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

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 Hélein's convergence theorem\cite{H}\cite{KL12}.