The Kazdan--Warner problem with (and via) symmetries

TL;DR

This paper extends the classification of scalar curvature possibilities to manifolds with symmetries, analyzing invariant metrics and providing explicit examples and density results.

Contribution

It offers a classification of scalar curvature functions on manifolds with symmetries, generalizing Yamabe's approach and studying invariant metrics under group actions.

Findings

Classification of scalar curvature functions for symmetric manifolds

Explicit examples of invariant metrics with prescribed scalar curvature

Density results for scalar curvature functions in symmetric settings

Abstract

After R.~Schoen completed the solution to the Yamabe problem, compact manifolds could be categorized into 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 compact connected Lie groups. Moreover, we approach the Kazdan--Warner problem for manifolds with isometric actions. Namely, we provide a detailed study of admissible functions as scalar curvature for invariant Riemannian metrics. We also conclude classification results of total spaces of fiber bundles with compact structure groups concerning scalar curvature. We provide explicit examples of our results' applications in large classes of manifolds, further obtaining density results.