Quasi-Einstein structures and Hitchin's equations

TL;DR

The paper develops a comprehensive rigidity and structural theory for quasi-Einstein manifolds, proving that non-gradient quasi-Einstein structures on closed manifolds with $m\ge 2$ or $m\le 2-n$ admit a Killing field $K$ with $[K,X]=0$, thereby generalising previous results and completing the compact-surface classification for $m\notin(0,2)$. It builds a bridge to Hitchin’s equations by showing that the $(m=-1,\lambda=0)$ case on surfaces arises as a symmetry reduction of ASDYM with gauge group $SU(2)$ and is equivalent to Hitchin data, with the rescaled metric $h=|X|^2g$ attaining constant negative curvature. The work also develops Einstein–Weyl frames for $m=2-n$, analyzes the two- and higher-dimensional topology of quasi-Einstein surfaces via Poincaré–Hopf, and provides explicit constructions of non-Killing examples through third-order PDE reductions and homothety-invariant ansätze. Collectively, these results illuminate the interplay between geometric analysis, integrable systems, and projective/metrizability structures, yielding new explicit examples on $S^2\times S^1$ and non-compact surfaces and clarifying rigidity phenomena across dimensions.

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 \cite{DL23} 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 $S^2 \times S^1$.

Theorems & Definitions (46)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• proof : Proof of Theorem \ref{['RQEi']}