Sasaki-Einstein Manifolds

TL;DR

The article surveys Sasaki–Einstein geometry, detailing constructions and obstructions that have advanced the field over the past decade. It unifies regular, quasi-regular, irregular, toric, and explicit-cohomogeneity constructions through the lens of Kähler cones, Reeb foliations, joins, and Hamiltonian forms, with central results such as $Ric_g=2(n-1)g$, cone Ricci-flatness, and $Hol^0(\bar{g})\subset SU(n)$. It highlights key families (e.g., $Y^{p,q}$, Labc-type, Brieskorn–Pham links) and the role of obstructions (Futaki, Bishop, Lichnerowicz) and stability notions in determining existence and uniqueness of Sasaki–Einstein metrics. The work underscores the deep connections between Sasakian geometry and algebraic geometry (orbifolds, toric geometry, K-stability) and outlines open problems, particularly in irregular and higher-rank toric cases, with potential impact on both mathematics and string theory.

Abstract

This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.

Theorems & Definitions (73)

• Definition 1.1
• Proposition 1.2
• Definition 1.3
• Proposition 1.4
• proof
• Theorem 1.5
• Theorem 1.6
• Definition 1.7
• Definition 1.8
• Proposition 1.9