Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 057, 21 pages      arXiv:1607.08558      https://doi.org/10.3842/SIGMA.2019.057

Ricci Flow and Volume Renormalizability

Eric Bahuaud ^a, Rafe Mazzeo ^b and Eric Woolgar ^c
a) Department of Mathematics, Seattle University, 901 12th Ave, Seattle, WA 98122, USA
^b) Department of Mathematics, Stanford University, Stanford, CA 94305, USA
^c) Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

Received December 06, 2018, in final form July 30, 2019; Published online August 07, 2019

Abstract
With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula\[\frac{{\rm d}}{{\rm d}t} {\rm RenV}\big(M^n, g(t)\big) = -\mathop{\vphantom{T}}^R \int_{M^n} (S(g(t))+n(n-1) ) {\rm d}V_{g(t)},\]where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and $\mathop{\vphantom{T}}^R \! \! \! \int (\cdot) {\rm d}V_g$ is Riesz renormalization. This extends our earlier work to a broader class of metrics.

Key words: Ricci flow; conformally compact metrics; asymptotically hyperbolic metrics; renormalized volume.

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