Compactness of surfaces in \(\mathbb{R}^n\) with small total curvature
The authors study the compactness and local structure of immersed surfaces in the unit ball of \(\mathbb{R}^n\) with uniformly bounded area and small total curvature. A fundamental convergence theorem is obtained in intrinsic \(L_p\)-topology and extrinsic \(W^{2,2}\)-weak topology. To prove this theorem, the authors introduce a new quantity called ``isothermal radius'' and estimate its lower bound. In the end, the authors give two applications, one is bi-Lipschitz parametrization and Leon Simon's decomposition theorem, and the other one is Hélein's convergence theorem.
scientific article; zbMATH DE number 7388788
scientific article