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arXiv · 2608.31008

Stability of gravitational instantons with a bounded Killing vector field

Tristan Ozuch
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Abstract

We prove that a complete ALF Ricci-flat 4-manifold carrying a bounded Killing vector field is linearly stable if and only if it is locally hyperk\"ahler. This applies uniformly to all known examples and in particular proves the instability of all of the metrics recently found by Li-Sun. The argument is based on infinitesimal Einstein-Maxwell deformations of the Ricci-flat metric associated with the Killing field. A separate simpler identity proves the instability of nonflat static Ricci-flat 4-metrics which include the static axisymmetric smooth Riemannian Myers/Korotkin-Nicolai metrics. To our knowledge, this proves the equivalence between linear stability and special holonomy and (anti-)selfduality for all known families of Ricci-flat 4-manifolds. In higher dimensions, for generalizations of AF manifolds, we show that stability forces the universal cover to split a line. This gives explicit destabilizing tensors on the Riemannian Myers-Perry instantons and the Riemannian Schwarzschild-Tangherlini metrics in all dimensions.

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Tristan Ozuch. 2026-08-31. Stability of gravitational instantons with a bounded Killing vector field. https://arxiv.org/abs/2608.31008

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