Gromov-Hausdorff convergence theory of surfaces

Abstract

In this paper, we use the viewpoint of Gromov-Haustorff convergence to give some new comprehension of well known theorem,it is Huber's classification theorem\cite{Huber}\cite{MS}for complete Riemannian surfaces immersed in ${\mathbb{R}}^n$ with finite total curvature( ${\int }_{\Sigma }|A{|}^2<+\infty$) it depend heavily on Müller and Šverák's Hardy-estimate\cite{MS} for the curvature form of surfaces immersed in ${\mathbb{R}}^n$ with finite total curvature.