A topological lower bound for the energy of a unit vector field on a closed hypersurface of the euclidean space. The 3-dimensional case

TL;DR

This paper establishes a topological lower bound for the energy of a unit vector field on a closed 3-dimensional hypersurface in Euclidean space, linking geometric energy to the degree of the Gauss map.

Contribution

It proves that the degree of the Gauss map provides a lower bound for the total bending functional, extending to the energy functional introduced by Wood.

Findings

Degree of the Gauss map bounds the total bending functional

Energy functional admits a topological lower bound

Results are specific to 3-dimensional hypersurfaces in Euclidean space

Abstract

In this short note we prove that the degree of the Gauss map {\nu} of a closed 3-dimensional hypersurface of the Euclidean space is a lower bound for the total bending functional B, introduced by G. Wiegmink. Consequently, the energy functional E introduced by C. M. Wood admits a topological lower bound.