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The renormalized volume of a 4-dimensional Ricci-flat ALE space

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arxiv 1901.03647 v2 pith:AP2HEEMA submitted 2019-01-11 math.DG

The renormalized volume of a 4-dimensional Ricci-flat ALE space

classification math.DG
keywords renormalizedvolumedimensionalonlyricci-flatspaceexamplesknown
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We introduce a natural definition of the renormalized volume of a 4-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of 4-dimensional Ricci-flat ALE spaces are Kronheimer's gravitational instantons and their quotients, which are also known to be the only possible examples of special holonomy. We calculate the renormalized volume of these spaces in terms of Kronheimer's period map.

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