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Hyperkähler ambient metrics associated with twistor CR manifolds

Taiji Marugame

TL;DR

This work constructs a neutral (signature $(2,2)$) hyperkähler ambient metric for twistor CR manifolds by embedding the twistor CR structure into the twistor space of an anti self-dual Poincaré–Einstein 4-manifold. Central to the construction are two integrable complex structures $bI$ and $bJ$ on the spinor bundle $S'$, whose interplay yields a Ricci-flat ambient metric $ ilde{g}$ on $S'ackslasholdsymbol{o}$ and a corresponding Kähler–Einstein metric on the projectivized spinor space $P(S')$ with KE form $oldsymbol ext{ω}_{ m KE}=iddbaroldsymbol{ m log} orm{oldsymbol{ ext{pi}}}^2$. By pulling back along defining functions and employing dilations, the construction yields a hyperkähler ambient metric for the twistor CR manifold $M\congP(S')|_\Sigma$, together with a smooth Cheng–Yau metric on $P(S')|_{Xackslash\Sigma}$ and a decomposition $g_{ m CY}=(-oldsymbol{\Lambda} g_+)oxplus(-g_{ m FS})$. In the flat model, the authors provide explicit coordinates and formulas, demonstrating that the ambient metric reduces to a flat neutral metric on $C^4$, with explicit potential $ ilde r$ and Kähler form. The results give a concrete example of ambient metrics with special holonomy in CR geometry and illuminate connections to Fefferman–Graham theory, Poincaré–Einstein geometry, and twistor theory.

Abstract

Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR 5-manifolds defined as $\mathbb{P}^1$-bundles over 3-dimensional conformal manifolds. In this paper, we embed a real analytic twistor CR manifold into the twistor space of the anti self-dual Poincaré-Einstein metric whose conformal infinity is the base conformal 3-manifold, and construct the associated Fefferman ambient metric as a neutral hyperkähler metric on the spinor bundle with the zero section removed. We also describe the structure of the Cheng--Yau type Kähler-Einstein metric which has the twistor CR manifold as the boundary at infinity.

Hyperkähler ambient metrics associated with twistor CR manifolds

TL;DR

This work constructs a neutral (signature ) hyperkähler ambient metric for twistor CR manifolds by embedding the twistor CR structure into the twistor space of an anti self-dual Poincaré–Einstein 4-manifold. Central to the construction are two integrable complex structures and on the spinor bundle , whose interplay yields a Ricci-flat ambient metric on and a corresponding Kähler–Einstein metric on the projectivized spinor space with KE form . By pulling back along defining functions and employing dilations, the construction yields a hyperkähler ambient metric for the twistor CR manifold , together with a smooth Cheng–Yau metric on and a decomposition . In the flat model, the authors provide explicit coordinates and formulas, demonstrating that the ambient metric reduces to a flat neutral metric on , with explicit potential and Kähler form. The results give a concrete example of ambient metrics with special holonomy in CR geometry and illuminate connections to Fefferman–Graham theory, Poincaré–Einstein geometry, and twistor theory.

Abstract

Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR 5-manifolds defined as -bundles over 3-dimensional conformal manifolds. In this paper, we embed a real analytic twistor CR manifold into the twistor space of the anti self-dual Poincaré-Einstein metric whose conformal infinity is the base conformal 3-manifold, and construct the associated Fefferman ambient metric as a neutral hyperkähler metric on the spinor bundle with the zero section removed. We also describe the structure of the Cheng--Yau type Kähler-Einstein metric which has the twistor CR manifold as the boundary at infinity.
Paper Structure (31 sections, 21 theorems, 247 equations)

This paper contains 31 sections, 21 theorems, 247 equations.

Table of Contents

  1. Introduction
  2. The spinor calculus
  3. Riemannian spinor bundles
  4. The curvature spinors
  5. Conformal spinor bundles
  6. The hyperkähler metric on $\mathbb{S}'\setminus\{\boldsymbol{o}\}$
  7. The complex structure $\mathbb{I}$ on $\mathbb{S}'\setminus\{\boldsymbol{o}\}$
  8. Another complex structure $\mathbb{J}$ on $\mathbb{S}'\setminus\{\boldsymbol{o}\}$
  9. The hyperkähler metric $\widetilde{g}$ on $\mathbb{S}'\setminus\{\boldsymbol{o}\}$
  10. The Kähler-Einstein metric on $\mathbb{P}(\mathbb{S}')$
  11. The Ricci-flat equation and the complex Monge--Ampère equation
  12. The structure of the Kähler-Einstein metric on $\mathbb{P}(\mathbb{S}')$
  13. Ambient metrics in conformal and CR geometries
  14. The Fefferman--Graham ambient metric
  15. The Poincaré-Einstein metric
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $(X, g)$ be a $4$-dimensional anti self-dual Einstein manifold with scalar curvature $R\neq0$, and $\mathbb{S}'$ the spinor bundle over $X$. Then, $\mathbb{S}'\setminus\{\boldsymbol{o}\}$ admits a hyperkähler metric which is positive definite when $R>0$ and has neutral signature when $R<0$.

Theorems & Definitions (36)

  • Theorem 1.1: Sw
  • Theorem 1.2
  • Theorem 3.1: PAHS
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • proof
  • Proposition 3.6
  • proof
  • ...and 26 more