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On the asymptotic geometry of gravitational instantons. (English) Zbl 1215.53043

The author investigates the geometry at infinity of gravitational instantons, or asymptotically flat hyper-Kähler four-manifolds. His main result states that gravitational instantons with cubic volume growth are asymptotically locally flat, that is, asymptotic to a circle fibration over a Euclidean space of dimension three, with fibers of asymptotically constant length. In Appendix A, the author sharpens his result from [Geom. Funct. Anal. 18, No. 5, 1696–1749 (2008; Zbl 1166.53028)] on the bound of the derivatives of the curvature of a Ricci flat manifold.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C80 Applications of global differential geometry to the sciences
53C29 Issues of holonomy in differential geometry

Citations:

Zbl 1166.53028
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