A note on renormalized volume functionals
Sun-Yung Alice Chang, Hao Fang, and C. Robin Graham
TL;DR
This paper explores new properties of renormalized volume functionals in asymptotically hyperbolic Einstein manifolds, providing formulas and analyzing their variations to identify extremal metrics.
Contribution
It introduces a formula for renormalized volume in even-dimensional cases and analyzes the second variation to characterize Einstein metrics as local extrema.
Findings
Derived new properties of renormalized volume functionals.
Provided a formula for the volume in terms of totally geodesic compactification.
Showed Einstein metrics of nonzero scalar curvature are local extrema.
Abstract
New properties are derived of renormalized volume functionals, which arise as coefficients in the asymptotic expansion of the volume of an asymptotically hyperbolic Einstein (AHE) manifold. A formula is given for the renormalized volume of an even-dimensional AHE manifold in terms of an arbitrary totally geodesic compactification. The second variation of renormalized volume functionals under conformal change is identified, and is used to show that Einstein metrics of nonzero scalar curvature are local extrema.
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