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 M\"{u}ller and \v{S}ver\'{a}k's Hardy-estimate\cite{MS} for the curvature form of surfaces immersed in $\mathbb{R}^n$ with finite total curvature.