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arXiv · 1304.5510

Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Renato G. Bettiol·Paolo Piccione
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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.

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Renato G. Bettiol, Paolo Piccione. 2013-04-19. Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions. https://doi.org/10.2140/pjm.2013.266.1

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