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Instability of conformally Kähler, Einstein metrics

Olivier Biquard, Tristan Ozuch

TL;DR

The paper proves instability for essentially all known four-dimensional Einstein metrics that are conformal to a Kähler metric and have nonnegative scalar curvature, provided they are not half-conformally flat. It develops a conformal framework for the second variation of the Hilbert-Einstein functional, centering on the conformally invariant operator $P=T^*T-W_+$ and a gradient structure via $T$, and constructs destabilizing tensors from harmonic anti-selfdual forms and Killing 2-forms. The main results cover compact examples such as the Page metric and Chen–Lebrun–Weber, as well as non-Kähler conformally Kähler gravitational instantons like Kerr, Taub-Bolt, and Chen-Teo, and extend to ALF Ricci-flat gravitational instantons with quadratic curvature decay, including Schwarzschild. This provides a uniform instability mechanism across known conformally Kähler Einstein 4-manifolds, impacting understanding of Ricci flow fixed points and the gravitational instanton landscape.

Abstract

We prove the instability of conformally Kähler, compact or ALF Einstein 4-manifolds with nonnegative scalar curvature which are not half conformally flat. This applies to all the known examples of gravitational instantons which are not hyperKähler and to the Chen-Lebrun-Weber metric in particular.

Instability of conformally Kähler, Einstein metrics

TL;DR

The paper proves instability for essentially all known four-dimensional Einstein metrics that are conformal to a Kähler metric and have nonnegative scalar curvature, provided they are not half-conformally flat. It develops a conformal framework for the second variation of the Hilbert-Einstein functional, centering on the conformally invariant operator and a gradient structure via , and constructs destabilizing tensors from harmonic anti-selfdual forms and Killing 2-forms. The main results cover compact examples such as the Page metric and Chen–Lebrun–Weber, as well as non-Kähler conformally Kähler gravitational instantons like Kerr, Taub-Bolt, and Chen-Teo, and extend to ALF Ricci-flat gravitational instantons with quadratic curvature decay, including Schwarzschild. This provides a uniform instability mechanism across known conformally Kähler Einstein 4-manifolds, impacting understanding of Ricci flow fixed points and the gravitational instanton landscape.

Abstract

We prove the instability of conformally Kähler, compact or ALF Einstein 4-manifolds with nonnegative scalar curvature which are not half conformally flat. This applies to all the known examples of gravitational instantons which are not hyperKähler and to the Chen-Lebrun-Weber metric in particular.
Paper Structure (3 sections, 6 theorems, 38 equations)

This paper contains 3 sections, 6 theorems, 38 equations.

Table of Contents

  1. Conformal invariance
  2. Instability of compact conformally Kähler metrics
  3. Instability of conformally Kähler ALF Ricci-flat metrics

Key Result

Theorem A

Let $(M^4,g)$ be an Einstein manifold which is conformal to a Kähler metric. Suppose Then if $(M^4,g)$ is not half-conformally flat, then it is unstable.

Theorems & Definitions (23)

  • Theorem A
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 4
  • Remark 5
  • Definition 6
  • Theorem 7
  • Example 8
  • ...and 13 more