# The weighted ambient metric

Jeffrey S. Case  109 McAllister Building
Penn State University
University Park, PA 16802
USA<EMAIL_ADDRESS>and  Ayush Khaitan  109 McAllister Building
Penn State University
University Park, PA 16802
USA<EMAIL_ADDRESS>
###### Abstract.

We prove the existence and uniqueness of weighted ambient metrics and weighted
Poincaré metrics for smooth metric measure spaces.

###### Key words and phrases:

ambient metric; Poincaré metric; smooth metric measure space

###### 2010 Mathematics Subject Classification:

Primary 53A30; Secondary 53A55, 31C12

##  1\. Introduction

The Fefferman–Graham ambient space [FeffermanGraham2012] is a formally Ricci
flat space canonically associated to a given conformal manifold. Among its
many applications are the classification of local scalar conformal invariants
[BaileyEastwoodGraham1994, Gover2001] and the construction of a family of
conformally covariant operators, called GJMS operators, with leading-order
term a power of the Laplacian [GJMS1992]. One can also use the ambient space
to construct an asymptotically hyperbolic, formally Einstein space with
conformal boundary a given conformal manifold. The resulting space, called a
Poincaré space, likewise has many applications, among them a proof that the
GJMS operators are formally self-adjoint [GrahamZworski2003] and the
construction and study of variational scalar invariants generalizing the
$\sigma_{2}$-curvatures [ChangFang2008, Graham2009, ChangFangGraham2012].

A _smooth metric measure space_ is a five-tuple $(M^{d},g,f,m,\mu)$ formed
from a Riemannian manifold $(M^{d},g)$, a positive function $f\in
C^{\infty}(M)$, a dimensional parameter $m\in[0,\infty]$, and a curvature
parameter $\mu\in\mathbb{R}$. Smooth metric measure spaces arise in many ways,
including as (possibly collapsed) limits of sequences of Riemannian manifolds
[CheegerColding2000a], as smooth manifolds satisfying curvature-dimension
inequalities [BakryEmery1985, Wei_Wylie], as the geometric framework
[CaseChang2013] for studying curved analogues of the Caffarelli–Silvestre
extension for defining the fractional Laplacian [CaffarelliSilvestre2007],
and, in the limiting case $m=\infty$, as a geometric framework for the
realization of the Ricci flow as a gradient flow [Perelman1].

A _weighted invariant_ is tensor-valued function on smooth metric measure
spaces such that
$T(N,\Phi^{\ast}g,\Phi^{\ast}f,m,\mu)=\Phi^{\ast}\bigl{(}T(M,g,f,m,\mu)\bigr{)}$
for any smooth metric measure space $(M^{d},g,f,m,\mu)$ and any diffeomorphism
$\Phi\colon N\mapsto M$. For example, the _Bakry–Émery Ricci curvature_ of
$(M^{d},g,f,m,\mu)$ is

$\operatorname{Ric}_{\phi}^{m}:=\operatorname{Ric}-\frac{m}{f}\nabla^{2}f$

and the _weighted scalar curvature_ of $(M^{d},g,f,m,\mu)$ is

$R_{\phi}^{m}:=R-\frac{2m}{f}\Delta f-\frac{m(m-1)}{f^{2}}\left(\lvert\nabla
f\rvert^{2}-\mu\right).$

A smooth metric measure space is often studied as an abstract analogue of the
base of the warped product $M^{d}\times_{f}F^{m}(\mu)$ of $(M^{d},g)$ with an
$m$-dimensional spaceform $F^{m}(\mu)$ of constant sectional curvature $\mu$,
in the sense that when $m$ is a nonnegative integer, most weighted invariants
are equivalent to Riemannian invariants of the warped product. For example, if
$m\in\mathbb{N}_{0}$, then $\operatorname{Ric}_{\phi}^{m}$ is the restriction
of the Ricci tensor of $M\times_{f}F^{m}(\mu)$ to horizontal (i.e. tangent to
$M$) vectors and $R_{\phi}^{m}$ is the scalar curvature of
$M\times_{f}F^{m}(\mu)$.

We say that two smooth metric measure spaces $(M^{d},g_{i},f_{i},m,\mu)$,
$i\in\\{1,2\\}$, are _pointwise conformally equivalent_ if there is a function
$u\in C^{\infty}(M)$ such that $g_{2}=e^{2u}g_{1}$ and $f_{2}=e^{u}f_{1}$.
This notion has many applications. For example, the (curved version of the)
Caffarelli–Silvestre extension [CaseChang2013, Case2019tr,
CaffarelliSilvestre2007] identifies a conformally covariant operator with
leading-order term a fractional power of the Laplacian [GrahamZworski2003] as
a generalized Dirichlet-to-Neumann operator associated to a weighted analogue
of the GJMS operators [CaseChang2013, Case2011t], and there is a conformally
invariant analogue of the Yamabe functional on smooth metric measure spaces
[Case2013y] which interpolates between the usual Yamabe functional
[Yamabe1960] and Perelman’s $\mathcal{F}$\- and $\mathcal{W}$-functionals
[Perelman1]. This latter perspective leads to an ad hoc construction
[Case2014s, Case2014sd, Case2016v] of some fully nonlinear analogues of
Perelman’s functionals.

Given the variety of applications of the Fefferman–Graham ambient space and of
the conformal geometry of smooth metric measure spaces, it is natural to ask
whether there is a canonical weighted ambient space associated to a smooth
metric measure space. We show that this is the case.

Roughly speaking, a _weighted ambient space_ is a metric measure structure
$(\widetilde{g},\widetilde{f})$ on $\mathbb{R}_{+}\times
M^{d}\times(-\epsilon,\epsilon)$ such that

  1. (i)

if $d+m$ is not an even integer, then
$\widetilde{\text{Ric}}_{\phi}^{m},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$,
where $\rho$ is the coordinate on $(-\epsilon,\epsilon)$ and

$\widetilde{F_{\phi}^{m}}:=\widetilde{f}\widetilde{\Delta}\widetilde{f}+(m-1)(|\widetilde{\nabla}\widetilde{f}|^{2}-\mu);$

  2. (ii)

if $d+m$ is an even integer, then

$\displaystyle\widetilde{\text{Ric}}_{\phi}^{m},\widetilde{F_{\phi}^{m}}$
$\displaystyle=O(\rho^{\frac{d+m}{2}-1}),$ $\displaystyle
g^{ij}(\widetilde{\text{Ric}}_{\phi}^{m})_{ij}-{m}{f}^{-2}\widetilde{F_{\phi}^{m}}$
$\displaystyle=O(\rho^{\frac{d+m}{2}}),$

where $i,j$ denote the coordinates on $M$.

