# Area minimizing vector fields on spheres

## Olga Gil-Medrano Universidad de Valencia

A vector field V on a n-dimensional round sphere S^n(r) defines a submanifold V(S^nof the tangent bundle TS^n. A natural question, that goes back to the pioneering work of H. Gluck and W. Ziller in the 80's, is to ask for the (absolute) minimizers of the area (i.e. the n-dimensional volume ) of V(S^n) among smooth unit vector fields. This volume is computed with respect to the natural metric on the tangent bundle as defined by Sasaki. Surprisingly, the problem is only completely solved for dimension three. Nevertheless, a number of important advances towards a possible solution have been obtained during these almost 30 years for spheres of odd dimension; the first part of the talk consists in a survey of these results. The second part is devoted to the special case of the 2-dimensional sphere S^2(rwhose unit tangent is diffeomorphic to the 3-dimensional real projective space. Since there is no globally defined unit vector field on S^2, the infimum is taken on a class of singular unit vector fields. These are vector fields defined on a dense open set and such that the closure of their image is a surface without boundary of T^1S^2 . In particular if the vector field is area minimizing it defines a minimal surface of T^1S^2. After giving the solution of the Gluck and Ziller problem for the unit 2-sphere, the talk will end by showing some results concerning the case of 2-spheres with radius r1 .