The symmetric Kazdan-Warner problem and applications

Abstract

After R. Schoen completed the solution of the Yamabe problem, compact manifolds could be allocated in three classes depending on whether they admit a metric with positive, non-negative or only negative scalar curvature. Here we follow Yamabe's first attempt to solve his problem through variational methods and provide an analogous equivalent classification for manifolds equipped with actions by non-discrete compact Lie groups. Moreover, we apply the method, and the results to classify total spaces of fibre bundles with compact structure groups (concerning scalar curvature), to conclude density results, and compare realizable scalar curvature functions between some exotic manifolds their standard counterpart. We also provide an extended range of prescribed scalar curvature functions of warped products, especially with Calabi-Yau manifolds, providing an upper bound for the first positive eigenvalue of the Laplacian under relatively mild conditions.

Yamabe problem in the presence of singular Riemannian Foliations

“…One of the approaches considered has been to find equivariant solutions with respect to a given compact Lie group action by isometries with positive dimensional orbits. The approach of finding equivariant solutions to the Yamabe equation has also led to an equivariant solution to the Kazdan-Warner problem in [11].…”

“…We point out that a general statement of the Palais' Principle of Symmetric criticality for an arbitrary foliated functional is probably false in general. Nonetheless Theorem G can be applied to the classical Yamabe equation in the context of the foliated Kazdan-Warner problem, that is, finding Riemannian metrics with scalar curvature equal to a prescribed F-invariant function for a singular Riemannian foliation (M, F) as in [11,Section 2]. This problem has also been considered for regular Riemannian foliations on closed manifolds in [49].…”