Note that
$\widetilde{R_{\phi}^{m}}=\widetilde{g}^{IJ}(\widetilde{\text{Ric}}_{\phi}^{m})_{IJ}-mf^{-2}\widetilde{F_{\phi}^{m}}$.
The improved orders of vanishing in the case that $d+m$ is even can be
interpreted as the statement that, modulo the $\rho$ components of
$\widetilde{g}^{IJ}(\widetilde{\text{Ric}}_{\phi}^{m})_{IJ}$, the weighted
scalar curvature $\widetilde{R}_{\phi}^{m}$ vanishes to one higher order in
$\rho$ than the Bakry–Émery Ricci tensor
$\widetilde{\operatorname{Ric}}_{\phi}^{m}$.

From Section 4.1 onwards, the metric measure structure
$(\widetilde{g},\widetilde{f})$ will be assumed to be of the form

(1.1)
$\displaystyle\begin{split}\widetilde{g}&=2\rho\,dt^{2}+2t\,d\rho\,dt+t^{2}g_{\rho}\\\
\widetilde{f}&=tf_{\rho}\end{split}$

Metrics of the form Equation 1.1 are called _straight_ and _normal_. See
Sections 3 and 4 for more details.

The primary purpose of this article is to prove the existence and uniqueness
of weighted ambient spaces.

######  Theorem 1.1.

Let $(M^{d},[g,f],m,\mu),d\geq 3$, be a conformal class of smooth metric
measure spaces. Then

  1. (i)

there exists a unique, up to ambient-equivalence, weighted ambient space
$(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ for
$(M^{d},[g,f],m,\mu)$; and

  2. (ii)

if $d+m$ is an even integer, the weighted obstruction tensor,

(1.2) $\displaystyle\mathcal{O}_{ij}$
$\displaystyle:=c_{d+m}{\left.\kern-1.2pt\partial_{\rho}^{\frac{d+m}{2}-1}\vphantom{\big{|}}\right|_{{\rho=0}}}(\widetilde{\text{Ric}_{\phi}^{m}})_{ij},$
$\displaystyle c_{d+m}$
$\displaystyle=(-2)^{(d+m)/2-1}\frac{((d+m)/2-1)!}{d+m-2},$

vanishes if and only if there is a weighted ambient space
$(\widetilde{g},\widetilde{f})$ such that
$\widetilde{\text{Ric}_{\phi}^{m}},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$.

When $m=0$, Theorem 1.1 recovers the Fefferman–Graham ambient space
[FeffermanGraham2012]. When $m$ is a positive integer, the existence and
uniqueness statement of Theorem 1.1 implies that
$\widetilde{\mathcal{G}}\times_{\widetilde{f}}F^{m}(\mu)$ is the
Fefferman–Graham ambient space of $M\times_{f}F^{m}(\mu)$. See Theorem 4.8 for
additional properties of the weighted obstruction tensor.

It is in general laborious to compute the asymptotic expansions of
$(\widetilde{g},\widetilde{f})$. However, one readily computes that, in the
case of a straight and normal weighted ambient space,

$\displaystyle g_{\rho}$ $\displaystyle=g+2\rho P_{\phi}^{m}+O(\rho^{2}),$
$\displaystyle f_{\rho}$ $\displaystyle=f+\frac{f}{m}\rho
Y_{\phi}^{m}+O(\rho^{2}),$

where

$P_{\phi}^{m}:=\frac{1}{d+m-2}\left(\operatorname{Ric}_{\phi}^{m}-J_{\phi}^{m}g\right)$

is the _weighted Schouten tensor_ ,

$J_{\phi}^{m}:=\frac{1}{2(d+m-1)}R_{\phi}^{m}$

is the _weighted Schouten scalar_ , and
$Y_{\phi}^{m}:=J_{\phi}^{m}-\operatorname{tr}_{g}P_{\phi}^{m}$.

We can also explicitly identify the weighted ambient space in three special
cases.

######  Theorem 1.2.

Let $(M^{d},g,f,m,\mu)$, $d\geq 3$, be a smooth metric measure space. Set
$\widetilde{\mathcal{G}}=\mathbb{R}_{+}\times M^{d}\times(-\epsilon,\epsilon)$
and consider the smooth metric measure space
$(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$, where
$(\widetilde{g},\widetilde{f})$ is of the form Equation 1.1 and

$\displaystyle\left(g_{\rho}\right)_{ij}$
$\displaystyle=g_{ij}+2\rho\left(P_{\phi}^{m}\right)_{ij}+\rho^{2}\left(P_{\phi}^{m}\right)_{ik}\left(P_{\phi}^{m}\right)_{j}^{k},$
$\displaystyle f_{\rho}$ $\displaystyle=f+\frac{f}{m}\rho Y_{\phi}^{m}.$

Then $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ is a
weighted ambient space if

  1. (i)

there is a constant $\lambda\in\mathbb{R}$ such that

$F_{\phi}^{m}=-\lambda f^{2}\hskip 14.22636pt\text{ and }\hskip
14.22636ptRic_{\phi}^{m}=\lambda g;$

  2. (ii)

$Rm=P_{\phi}^{m}\wedge g$ and $dP_{\phi}^{m}=0$; or

  3. (iii)

$f=1$ and there is a constant $\lambda\in\mathbb{R}$ such that

$\text{Ric}_{\phi}^{m}=\lambda g\hskip 14.22636pt\text{and}\hskip
14.22636ptR=d\lambda.$

The first case of Theorem 1.2 is the case of a quasi-Einstein manifold
[Case_Shu_Wei]; the second case is the case of a smooth metric measure space
which is locally conformally flat in the weighted sense [Case2014s]; and the
third case is a weighted interpretation of the explicit ambient metric of
Gover and Leitner [GoverLeitner2009].

