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On the Compactness Theorem for Embedded Minimal Surfaces in 3-manifolds with Locally Bounded Area and Genus

Brian White

TL;DR

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, this work proves subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity.

Abstract

Given a sequence of properly embedded minimal surfaces in a $3$-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where the convergence is not smooth.

On the Compactness Theorem for Embedded Minimal Surfaces in 3-manifolds with Locally Bounded Area and Genus

TL;DR

Given a sequence of properly embedded minimal surfaces in a -manifold with local bounds on area and genus, this work proves subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity.

Abstract

Given a sequence of properly embedded minimal surfaces in a -manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where the convergence is not smooth.

Paper Structure

This paper contains 5 sections, 10 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The Main Theorems
  3. The Annulus Lemma
  4. Proof of the No -Tilt Theorem
  5. Appendix

Key Result

Theorem 1.1

Let $\Omega$ be an open subset of a Riemannian $3$-manifold. Let $g_i$ be a sequence of smooth Riemannian metrics on $\Omega$ converging smoothly to a Riemannian metric $g$. Let $M_i\subset \Omega$ be a sequence of properly embedded surfaces such that $M_i$ is minimal with respect to $g_i$. Suppose In the second case, if $\Sigma$ is two-sided, then it must be stable. Now suppose that $\Omega$ is

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 2.1: Compactness Theorem
  • proof
  • Theorem 2.2: Blow-up Theorem
  • proof
  • Theorem 2.3: No -Tilt Theorem
  • Theorem 2.4
  • proof
  • Lemma 3.1: Annulus Lemma
  • proof
  • ...and 11 more