Extremal Sasakian Geometry on S3-bundles over Riemann Surfaces

In this paper, we study the Sasakian geometry on S³-bundles over a Riemann surface Σg of genus g>0 with an emphasis on extremal Sasaki metrics. We prove the existence of a countably infinite number of inequivalent contact structures on the total space of such bundles that admit 2D Sasaki cones each with a Sasaki metric of constant scalar curvature (CSC). This CSC Sasaki metric is most often irregular. We further study the extremal subset $\mathfrak e$ in the Sasaki cone showing that if 0