As in Riemannian geometry, the weighted ambient space gives rise to a
canonical weighted Poincaré space associated to a given smooth metric measure
space.

######  Theorem 1.3.

Let $\left(M^{d},[g,f],m,\mu\right)$, $d\geq 3$, be a conformal class of
smooth metric measure spaces. Then there exists an even weighted Poincaré
space for it. Moreover, if $\left(g_{+}^{1},f_{+}^{1}\right)$ and
$\left(g_{+}^{2},f_{+}^{2}\right)$ are two even weighted Poincaré spaces for
$\left(M^{d},g,f,m,\mu\right)$ defined on $\left(M_{+}^{1}\right)^{\circ}$ and
$\left(M_{+}^{2}\right)^{\circ}$, respectively, then there are open sets
$U_{1}\subset M_{+}^{1}$ and $U_{2}\subset M_{+}^{2}$ containing
$M\times\\{0\\}$ and an even diffeomorphism $\phi:U_{1}\rightarrow$ $U_{2}$
such that $\phi|_{M\times\\{0\\}}$ is the identity map, and

  1. (i)

if $m+d$ is not an even integer, then

$\left(g_{+}^{1}-\phi^{*}g_{+}^{2},-2(f_{+}^{1}-\phi^{*}f_{+}^{2})\right)=O\left(r^{\infty}\right);$

  2. (ii)

if $d+m$ is an even integer, then

$\left(g_{+}^{1}-\phi^{*}g_{+}^{2},-2(f_{+}^{1}-\phi^{*}f_{+}^{2})\right)=O_{\alpha\beta}^{1,+}\left(r^{d+m-2}\right).$

See Section 5 for the relevant definitions.

In a follow-up article [Khaitan2022], Khaitan will use Theorems 1.1, 1.2 and
1.3 to give a rigorous construction of the weighted GJMS operators of all
orders up to the obstruction, and to prove that they are formally self-
adjoint. He will also give explicit formulas for the weighted GJMS operators
for smooth metric measure spaces satisfying the first or last conditions of
Theorem 1.2. This makes rigorous the factorization of the weighted GJMS
operators established by Case and Chang [CaseChang2013] in the latter case by
formally arguing via warped products.

This article is organized as follows: In Section 2, we recall some weighted
invariants of smooth metric measure spaces. In Section 3, we define weighted
ambient spaces and discuss weighted ambient-equivalence. In Section 4, we
prove Theorem 1.1. In Section 5, we define weighted Poincarè spaces and prove
Theorem 1.3. In Section 6, we write down the first couple of terms of the
expansion of the weighted ambient metric. In Section 7, we prove Theorem 1.2.

##  2\. Smooth metric measure spaces

We recall some weighted invariants relevant to the geometry of smooth metric
measure spaces [Case2014s]. Let $(M^{d},g,f,m,\mu)$ be a smooth metric measure
space. This data determines a volume element

$dv_{\phi}:=f^{m}\,\operatorname{dvol}_{g}.$

When $m>0$, we set $\phi:=-m\ln f$, so that

$dv_{\phi}=e^{-\phi}\,\operatorname{dvol}_{g}.$

In terms of $\phi$, it holds that

$\displaystyle\operatorname{Ric}_{\phi}^{m}$
$\displaystyle=\operatorname{Ric}+\nabla^{2}\phi-\frac{1}{m}d\phi\otimes
d\phi,$ $\displaystyle R_{\phi}^{m}$
$\displaystyle=R+2\Delta\phi-\frac{m+1}{m}\lvert\nabla\phi\rvert^{2}+m(m-1)\mu
e^{2\phi/m}.$

Recall that _weighted Schouten tensor $P_{\phi}^{m}$_ and the _weighted
Schouten scalar $J_{\phi}^{m}$_ of $(M^{d},g,f,m,\mu)$ are

$\displaystyle P_{\phi}^{m}$
$\displaystyle:=\frac{1}{d+m-2}(\operatorname{Ric}_{\phi}^{m}-J_{\phi}^{m}g),$
$\displaystyle J_{\phi}^{m}$ $\displaystyle:=\frac{1}{2(d+m-1)}R_{\phi}^{m}.$

A _weighted Einstein manifold_ is a smooth metric measure space
$(M^{d},g,f,m,\mu)$ such that

$P_{\phi}^{m}=\lambda g$

for some constant $\lambda\in\mathbb{R}$. A weighted Einstein manifold is
_quasi-Einstein_ [Case_Shu_Wei] if additionally

$J_{\phi}^{m}=(d+m)\lambda.$

The _weighted Weyl tensor $A_{\phi}^{m}$_ and the _weighted Cotton tensor
$dP_{\phi}^{m}$_ of a smooth metric measure space $(M^{d},g,f,m,\mu)$ are

$\displaystyle A_{\phi}^{m}(x,y,z,w)$
$\displaystyle:=(\operatorname{Rm}-P_{\phi}^{m}\wedge g)(x,y,z,w),$
$\displaystyle dP_{\phi}^{m}(x,y,z)$
$\displaystyle:=\nabla_{x}P_{\phi}^{m}(y,z)-\nabla_{y}P_{\phi}^{m}(x,z),$

where $h\wedge k$ denotes the Kulkarni–Nomizu product

$(h\wedge k)(x,y,z,w):=h(x,z)k(y,w)+h(y,w)k(x,z)-h(x,w)k(y,z)-h(y,z)k(x,w).$

The _weighted Bach tensor_ of $(M^{d},g,f,m,\mu)$ is

$B_{\phi}^{m}:=\delta_{\phi}dP_{\phi}^{m}-\frac{1}{m}\operatorname{tr}dP_{\phi}^{m}\otimes
d\phi+\left\langle
A_{\phi}^{m},P_{\phi}^{m}-\frac{Y_{\phi}^{m}}{m}g\right\rangle,$

where $Y_{\phi}^{m}:=J_{\phi}^{m}-\operatorname{tr}P_{\phi}^{m}$, the
contractions are

