Communications in Analysis and Geometry

Volume 27 (2019)

Number 6

Complete hypersurfaces in Euclidean spaces with finite strong total curvature

Pages: 1251 – 1279



Manfredo do Carmo (Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, RJ, Brazil)

Maria Fernanda Elbert (Instituto de Matemática, Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, RJ, Brazil)


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 Müller–Šverák [18]. 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