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

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