$\displaystyle(\operatorname{tr}dP_{\phi}^{m})(x)$
$\displaystyle:=\sum_{i=1}^{d}dP_{\phi}^{m}(e_{i},x,e_{i}),$
$\displaystyle\left\langle
A_{\phi}^{m},P_{\phi}^{m}-\frac{Y_{\phi}^{m}}{m}g\right\rangle(x,y)$
$\displaystyle:=\sum_{i,j=1}^{d}A_{\phi}^{m}(e_{i},x,e_{j},y)\left(P_{\phi}^{m}-\frac{Y_{\phi}^{m}}{m}g\right)(e_{i},e_{j})$

for $\\{e_{i}\\}$ an orthonormal basis, and the weighted divergence is

$(\delta_{\phi}dP_{\phi}^{m})(x,y):=-dP_{\phi}^{m}(\nabla\phi,x,y)+\sum_{i=1}^{d}\nabla_{e_{i}}dP_{\phi}^{m}(e_{i},x,y).$

The _weighted Bianchi identity_ [Case2014sd] is

(2.1)
$\delta_{\phi}\operatorname{Ric}_{\phi}^{m}-\frac{1}{2}dR_{\phi}^{m}-\frac{1}{f^{2}}F_{\phi}^{m}d\phi=0.$

A smooth metric measure space $(M^{d},g,f,m,\mu)$ is _locally conformally flat
in the weighted sense_ [Case2014s] if

  1. (i)

$d+m\in\\{1,2\\}$;

  2. (ii)

$d+m=3$ and $dP_{\phi}^{m}=0$; or

  3. (iii)

$d+m\not\in\\{1,2,3\\}$ and $A_{\phi}^{m}=0$.

Note that [Case2014s] if $d+m\geq 3$, then $(M^{d},g,f,m,\mu)$ is locally
conformally flat in the weighted sense if and only if
$A_{\phi}^{m}=dP_{\phi}^{m}=0$.

We require some formulas satisfied by smooth metric measure spaces which are
locally conformally flat in the weighted sense.

######  Lemma 2.1 ([Case2014s]*Lemma 3.2).

Let $\left(M^{d},g,v,m,\mu\right)$ be a smooth metric measure space that is
locally conformally flat in the weighted sense. Then

(2.2a)
$\displaystyle\begin{split}0&=P_{\phi}^{m}(\nabla\phi)+dY_{\phi}^{m}-\frac{1}{m}Y_{\phi}^{m}d\phi,\end{split}$
(2.2b)
$\displaystyle\begin{split}0&=mP_{\phi}^{m}-\nabla^{2}\phi+\frac{1}{m}d\phi\otimes
d\phi+Y_{\phi}^{m}g,\end{split}$ (2.2c)
$\displaystyle\begin{split}0&=dP_{\phi}^{m}.\end{split}$

##  3\. Weighted Ambient spaces

Let $(M^{d},g,f,m,\mu)$ be a smooth metric measure space with $d\geq 3$, and
let $\mathcal{G}$ be the ray subbundle of $TM\oplus\mathbb{R}$ consisting of
all triples $(h,u,x)$ such that $h=s^{2}g_{x}$ and $u=sf(x)$ for some
$s\in\mathbb{R}_{+}$ and $x\in M$. We define the dilation
$\delta_{s}\colon\mathcal{G}\mapsto\mathcal{G}$ by
$(h,u,x)\mapsto(s^{2}h,su,x)$. Let
$T:={\left.\kern-1.2pt\frac{d}{ds}\delta_{s}\vphantom{\big{|}}\right|_{{s=1}}}$
denote the infinitesimal generator of dilations. Also, let
$\pi\colon(h,u,x)\mapsto x$ be the projection from $\mathcal{G}$ to $M$. There
is a canonical metric measure structure $(\boldsymbol{g},\boldsymbol{f})$ on
$\mathcal{G}$ defined by $\boldsymbol{g}(X,Y)=h(\pi_{*}X,\pi_{*}Y)$ and
$\boldsymbol{f}(h,u,x)=u(x)$.

Given $(g,f)$, define a coordinate chart
$\mathcal{G}\mapsto\mathbb{R}_{+}\times M$, by
$(t^{2}g_{x},tf,x)\mapsto(t,x)$. Then $T=t\partial_{t}$.

Consider the embedding $\iota\colon\mathcal{G}\
\hookrightarrow\mathcal{G}\times\mathbb{R},(t,x)\mapsto(t,x,0)$. Each point in
$\mathcal{G}\times\mathbb{R}$ can be written as $(t,x,\rho)$. We extend the
dilation to $\mathcal{G}\times\mathbb{R}$ as
$\delta_{s}(t,x,\rho)=(st,x,\rho)$.

We now define weighted pre-ambient spaces and special cases thereof.

######  Definition 3.1.

$(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ is called a
_weighted pre-ambient space_ if

  1. (i)

$\widetilde{\mathcal{G}}$ is a dilation-invariant neighborhood of
$\mathcal{G}\times\\{0\\}$;

  2. (ii)

$\widetilde{g}$ is a smooth metric of signature $(d+1,1)$ and $\widetilde{f}$
a smooth function on $\widetilde{\mathcal{G}}$;

  3. (iii)

$\delta_{s}^{*}\widetilde{g}=s^{2}\widetilde{g}$ and
$\delta_{s}^{*}\widetilde{f}=s\widetilde{f}$; and

  4. (iv)

$(\iota^{*}\widetilde{g},\iota^{*}\widetilde{f})=(\boldsymbol{g},\boldsymbol{f})$.

######  Remark 3.2.

Our results can be extended to pseudo-Riemannian metrics in the usual way (cf.
[FeffermanGraham2012]).

######  Definition 3.3.

A weighted pre-ambient space
$(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ is said to be in
_normal form_ relative to a metric measure structure $(g,f)$ if

  1. (i)

for each fixed $z\in\mathcal{G}$, the set of $\rho\in\mathbb{R}$ such that
$(z,\rho)\in\widetilde{\mathcal{G}}$ is an open interval $I_{z}$ containing
$0$;

  2. (ii)

for each $z\in\mathcal{G}$, the curve on $I_{z}$ defined as
$\rho\mapsto(z,\rho)$ is a geodesic in $\widetilde{\mathcal{G}}$; and

  3. (iii)

on $\mathcal{G}$, it holds that $\widetilde{g}=\boldsymbol{g}+2t\,d\rho\,dt$.

