Almgren's center manifold in a simple setting

TL;DR

This paper simplifies and explains core ideas behind Almgren's center manifold and De Giorgi's epsilon-regularity theorem in the context of area minimizing surfaces, highlighting key PDE estimates.

Contribution

It provides a simplified setting to illustrate fundamental PDE estimates underlying Almgren's center manifold and De Giorgi's regularity theorem.

Findings

Simplified proof of Almgren's center manifold in a basic setting

Illustration of key PDE estimates in regularity theory

Clarification of fundamental ideas in area minimizing surface regularity

Abstract

We aim at explaining the most basic ideas underlying two fundamental results in the regularity theory of area minimizing oriented surfaces: De Giorgi's celebrated $\varepsilon$-regularity theorem and Almgren's center manifold. Both theorems will be proved in a very simplified situation, which however allows to illustrate some of the most important PDE estimates.