Renormalized volume and the evolution of APEs

TL;DR

This paper investigates how the renormalized volume of asymptotically Poincare-Einstein metrics evolves under normalized Ricci flow, revealing monotonicity properties and connections to the Hawking-Page phase transition in four dimensions.

Contribution

It establishes the monotonic decrease of renormalized volume under Ricci flow for certain initial conditions and relates these results to phase transitions in four-dimensional gravity.

Findings

Renormalized volume decreases monotonically if initial scalar curvature condition holds.

Derived the formula for the time derivative of renormalized volume under Ricci flow.

Connected the evolution of renormalized volume to the Hawking-Page phase transition in 4D.

Abstract

We study the evolution of the renormalized volume functional for asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized Ricci flow. In particular, we prove that the time derivative of the renormalized volume along the flow is the negative integral of scal(g(t)) + n(n-1) over the manifold. This implies that if scal(g(0))+n(n-1) is non-negative at t=0, then the renormalized volume decreases monotonically. We also discuss how, when n=4, our results describe the Hawking-Page phase transition. Differences in renormalized volumes give rigorous meaning to the Hawking-Page difference of actions and describe the free energy liberated in the transition.