######  Definition 3.4.

A weighted pre-ambient space
$(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ for
$(M^{d},[g,f],m,\mu)$ is be said to be straight if any of the following
equivalent properties hold:

  1. (i)

for each $p\in\widetilde{\mathcal{G}}$. the dilation orbit
$s\mapsto\delta_{s}p$ is a geodesic for $\widetilde{g}$.

  2. (ii)

$\widetilde{g}(2T,.)=d(\widetilde{g}(T,T));$ or

  3. (iii)

the infinitesimal dilation field $T$ satisfies
$\widetilde{\nabla}T=\mathrm{Id}$.

Given a point $(t,x,\rho)\in\mathcal{G}\times\mathbb{R}$, we refer to the $t$
coordinate as the $0$ coordinate, and the $\rho$ coordinate as the $\infty$
coordinate. Coordinates in $M$ are denoted with lowercase latin characters
$(i,j,k,\dots)$.

The metric of a normal weighted pre-ambient space takes a special form.

######  Lemma 3.5.

Let $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ be a
weighted pre-ambient space such that for each $z\in\mathcal{G}$, the set of
all $\rho\in\mathbb{R}$ such that $(z,\rho)\in\widetilde{\mathcal{G}}$ is an
open interval containing $0$. Then $(\widetilde{g},\widetilde{f})$ is a normal
metric measure structure if and only if $\widetilde{g}_{0\infty}=t$,
$\widetilde{g}_{\infty i}=0$ and $\widetilde{g}_{\infty\infty}=0$.

###### Proof.

A normal metric measure structure has

$\widetilde{g}_{\infty\infty}\big{|}_{\rho=0}=\widetilde{g}_{i\infty}\big{|}_{\rho=0}=0\,\,\text{
and }\,\,\widetilde{g}_{0\infty}\big{|}_{\rho=0}=t.$

Moreover, it also has geodesic $\rho$-lines. The $\rho$-lines are geodesics if
and only if the Christoffel symbols $\widetilde{\Gamma}_{\infty\infty I}$,
$I\in\\{0,i,\infty\\}$, vanish. Taking $I=\infty$ gives
$\partial_{\rho}\widetilde{g}_{\infty\infty}=0$, and hence
$\widetilde{g}_{\infty\infty}=0.$ Now taking $I=i$ and $I=0$ gives
$\widetilde{g}_{\infty i}=0$ and $\widetilde{g}_{\infty 0}=t$.

The converse follows similarly. ∎

Defining a weighted ambient space required some additional notation.

######  Definition 3.6.

Let $(\widetilde{S}_{IJ},\widetilde{h})$ be a pair of a symmetric 2-tensor
$\widetilde{S}_{IJ}$ and a smooth function $\widetilde{h}$ on an open
neighborhood of $\mathcal{G}\times\\{0\\}\subset\mathcal{G}\times\mathbb{R}$.
For $k,m\geq 0$, we write
$(\widetilde{S}_{IJ},\widetilde{h})=O_{IJ}^{k,+}(\rho^{m})$ if

  1. (i)

$(\widetilde{S}_{IJ},\widetilde{h})=O(\rho^{m})$;

  2. (ii)

$\widetilde{S}_{00},\widetilde{S}_{0i}=O(\rho^{m+1})$; and

  3. (iii)

$mf^{-k}\widetilde{h}-g^{ij}\widetilde{S}_{ij}=O(\rho^{m+1})$.

We now define a weighted ambient space.

######  Definition 3.7.

A _weighted ambient space_ for $(M^{d},g,f,m,\mu)$ is a weighted pre-ambient
space $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ such that

  1. (i)

if $d+m\notin 2\mathbb{N}$, then
$(\widetilde{\text{Ric}_{\phi}^{m}},\widetilde{F_{\phi}^{m}})=O_{IJ}(\rho^{\infty})$;

  2. (ii)

if $d+m\in 2\mathbb{N}$, then
$(\widetilde{\text{Ric}_{\phi}^{m}},\widetilde{F_{\phi}^{m}})=O^{2,+}_{IJ}(\rho^{\frac{d+m}{2}-1}).$

A key step in the proof of Theorem 1.1 is the construction and classification
of straight and normal metrics [FeffermanGraham2012].

######  Theorem 3.8.

Let $(M^{d},g,f,m,\mu)$, $d\geq 3$, be a smooth metric measure space. Then
there exists a straight and normal weighted ambient space
$(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ for it.
Moreover, if
$(\widetilde{\mathcal{G}}_{j},\widetilde{g}_{j},\widetilde{f}_{j},m,\mu)$,
$j\in\\{1,2\\}$, are two such weighted ambient spaces, then

  1. (i)

if $d+m\notin 2\mathbb{N}$, then
$(\widetilde{g}_{1}-\widetilde{g}_{2},-2(\widetilde{f}_{1}-\widetilde{f}_{2}))=O(\rho^{\infty})$;
and

  2. (ii)

if $d+m\in 2\mathbb{N}$, then
$(\widetilde{g}_{1}-\widetilde{g}_{2},-2(\widetilde{f}_{1}-\widetilde{f}_{2}))=O_{IJ}^{1,+}(\rho^{\frac{d+m}{2}-1})$.

We prove Theorem 3.8 in Section 4.

A geometric interpretation of the improved orders of vanishing in the case
that $d+m\in 2\mathbb{N}$ is that the weighted volume element
$\widetilde{f}^{m}\operatorname{dvol}_{\widetilde{g}}$ vanishes to one higher
order in $\rho$ than the weighted ambient space
$(\widetilde{g},\widetilde{f})$.

The following definition generalizes the notion of uniqueness from Theorem 3.8
to general pre-ambient spaces.

