# A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

## TL;DR

This paper establishes a topological lower bound for the energy of unit vector fields on closed Euclidean hypersurfaces, linking it to the degree of the Gauss map, and explores properties of certain functionals related to the immersion degree.

## Contribution

It introduces a new lower bound for the energy of vector fields based on the Gauss map degree and studies the properties of functionals on hypersurfaces, including minimization by Hopf flows.

## Key findings

- Lower bound for energy depends on Gauss map degree
- Hopf flows minimize the functional  on spheres
- Functionals  share properties related to immersion degree

## Abstract

For a unit vector field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\geq 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the unit sphere $\mathbb{S}^{2n+1}$, immersed with degree one, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals $\mathcal{B}_k$ on a compact Riemannian manifold $M^{m}$, $1\leq k\leq m$, and show that, when the underlying manifold is a closed hypersurface, these functionals possess similar properties regarding the degree of the immersion. In addition, we prove that Hopf flows minimize $\mathcal{B}_n$ on $\mathbb{S}^{2n+1}$.