A Note on the Infimum of Energy of Unit Vector Fields on a Compact Riemannian Manifold |
| |
Authors: | Giovanni Nunes Jaime Ripoll |
| |
Institution: | 1.Universidade Federal de Pelotas—IFM,Pelotas,Brazil;2.Universidade Federal do R.G. do Sul,Instituto de Matemática,Porto Alegre,Brazil |
| |
Abstract: | Our main result in this paper establishes that if G is a compact Lie subgroup of the isometry group of a compact Riemannian manifold M acting with cohomogeneity one in M and either G has no singular orbits or the singular orbits of G have dimension at most n−3, then the unit vector field N orthogonal to the principal orbits of G is weakly smooth and is a critical point of the energy functional acting on the unit normal vector fields of M. A formula for the energy of N in terms of the of integral of the Ricci curvature of M and of the integral of the square of the mean curvature of the principal orbits of G is obtained as well. In the case that M is the sphere and G the orthogonal group it is known that that N is minimizer. It is an open question if N is a minimizer in general. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|