######  Definition 3.9.

We say that two weighted pre-ambient spaces
$(\widetilde{\mathcal{G}}_{1},\widetilde{g}_{1},\widetilde{f}_{1},m,\mu)$ and
$(\widetilde{\mathcal{G}}_{2},\widetilde{g}_{2},\widetilde{f}_{2},m,\mu)$ for
$(M^{d},[g,f],m,\mu)$ are _ambient-equivalent_ if there exist open sets
$\mathcal{U}_{1}\subset\widetilde{\mathcal{G}}_{1}$ and
$\mathcal{U}_{2}\subset\widetilde{\mathcal{G}}_{2}$, and a diffeomorphism
$\phi:\mathcal{U}_{1}\rightarrow\mathcal{U}_{2}$ such that

  1. (i)

$\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ both contain
$\mathcal{G}\times\\{0\\}$;

  2. (ii)

$\mathcal{U}_{1}$ and $\mathcal{U}_{2}$ are dilation-invariant and $\phi$
commutes with dilations;

  3. (iii)

the restriction of $\phi$ to $\mathcal{G}\times\\{0\\}$ is the identity map;
and

  4. (iv)

     * (a)

if $d+m\notin 2\mathbb{N}$, then
$(\widetilde{g}_{1}-\phi^{*}\widetilde{g}_{2},-2(\widetilde{f}_{1}-\phi^{*}\widetilde{f}_{2}))=O(\rho^{\infty})$;

     * (b)

if $d+m\in 2\mathbb{N}$, then
$(\widetilde{g}_{1}-\phi^{*}\widetilde{g}_{2},-2(\widetilde{f}_{1}-\phi^{*}\widetilde{f}_{2}))=O_{IJ}^{1,+}(\rho^{\frac{d+m}{2}-1})$.

Using Theorem 4.4 below, we see that every weighted pre-ambient space is
ambient-equivalent to a straight and normal weighted pre-ambient space.

######  Theorem 3.10.

Let $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ be a
weighted ambient space for the conformal class $(M^{d},[g,f],m,\mu),d\geq 3,$
of smooth metric measure spaces. Then it is ambient-equivalent to a weighted
ambient space in straight and normal form relative to $(g,f)$.

###### Proof.

Following the proof in [FeffermanGraham2012]*Proposition 2.8, we observe that
any weighted pre-ambient space is ambient-equivalent to a normal weighted pre-
ambient space. Moreover, it follows from Theorem 4.4 below that a normal
weighted ambient space is ambient-equivalent to a straight and normal ambient
space through the identity map. ∎

The uniqueness of Theorem 3.8 implies the uniqueness of weighted ambient
spaces up to ambient-equivalence (cf. [FeffermanGraham2012]*Proposition 2.9).

######  Theorem 3.11.

Any two weighted ambient spaces for $(M,[g,f],m,\mu)$,$d\geq 3$, are ambient-
equivalent.

###### Proof.

We pick a representative $(g,f)$ and invoke Theorem 3.8. Then there exists a
straight and normal weighted ambient space
$(\widetilde{\mathcal{G}}_{1},\widetilde{g}_{1},\widetilde{f}_{1},m,\mu)$ for
$(M,g,f,m,\mu)$. Now let
$(\widetilde{\mathcal{G}}_{2},\widetilde{g}_{2},\widetilde{f}_{2},m,\mu)$ be a
weighted ambient space for $(M,[g,f],m,\mu)$. Applying Theorem 3.10, we find
that $(\widetilde{\mathcal{G}}_{2},\widetilde{g}_{2},\widetilde{f}_{2},m,\mu)$
is ambient-equivalent to a weighted ambient space in straight and normal form
relative to $(g,f)$. By Theorem 3.8, this space is ambient-equivalent to
$(\widetilde{\mathcal{G}}_{1},\widetilde{g}_{1},\widetilde{f}_{1},m,\mu)$. ∎

##  4\. Formal Theory

We prove Theorem 3.8 by iteratively constructing a power series solution to
$\widetilde{\text{Ric}_{\phi}^{m}}(\widetilde{g}),\widetilde{F_{\phi}^{m}}=O(\rho^{j})$.
This process is only obstructed at order $O(\rho^{\frac{d+m}{2}-1})$ when
$d+m\in 2\mathbb{N}$.

We first give a necessary condition for a weighted ambient space to be normal.

######  Lemma 4.1.

Let $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ be a normal
weighted ambient space. Then

(4.1) $\widetilde{g}=adt^{2}+2b_{i}dx^{i}dt+2td\rho dt+t^{2}(g_{ij})_{\rho},$

with $a=2\rho+O(\rho^{2})$ and $b_{i}=O(\rho^{2})$.

###### Proof.

Lemma 3.5 implies that $\widetilde{g}$ has the form Equation 4.1. Since
$\widetilde{g}$ is normal,
${\left.\kern-1.2pta\vphantom{\big{|}}\right|_{{\rho=0}}}=0$ and
${\left.\kern-1.2ptb_{i}\vphantom{\big{|}}\right|_{{\rho=0}}}=0$. By
definition,

(4.2)
$\widetilde{(\operatorname{Ric}_{\phi}^{m})}_{IJ}=\frac{1}{2}\widetilde{g}^{KL}\left(\partial^{2}_{IL}\widetilde{g}_{JK}+\partial^{2}_{JK}\widetilde{g}_{IL}-\partial^{2}_{KL}\widetilde{g}_{IJ}-\partial^{2}_{IJ}\widetilde{g}_{KL}\right)\\\
+\widetilde{g}^{KL}\widetilde{g}^{PQ}\left(\widetilde{\Gamma}_{ILP}\widetilde{\Gamma}_{JKQ}-\widetilde{\Gamma}_{IJP}\widetilde{\Gamma}_{KLQ}\right)-\frac{m}{\widetilde{f}}\big{[}\partial^{2}_{IJ}\widetilde{f}-\widetilde{\Gamma}^{K}_{IJ}\partial_{K}\widetilde{f}\big{]}.$

Evaluating at $\rho=0$ yields

$(\widetilde{\operatorname{Ric}_{\phi}^{m}})_{00}=\frac{d+m}{2t^{2}}(2-\partial_{\rho}a)+O(\rho),$
$(\widetilde{\operatorname{Ric}_{\phi}^{m}})_{0i}=-\frac{d+m}{2t}\partial_{\rho}b_{i}+O(\rho).$

We conclude that $a=2\rho+O(\rho^{2})$ and $b_{i}=O(\rho^{2})$. ∎

We now take a brief digression to discuss straight weighted ambient spaces.
First, notice that a simple choice of certain components of $\widetilde{g}$
makes $(\widetilde{\text{Ric}}_{\phi}^{m})_{0I}$ vanish.

