The Weighted Ambient Metric

Abstract

We construct the ambient metric for smooth metric measure space (Mⁿ, g, f, m, µ) with m ∈ [0, ∞]. With this metric, we construct the GJMS operators and renormalized volume coefficients. We write down the explicit ambient metric for Einstein spaces and locally conformally flat spaces. We also study the total functional and its critical points. We show that Einstein metrics are stable critical points of these functionals. Using the ambient metric for manifolds with density, we construct families of fully non-linear analogues of Perelman’s F-functional and W-functional, and study their monotonicity under the gradient Ricci flow.