Ricci Flow and Volume Renormalizability

TL;DR

This paper studies the behavior of asymptotically hyperbolic metrics under Ricci flow, focusing on volume renormalization and curvature variations, extending previous results to a broader class of metrics.

Contribution

It proves that certain asymptotic expansions are preserved under normalized Ricci flow and derives a new variation formula for the renormalized volume involving scalar curvature.

Findings

Preservation of even asymptotic expansions under Ricci flow.

Derivation of a variation formula for renormalized volume.

Extension of previous results to a broader class of metrics.

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.