######  Lemma 4.2.

Let $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ be a
weighted pre-ambient space. If $\widetilde{g}$ has the form

(4.3) $\widetilde{g}_{IJ}=\begin{pmatrix}2\rho&0&t\\\ 0&t^{2}g_{\rho}&0\\\
t&0&0\end{pmatrix},$

where $g_{\rho}$ is a one-parameter family of metrics on $M$, then
$(\widetilde{\text{Ric}}_{\phi}^{m})_{0I}=0$.

###### Proof.

This follows by direct computation. ∎

The relevance of Lemma 4.2 stems from the following equivalent
characterizations of straight normal pre-ambient metrics.

######  Proposition 4.3.

Let $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu),d\geq 3$, be
in normal form relative to $(g,f)$. Then the following conditions are
equivalent:

  1. (i)

$\widetilde{g}_{00}=2\rho$ and $\widetilde{g}_{0i}=0$;

  2. (ii)

for each $p\in\widetilde{\mathcal{G}}$. the dilation orbit
$s\mapsto\delta_{s}p$ is a geodesic for $\widetilde{g}$;

  3. (iii)

$\widetilde{g}(2T,.)=d(\widetilde{g}(T,T))$;

  4. (iv)

the infinitesimal dilation field $T$ satisfies
$\widetilde{\nabla}T=\operatorname{Id}$.

###### Proof.

The proof is identical to that of [FeffermanGraham2012]*Proposition 3.4. ∎

The following result implies that, up to ambient-equivalence, we may restrict
our attention to metrics of the form of Equation 4.3.

######  Theorem 4.4.

A normal weighted ambient space is ambient-equivalent to a straight and normal
ambient space.

###### Proof.

Let $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ be a normal
weighted ambient space. By Lemma 4.1, we may write

$\widetilde{g}=\begin{pmatrix}2\rho&0&t\\\ 0&t^{2}g_{\rho}&0\\\
t&0&0\end{pmatrix}+\rho^{n}\begin{pmatrix}a&tb_{i}&0\\\ tb_{j}&0&0\\\
0&0&0\end{pmatrix}$

for some $n\geq 2$. Direct computation using Lemma 4.2 yields

(4.4) $\displaystyle t^{2}(\widetilde{\text{Ric}}_{\phi}^{m})_{00}$
$\displaystyle=n\left(n-1-\frac{d+m}{2}\right)\rho^{n-1}\psi_{00}+O\left(\rho^{n}\right),$
$\displaystyle t(\widetilde{\text{Ric}}_{\phi}^{m})_{0i}$
$\displaystyle=n\left(n-1-\frac{d+m}{2}\right)\rho^{n-1}\psi_{0i}+\frac{n}{2}\rho^{n-1}\partial_{i}\psi_{00}+O\left(\rho^{n}\right),$

(cf. [FeffermanGraham2012]*Equation (3.11)). We conclude that if $d+m\notin
2\mathbb{N}$, then $a=b_{i}=0$; and if $d+m\in 2\mathbb{N}$, then
$n\geq\frac{d+m}{2}$. Therefore $\widetilde{g}$ is ambient-equivalent to a
metric of the form of Equation 4.3. The conclusion follows from Proposition
4.3. ∎

We now iteratively construct a straight and normal weighted ambient space. Let
$n\in\mathbb{N}$ be such that there is a metric

$\widetilde{g}_{IJ}^{(n-1)}=\begin{pmatrix}2\rho&0&t\\\ 0&t^{2}g_{\rho}&0\\\
t&0&0\end{pmatrix}$

and a function $\widetilde{f}^{(n-1)}=tf_{\rho}$ such that

$\displaystyle(\widetilde{\operatorname{Ric}_{\phi}^{m}})^{(n-1)}$
$\displaystyle=O(\rho^{n-1}),$ $\displaystyle\widetilde{F_{\phi}^{m}}^{(n-1)}$
$\displaystyle=O(\rho^{n-1}).$

Note that the existence of $\widetilde{g}_{IJ}^{(1)}$ and
$\widetilde{f}^{(1)}$ trivially holds. Let
$\widetilde{g}^{(n)}_{IJ}=\widetilde{g}^{(n-1)}_{IJ}+\Phi_{IJ}$ and
$\widetilde{f}^{(n)}=\widetilde{f}^{(n-1)}+\rho^{n}t\upsilon$, where

$\displaystyle\Phi_{IJ}$ $\displaystyle=\rho^{n}\begin{pmatrix}0&0&0\\\
0&t^{2}\psi_{ij}&0\\\ 0&0&0\end{pmatrix}$

and $\psi_{ij},\upsilon$ depend only on $x$ and $\rho$. We seek $\psi_{IJ}$
and $\upsilon$ such that

$\displaystyle\widetilde{(\text{Ric}_{\phi}^{m})}^{(n)}$
$\displaystyle=O(\rho^{n}),$ $\displaystyle\widetilde{F_{\phi}^{m}}^{(n)}$
$\displaystyle=O(\rho^{n}).$

Note that the inverse of $\widetilde{g}^{(n)}$, calculated modulo
$O(\rho^{2})$, is

$\widetilde{g}^{IJ}=\begin{pmatrix}0&0&t^{-1}\\\ 0&t^{-2}g^{ij}&0\\\
t^{-1}&0&-{2\rho}t^{-2}\end{pmatrix}.$

The Christoffel symbols for $\widetilde{g}^{(n)}$ modulo $O(\rho^{n})$ are
(cf. [FeffermanGraham2012])

(4.5) $\displaystyle\widetilde{\Gamma}_{IJ}^{0}$
$\displaystyle=\begin{pmatrix}0&0&0\\\
0&-\frac{1}{2}t\left(\partial_{\rho}g_{ij}+n\rho^{n-1}\psi_{ij}\right)&0\\\
0&0&0\end{pmatrix},$ $\displaystyle\widetilde{\Gamma}_{IJ}^{m}$
$\displaystyle=\begin{pmatrix}0&t^{-1}\delta_{j}^{m}&0\\\
t^{-1}\delta_{i}^{m}&\Gamma_{ij}^{m}&\frac{1}{2}g^{mk}\partial_{\rho}g_{ik}+\frac{n}{2}\rho^{n-1}\psi_{i}^{m}\\\
0&\frac{1}{2}g^{mk}\partial_{\rho}g_{jk}+\frac{n}{2}\rho^{n-1}\psi_{j}^{m}&0\end{pmatrix},$
$\displaystyle\widetilde{\Gamma}_{IJ}^{\infty}$
$\displaystyle=\begin{pmatrix}0&0&t^{-1}\\\
0&\rho\partial_{\rho}g_{ij}-g_{ij}&0\\\ t^{-1}&0&0\end{pmatrix}.$

