Gromov-Hausdorff convergence theory of surfaces

TL;DR

This paper applies Gromov-Hausdorff convergence to provide new insights into Huber's classification theorem for complete Riemannian surfaces with finite total curvature, emphasizing the role of Hardy estimates.

Contribution

It offers a novel perspective on Huber's theorem using Gromov-Hausdorff convergence and builds on Hardy estimates for surfaces with finite total curvature.

Findings

New understanding of Huber's classification via Gromov-Hausdorff convergence

Connection between curvature estimates and surface classification

Extension of Hardy-estimate applications in geometric analysis

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.