Existence and non-existence of constant scalar curvature and extremal Sasaki metrics

We discuss the existence and non-existence of constant scalar curvature, as well extremal, Sasaki metrics. prove that natural Sasaki–Boothby–Wang manifold over admissible projective bundles local products non-negative CSC Kähler metrics, described in [3], always has a curvature (CSC) metric its Sasaki-Reeb cone. Moreover, we give examples show extremal Sasaki–Reeb cone, defined set vector fields admitting compatible metric, is not necessarily connected it can be empty even non-Gorenstein case. also by example non-empty cone need contain which answers question posed [16]. The paper contains an appendix where explore metrics weighted [43], on manifolds