Complete hypersurfaces in Euclidean spaces with finite strong total curvature

摘要

We prove that finite strong total curvature (see definition in Section 2) complete hypersurfaces of (n + 1)-euclidean space are proper and diffeomorphic to a compact manifold minus finitely many points. With an additional condition, we also prove that the Gauss map of such hypersurfaces extends continuously to the punctures. This is related to results of White [22] and and Muller-Sverak [18]. Further properties of these hypersurfaces are presented, including a gap theorem for the total curvature.