Yamabe problem in the presence of singular Riemannian Foliations

TL;DR

This paper proves the existence of multiple solutions to Yamabe problems with symmetries from singular Riemannian foliations, revealing new solution behaviors on spheres that are not derived from group actions or isoparametric functions.

Contribution

It introduces a novel approach using variational methods and symmetries from singular Riemannian foliations to find solutions to Yamabe problems, including sign-changing solutions with new qualitative properties.

Findings

Existence of infinitely many sign-changing solutions constant along foliation leaves.

Identification of a positive minimal energy solution among symmetric solutions.

Development of a Sobolev embedding theorem for singular Riemannian foliations.

Abstract

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.