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Classification results for conformally K\"ahler gravitational instantons

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arxiv 2310.13197 v4 pith:KAFAO6SR submitted 2023-10-19 math.DG math-phmath.MP

Mingyang Li This is my paper · ORCID
classification math.DGmath-phmath.MP
keywords ahlerhermitiannon-kfamilygravitationalasymptoticconformallyinfinity
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abstract

We investigate the asymptotic geometry of Hermitian non-K\"ahler Ricci-flat metrics with finite $\int|Rm|^2$ at infinity. Specifically, we prove: 1. Any such metric is asymptotic to an ALE, ALF-A, AF, skewed special Kasner, ALH* model at infinity. 2. Any Hermitian non-K\"ahler gravitational instanton with non-Euclidean volume growth is one of the following: the Kerr family, the Chen-Teo family, the Taub-bolt space, the reversed Taub-NUT space. This particularly confirms a conjecture by Aksteiner-Andersson. It includes the well-known Kerr family from general relativity. 3. All Hermitian non-K\"ahler gravitational instantons can be compactified to log del Pezzo surfaces. This explains a curious relation to compact Hermitian non-K\"ahler Einstein 4-manifolds. For a 4-dimensional Ricci-flat metric, being Hermitian non-K\"ahler is equivalent to being non-trivially conformally K\"ahler.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Conformal K\"{a}hler rigidity of Einstein four-manifolds

    math.DG 2026-07 conditional novelty 8.0 of 10

    For compact Einstein four-manifolds, the spectral gap α>β in the self-dual Weyl curvature forces β=γ=−α/2, so the metric is conformally Kähler with positive scalar curvature (after at most a double cover).

  2. A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds

    math.DG 2026-05 unverdicted novelty 7.0 of 10

    The mass of toric ALE or ALF 4-manifolds with nonnegative scalar curvature is at least the mass of the corresponding toric gravitational instanton plus a term from its conical defects, with equality only when the mani...

  3. Mass Lower Bounds for Asymptotically Locally Flat Manifolds

    math.DG 2025-09 conditional novelty 7.0 of 10

    For ALF 4-manifolds with an almost free circle symmetry and nonnegative scalar curvature, mass is nonnegative and at least (ℓ/16) times the degree of the asymptotic circle bundle; in the AF case, homology-trivial coor...

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