Multiparameter sweepouts and the existence of minimal hypersurfaces

Fernando Marques , Princeton University
Rutgers - Hill Center, Room 705

It follows from the work of Almgren in the 1960s that the space of unoriented closed hypersurfaces, in a compact Riemannian manifold M, endowed with the flat topology, is weakly homotopically equivalent to the infinite dimensional real projective space.  Together with Andre Neves, we have used this nontrivial structure, and previous work of Gromov and Guth on the associated multiparameter sweepouts, to prove the existence of infinitely many smooth embedded closed minimal hypersurfaces in manifolds with positive Ricci curvature and dimension at most 7.  This is motivated by a conjecture of Yau (1982). We will discuss this result, the higher dimensional case and current work in progress on  the problem of the Morse index.