Extremal Sasakian Geometry on S^3-bundles over Riemann Surfaces

TL;DR

This paper investigates extremal Sasakian metrics on S^3-bundles over Riemann surfaces, demonstrating the existence of infinitely many inequivalent contact structures with constant scalar curvature Sasaki metrics, often irregular, and analyzing their properties.

Contribution

It proves the existence of infinitely many inequivalent contact structures with CSC Sasaki metrics on S^3-bundles over Riemann surfaces, expanding understanding of extremal Sasakian geometry.

Findings

Countably infinite inequivalent contact structures with CSC Sasaki metrics.

Most CSC Sasaki metrics are irregular.

Exhaustion of the extremal subset in the Sasaki cone for genus g<5.

Abstract

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.