Extremal Sasakian Geometry on $T^2\times S^3$ and Related Manifolds

TL;DR

This paper demonstrates the existence of extremal Sasakian structures on multiple contact structures of $T^2\times S^3$ and related manifolds, revealing a rich geometric landscape with many such structures.

Contribution

It proves the existence of extremal Sasakian structures on infinitely many contact structures on $T^2\times S^3$ and related manifolds, expanding understanding of Sasakian geometry.

Findings

Extremal Sasakian structures exist on infinitely many contact structures.

These structures form bouquets and fill the Sasaki cones in most cases.

One case has no extremal metrics, indicating limitations in certain geometries.

Abstract

We prove the existence of extremal Sasakian structures occurring on a countably infinite number of distinct contact structures on $T^2\times S^3$ and certain related manifolds. These structures occur in bouquets and exhaust the Sasaki cones in all except one case in which there are no extremal metrics.