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

TL;DR

This paper proves a compactness theorem for sequences of embedded minimal surfaces in 3-manifolds with local area and genus bounds, showing subsequential smooth convergence away from a discrete set and analyzing blow-up behavior.

Contribution

It establishes a new compactness result for minimal surfaces with local bounds and characterizes the blow-up limits near singular points.

Findings

Subsequential convergence to a smooth embedded limit surface

Smooth convergence away from a discrete set

Analysis of blow-up limits near singularities

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.