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arXiv · 2510.04928

Poincaré-Einstein 4-manifolds with conformally Kähler geometry

Mingyang Li·Hongyi Liu
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Abstract

We study 4-dimensional Poincaré-Einstein manifolds whose conformal class contains a Kähler metric. Such Einstein metrics are non-Kähler and admit a Killing field extending to the conformal infinity, and the Einstein equation reduces to a Toda-type equation. When the Killing field integrates to an $\mathbb{S}^1$-action, we formulate a Dirichlet boundary value problem and establish existence and uniqueness theory. This construction provides a non-perturbative realization of infinite-dimensional families of new Poincaré-Einstein metrics whose conformal infinities are of non-positive Yamabe type.

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Mingyang Li, Hongyi Liu. 2025-10-06. Poincaré-Einstein 4-manifolds with conformally Kähler geometry. https://arxiv.org/abs/2510.04928

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