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 H\'{ e}lein's convergence theorem\cite{ H}\cite{ KL12}.
Categories: math.DG