Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

TL;DR

This paper demonstrates the existence of multiple solutions to the Yamabe problem on collapsing Riemannian submersions, showing at least three constant scalar curvature metrics in certain conformal classes as fibers shrink.

Contribution

It establishes the multiplicity of solutions to the Yamabe problem in collapsing fibered manifolds, especially for homogeneous metrics obtained via Cheeger deformation.

Findings

At least 3 solutions exist for infinitely many collapsing metrics

Solutions occur in conformal classes as fibers shrink to zero

Results apply to homogeneous fibrations with positive scalar curvature

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 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 homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.