A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface
Tom 125 / 2020
Annales Polonici Mathematici 125 (2020), 203-213
MSC: Primary 57R25; Secondary 47H11, 58E20.
DOI: 10.4064/ap200203-14-5
Opublikowany online: 9 November 2020
Streszczenie
For a unit vector field on a closed immersed Euclidean hypersurface , 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 1, 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}.
