Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Renato G. Bettiol and Paolo Piccione

Vol. 266 (2013), No. 1, 1–21

DOI: 10.2140/pjm.2013.266.1

Abstract

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 → 0 (that is, 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 accumulating at 0. This holds, for instance, for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.

Keywords

Yamabe problem, constant scalar curvature metrics, equivariant bifurcation, Riemannian submersions, Cheeger deformation, homogeneous fibration

Mathematical Subject Classification 2010

Primary: 53C30, 58J55

Secondary: 53A30, 53C20, 53C21, 58E50, 58J50

Milestones

Received: 6 August 2012

Accepted: 17 January 2013

Published: 23 September 2013