Extremal Sasakian Geometry on S³-bundles over Riemann Surfaces

In this paper we study the Sasakian geometry on S^3-bundles over a Riemann surface of genus g>0 with 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 2-dimensional 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 in the Sasaki cone showing that if 0<g<5 it exhausts the entire cone. Examples are given where exhaustion fails.