Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Renato Bettiol, Paolo Piccione
2013 Pacific Journal of Mathematics  
Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain conditions, there are at least 3 unit volume constant scalar curvature metrics in the conformal class [g_t] for infinitely many t's accumulating at 0. This holds, e.g., for
more » ... e.g., for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.
doi:10.2140/pjm.2013.266.1 fatcat:pldsxrcffndwrpri5wttqu4sje