The weighted ambient metric for manifolds with density

TL;DR

This paper develops a weighted ambient metric framework for manifolds with density, linking it to singular Ricci flows and extending Perelman's functionals, with applications to geometric analysis.

Contribution

It introduces a weighted ambient metric for manifolds with density, establishing a correspondence with singular Ricci flows and extending key geometric functionals.

Findings

Existence and uniqueness of weighted ambient metric.

Construction of ambient space from singular Ricci flows and vice versa.

Extension of Perelman's monotonicity to new functionals.

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.