Quasi-Einstein structures and Hitchin's equations

2025/04/25 by Colling, Alex, Dunajski, Maciej · 2 citations
#Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.2504.18475

Abstract

We prove (Theorem 1.1.) that a class of quasi-Einstein structures on closed manifolds must admit a Killing vector field. This extends the rigidity theorem obtained in \citeDL23 for the extremal black hole horizons and completes the classification of compact quasi-Einstein surfaces in this class. We also explore special cases of the quasi-Einstein equations related to integrability and the Hitchin equations, as well as to Einstein-Weyl structures and Kazdan-Warner type PDEs. This leads to novel explicit examples of quasi-Einstein structures on (non-compact) surfaces and on S² × S¹.

Cited by

• Symmetries of extremal horizons
• On Compact Quasi-Einstein Metrics of Constant Scalar Curvature

Related

• Rigidity of quasi-Einstein metrics: The incompressible case
• Killing Fields on Compact m-Quasi-Einstein Manifolds
• Intrinsic rigidity of extremal horizons
• New quasi-Einstein metrics on a two-sphere
• Quasi-Einstein structures and almost cosymplectic manifolds