Complete hypersurfaces in Euclidean spaces with strong finite total curvature

TL;DR

This paper proves that complete hypersurfaces with strong finite total curvature in Euclidean space are proper, topologically equivalent to a compact manifold minus points, and under certain conditions, have a Gauss map extending continuously to these points.

Contribution

It establishes topological and geometric properties of hypersurfaces with strong finite total curvature, including properness and Gauss map extension.

Findings

Hypersurfaces are proper and diffeomorphic to a compact manifold minus finitely many points.

Under additional conditions, the Gauss map extends continuously to punctures.

Provides a characterization of hypersurfaces with strong finite total curvature.

Abstract

We prove that strong finite total curvature 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.