explicit, quantitatively controlled examples of minimal surfaces with nontrivial singular sets in a product manifold, contributing to understanding stability and boundary-interaction in triple-junction minimal surfaces.

Abstract

We construct stationary perturbations of a specific minimal surface with a circle of triple junctions in $\mathbb{R}^2 \times \mathbb{S}^1$, that satisfy given boundary data.

Perturbations of a minimal surface with triple junctions in $\mathbb{R}^2 \times \mathbb{S}^1$

TL;DR

This work addresses constructing stationary perturbations of a minimal surface with a circle of triple junctions in that match prescribed boundary data. It adopts a non-parametric graph representation over the model surface , formulates the stationary condition as a quasilinear PDE system, and analyzes its linearization by splitting into Dirichlet and mixed boundary problems. Schauder estimates and a contraction-mapping argument yield existence and regularity of -stationary perturbations for small boundary data, with the spine remaining a circle of triple junctions. The results provide explicit, quantitatively controlled examples of minimal surfaces with nontrivial singular sets in a product manifold, contributing to understanding stability and boundary-interaction in triple-junction minimal surfaces.

Abstract

We construct stationary perturbations of a specific minimal surface with a circle of triple junctions in , that satisfy given boundary data.

Paper Structure

This paper contains 5 sections, 7 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Notation and the main result
  3. Parametrization of stationary varifolds as TEXT-perturbations of TEXT
  4. The Linearization of (\ref{['pde']})
  5. (\ref{['pde']}) with Dirichlet boundary condition and the proof of Theorem \ref{['thm_main']}

Key Result

Theorem 1.1

Given any $\delta \in (0, \frac{1}{2})$ and $\alpha \in (0, 1)$, there exists $\varepsilon > 0$ satisfying the following: for all $\underline{\varphi} \in \underline{\mathcal{C}}^{2, \alpha}(\mathbb{S}^1)$ with $\|\underline{\varphi}\|_{2, \alpha} < \varepsilon$, there exists $\underline{u} \in \und

Theorems & Definitions (15)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 5 more