A Gromov-Hausdorff convergence theorem of surfaces in $\mathbb{R}^n$ with small total curvature

TL;DR

This paper investigates the compactness and local structure of surfaces in Euclidean space with small total curvature, introducing the isothermal radius to establish convergence theorems and analyze surface behavior.

Contribution

It introduces the isothermal radius as a new tool to prove compactness and convergence results for surfaces with small total curvature in Euclidean space.

Findings

Established a compactness theorem in intrinsic and extrinsic topologies.

Provided a new perspective on Leon Simon's decomposition theorem.

Proved a non-collapsing version of Hélain's convergence theorem.

Abstract

In this paper, we mainly study the compactness and local structure of immersing surfaces in $\mathbb{R}^n$ with local uniform bounded area and small total curvature $\int_{\Sigma\cap B_1(0)} |A|^2$. A key ingredient is a new quantity which we call isothermal radius. Using the estimate of the isothermal radius we establish a compactness theorem of such surfaces in intrinsic $L^p$-topology and extrinsic $W^{2,2}$-weak topology. As applications, we can explain Leon Simon's decomposition theorem\cite{LS} in the viewpoint of convergence and prove a non-collapsing version of H\'{e}lein's convergence theorem\cite{H}\cite{KL12}.