Communications in Analysis and Geometry
Volume 27 (2019)
Complete hypersurfaces in Euclidean spaces with finite strong total curvature
Pages: 1251 – 1279
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  and and Müller–Šverák . Further properties of these hypersurfaces are presented, including a gap theorem for the total curvature.
Both authors are partially supported by CNPq and Faperj.
Received 26 April 2016
Accepted 20 August 2017
Published 12 December 2019