Contact Structures of Sasaki Type and their Associated Moduli

TL;DR

This survey explores the moduli spaces of Sasakian contact structures, focusing on extremal, constant scalar curvature, and Sasaki-Einstein metrics, to deepen understanding of their geometric classifications.

Contribution

It provides a comprehensive overview of the moduli of Sasakian structures and metrics, highlighting recent developments and open problems in the field.

Findings

Overview of moduli spaces of Sasakian structures

Discussion of extremal and constant scalar curvature Sasaki metrics

Analysis of Sasaki-Einstein metrics and their classifications

Abstract

This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.