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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.

Related project(s):
52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II