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Classification results for conformally K\"ahler gravitational instantons
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Classification results for conformally K\"ahler gravitational instantons
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.
Forward citations
Cited by 7 Pith papers
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Conformal K\"{a}hler rigidity of Einstein four-manifolds
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).
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On the existence of toric ALE and ALF gravitational instantons
Existence and uniqueness of toric ALE and ALF Ricci-flat gravitational instantons for admissible rod structures, with toric self-dual instantons being multi-Eguchi-Hanson or multi-Taub-NUT.
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On toric self-dual Einstein gravitational instantons
Toric self-dual Einstein instantons with negative cosmological constant satisfying a global conformal Kähler extension condition are precisely the Calderbank-Pedersen-Singer multipole solutions.
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A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds
The mass of any toric ALE/ALF 4-manifold with nonnegative scalar curvature is at least the mass of its corresponding toric gravitational instanton, corrected by conical angle defects, and equality holds only for the i...
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A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds
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...
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Mass Lower Bounds for Asymptotically Locally Flat Manifolds
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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On Chen-Teo geometries with cosmological constant
Einstein metrics extending Chen-Teo geometries with cosmological constant are either Plebański-Demiański or anti-self-dual Weyl cases, yielding for lambda<0 a conformal infinity between asymptotically hyperbolic ends,...
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