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Poincaré-Einstein 4-manifolds with conformally Kähler geometry

Mingyang Li, Hongyi Liu

TL;DR

The paper constructs infinite families of Poincaré-Einstein 4-manifolds whose conformal infinities are conformally Kähler, by exploiting a regular Killing field and Kähler reduction to reduce the Einstein equations to Toda-type PDEs. It distinguishes Type I (ASD-conformal-Kähler) and Type II (conformally Kähler via extremal Kähler metrics) and develops a canonical conformal change to obtain natural infinities, proving Morse-Bott structure for the moment map and deriving decoupled cohomogeneity-one solutions that recover classical examples such as AdS-Schwarzschild and Page–Pope. A central result is a Dirichlet boundary value problem for the function $w$ on $[0, frac{1}{2}] imesoldsymbol{ extSigma}$ with integral constraint, shown to admit a unique solution, which yields smooth conformally Kähler PE fillings on the total space of complex line bundles over Riemann surfaces. The authors prove that every regular conformally Kähler PE filling of this type arises from their PDE construction, and they establish positivity of the key reduction function $W$, ensuring a valid PE metric. This work provides non-perturbative, parameterized families of PE metrics with conformal infinities of non-positive Yamabe type, expanding the landscape of known PE fill-ins.

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.

Poincaré-Einstein 4-manifolds with conformally Kähler geometry

TL;DR

The paper constructs infinite families of Poincaré-Einstein 4-manifolds whose conformal infinities are conformally Kähler, by exploiting a regular Killing field and Kähler reduction to reduce the Einstein equations to Toda-type PDEs. It distinguishes Type I (ASD-conformal-Kähler) and Type II (conformally Kähler via extremal Kähler metrics) and develops a canonical conformal change to obtain natural infinities, proving Morse-Bott structure for the moment map and deriving decoupled cohomogeneity-one solutions that recover classical examples such as AdS-Schwarzschild and Page–Pope. A central result is a Dirichlet boundary value problem for the function on with integral constraint, shown to admit a unique solution, which yields smooth conformally Kähler PE fillings on the total space of complex line bundles over Riemann surfaces. The authors prove that every regular conformally Kähler PE filling of this type arises from their PDE construction, and they establish positivity of the key reduction function , ensuring a valid PE metric. This work provides non-perturbative, parameterized families of PE metrics with conformal infinities of non-positive Yamabe type, expanding the landscape of known PE fill-ins.

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 -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.

Paper Structure

This paper contains 20 sections, 30 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Poincaré-Einstein metrics with conformally Kähler geometry
  3. Type I with conformally Kähler geometry
  4. Type II with conformally Kähler geometry
  5. Regularity of conformal infinity
  6. Type I
  7. Type II
  8. The canonical conformal change
  9. Morse-Bott property for
  10. Decoupled solutions and cohomogeneity-one Einstein metrics
  11. Decoupled solutions with base
  12. Decoupled solutions with base when
  13. Decoupled solutions with base
  14. Regularity and conclusion
  15. The Dirichlet boundary value problem
  16. ...and 5 more sections

Key Result

Theorem 1.3

Given a closed, oriented, smooth surface $(\Sigma_{\mathop{\mathrm{\mathtt{g}}}\nolimits},g^\natural)$ of genus $\mathop{\mathrm{\mathtt{g}}}\nolimits \geq 1$, area $\mathop{\mathrm{\mathfrak{a}}}\nolimits$, and Euler characteristic $\chi = 2-2\mathop{\mathrm{\mathtt{g}}}\nolimits$, suppose $k \in \

Theorems & Definitions (62)

  • Definition 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: derdzinski
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Example 2.4: Hyperbolic 4-ball
  • ...and 52 more