The Symmetric Kazdan–warner Problem and Applications

Abstract

After R. Schoen completed the solution of the Yamabe problem, compact manifolds could be categorized 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 fiber bundles with compact structure groups (concerning scalar curvature), to conclude density results and compare realizable scalar curvature functions between some exotic manifolds and their standard counterpart. We also provide an extended range of prescribed scalar curvature functions of warped products, especially with Calabi–Yau manifolds.