Gravitational instantons with faster than quadratic curvature decay. I

Chen, Gao; Chen, Xiuxiong

doi:10.4310/acta.2021.v227.n2.a2

Cited by 20 publications

(38 citation statements)

References 31 publications

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“…These manifolds, originally introduced by Hawking in [Haw77], are widely studied for reasons lying at the intersection of Riemannian Geometry, Analysis and Mathematical Physics. Dropping any attempt to be complete, we address the interested reader to [Min10] for a nice mathematical presentation of this subject, as well as to the classification results contained in [CC15a,CC15b,CC16] and in the PhD thesis [Che17].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

Fogagnolo¹,

Mazzieri²

2020

Preprint

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The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull Ω * of a set Ω is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where Ω has C 1,α -boundary, the area of ∂Ω * is recovered as the limit of the p-capacities of Ω, as p → 1 + . Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided 3 ≤ n ≤ 7.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

Fogagnolo¹,

Mazzieri²

2020

Preprint

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“…It indicates that to estimate the rotational part, we only need to estimate its effect on one direction. By using this this obeservation, Chen and Chen have proved in [12] the following result.…”

Section: Table 1: Torus and Monodromymentioning

confidence: 89%

“…Recall that a gravitational instanton is a noncompact complete hyperkähler 4-manifold (M, g) with some curvature decay condition. It follows from [12] that any gravitational instanton with fasterthan-quadratic curvature decay falls into one of the categories: ALE, ALF, ALG and ALH, where the volume growth are r 4 , r 3 , r 2 and r, respectively. In the Appendix B, we prove that for complete hyperkähler 4-manifolds with (AF), the same classification also holds.…”

Section: Gravitational Instantonsmentioning

confidence: 99%

“…In this appendix, we prove that any asymptotically flat hyperkähler 4-manifold is a TALE manifold (Theorem B.2) by slightly generalizing the work of Chen-Chen [12].…”

Section: Table 1: Torus and Monodromymentioning

confidence: 99%

“…Theorem B.1 (Theorem 3.4 of [12]). For any complete hyperkähler 4-manifold (M 4 , g) with (AF), there exists a constant C H > 1 such that for any geodesic loop γ based at q with r = r(q) ≥ 2 and L(γ…”

Section: Table 1: Torus and Monodromymentioning

confidence: 99%

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On the geometry of asymptotically flat manifolds

Chen,

2019

Preprint

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In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a refined torus fibration over an ALE manifold. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat 4-manifolds with curvature decay and controlled holonomy. As an application, we show that any complete asymptotically flat Ricci-flat metric on a 4-manifold which is homeomorphic to R 4 must be isometric to the Euclidean or the Taub-NUT metric, provided that the tangent cone at infinity is not R × R + .

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