The weighted ambient metric for manifolds with density

TL;DR

This work develops a weighted Fefferman–Graham ambient framework for manifolds with density, proving existence and ambient-equivalence uniqueness of the weighted ambient metric and establishing a dual correspondence with singular gradient Ricci flow spacetimes. It then leverages this ambient setup to construct infinite families of fully nonlinear analogues of Perelman’s $\mathcal{F}$ and $\mathcal{W}$ functionals, together with associated weighted GJMS operators and renormalized volume coefficients, all shown to be variational and, in many cases, monotone along the weighted Ricci flows. The authors provide explicit ambient realizations for key examples—gradient Ricci solitons, locally conformally flat, and Einstein manifolds—highlighting the practical computability of the coefficients and operators in canonical geometries. A central technical achievement is the ambient-driven Ricci-flow vector field, enabling short proofs of monotonicity and a robust link between flow evolution and ambient geometry, with potential independent interest in geometric analysis and conformal geometry. Overall, the paper extends ambient metric theory to manifolds with density, enabling systematic construction and analysis of nonlinear, variational, and monotone functionals in the weighted setting, with broad implications for geometric flows and invariant theory.

Abstract

We prove the existence and uniqueness of a weighted analogue of the Fefferman-Graham ambient metric for manifolds with density. We then show that this ambient metric forms the natural geometric framework for the singular Ricci flow: given a singular gradient Ricci flow spacetime in the Kleiner-Lott sense, we construct a unique global ambient half-space from it. We also prove the converse, that every global ambient space contains a singular gradient Ricci flow spacetime, thereby completing the correspondence. Our main application is the construction of infinite families of fully non-linear analogues of Perelman's $\mathcal{F}$ and $\mathcal{W}$ functionals. We extend Perelman's monotonicity result to these two families of functionals under several conditions, including for shrinking solitons and Einstein manifolds. We do so by constructing a "Ricci flow vector field" in the ambient space, which may be of independent research interest. We also prove that the weighted GJMS operators associated with the weighted ambient metric are formally self-adjoint, and that the associated weighted renormalized volume coefficients are variational.

Theorems & Definitions (88)

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