Denote $\widetilde{R}_{IJ}^{(n)}=\widetilde{(Ric_{\phi}^{m})}_{IJ}^{(n)}$. It
follows that

(4.6a)
$\displaystyle\begin{split}&\widetilde{R}_{ij}^{(n)}=\widetilde{R}_{ij}^{(n-1)}+n\rho^{n-1}\bigg{[}\big{(}n-\frac{d+m}{2}\big{)}\psi_{ij}-\frac{1}{2}g^{kl}\psi_{kl}g_{ij}-\frac{m}{f}\upsilon
g_{ij}\bigg{]}+O(\rho^{n}),\end{split}$ (4.6b)
$\displaystyle\begin{split}&\widetilde{F_{\phi}^{m}}^{(n)}=\widetilde{F_{\phi}^{m}}^{(n-1)}+n\rho^{n-1}f\left[\frac{1}{2}fg^{kl}\psi_{kl}+\upsilon(d+2m-2n)\right]+O(\rho^{n}),\end{split}$
(4.6c)
$\displaystyle\widetilde{R}_{\infty\infty}^{(n)}=\widetilde{R}_{\infty\infty}^{(n-1)}-n(n-1)\rho^{n-2}\left[\frac{1}{2}g^{kl}\psi_{kl}+mf^{-1}\upsilon\right]+O(\rho^{n-1}).$

###  4.1. Solving $\widetilde{R}_{ij}=\widetilde{F_{\phi}^{m}}=O(\rho^{n})$

The following theorem specifies the result of recursively determining
$(\widetilde{g}^{(n)},\widetilde{f}^{(n)})$ by the requirement
$\widetilde{R}_{IJ}^{(n)}=O(\rho^{n})$.

######  Theorem 4.5.

Let $(M^{d},g,f,m,\mu),d\geq 3$, be a smooth metric measure space. Set
$\widetilde{\mathcal{G}}:=\mathbb{R}_{+}\times M\times(-\epsilon,\epsilon)$.

  1. (i)

If $d+m\notin\mathbb{N}$, then there is a metric measure structure
$(\widetilde{g},\widetilde{f})$ on $\widetilde{\mathcal{G}}$, unique modulo
$O(\rho^{\infty})$, such that
$\widetilde{R}_{IJ},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$.

  2. (ii)

If $d+m\in 2\mathbb{N}$, then

    1. (a)

there is a metric measure structure $(\widetilde{g},\widetilde{f})$ on
$\widetilde{\mathcal{G}}$, unique modulo $O^{+}(\rho^{\frac{d+m}{2}})$, such
that
$(\widetilde{R}_{IJ},\widetilde{F_{\phi}^{m}})=O_{IJ}^{2,+}(\rho^{\frac{d+m}{2}-1})$;

    2. (b)

if

$\partial_{\rho}^{(d+m)/2-1}(\widetilde{\text{Ric}_{\phi}^{m}})_{ij}=0,$

then there is a metric measure structure $(\widetilde{g},\widetilde{f})$ on
$\widetilde{\mathcal{G}}$ such that
$\widetilde{R}_{IJ},\widetilde{F_{\phi}^{m}}=O(\rho^{d+m-1})$;

    3. (c)

if

(4.7)
$\frac{m}{f^{2}}\widetilde{F_{\phi}^{m}}^{(d+m-1)}-g^{ij}\widetilde{R}_{ij}^{(d+m-1)}=O(\rho^{d+m}),$

then there is a metric measure structure $(\widetilde{g},\widetilde{f})$ on
$\widetilde{\mathcal{G}}$ such that
$\widetilde{R}_{IJ},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$.

  3. (iii)

If $d+m\in\mathbb{N}\setminus 2\mathbb{N}$, then

    1. (a)

there is a metric measure structure $(\widetilde{g},\widetilde{f})$ on
$\widetilde{\mathcal{G}}$, unique modulo $O(\rho^{d+m})$, such that
$\widetilde{R}_{IJ},\widetilde{F_{\phi}^{m}}=O(\rho^{d+m-1})$;

    2. (b)

if Equation 4.7 holds, then there is a metric measure structure
$(\widetilde{g},\widetilde{f})$ on $\widetilde{\mathcal{G}}$ such that
$\widetilde{R}_{IJ},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$.

######  Remark 4.6.

We say that $(\widetilde{g},\widetilde{f})$ is unique modulo $O^{+}(\rho^{k})$
if it is unique modulo $O(\rho^{k})$ and
$\frac{1}{2}g^{ij}\widetilde{g}_{ij}+\frac{m}{f}\widetilde{f}$ is unique
modulo $O(\rho^{k+1})$.

###### Proof.

On studying LABEL:ricij and LABEL:rphi, we observe that if
$n\neq\frac{d+m}{2}$, then there is a unique choice of the trace-free part
$\psi_{ij}-\frac{1}{d}g^{kl}\psi_{kl}g_{ij}$ of $\psi_{ij}$ which makes the
trace-free part of $\widetilde{R}^{(n)}_{ij}$ vanish modulo $O(\rho^{n})$.
However, if $n=\frac{d+m}{2}$, then we may not be able to make the trace-free
part of $\widetilde{R}_{ij}^{(n)}$ vanish modulo $O(\rho^{n})$. Additionally,
LABEL:ricij and LABEL:rphi imply that

(4.8) $\displaystyle
g^{ij}\widetilde{R}_{ij}^{(n)}=g^{ij}\widetilde{R}_{ij}^{(n-1)}+n\rho^{n-1}\frac{(2n-2d-m)}{2}g^{kl}\psi_{kl}-\frac{dmn}{f}\upsilon\rho^{n-1}+O(\rho^{n}),$
$\displaystyle\widetilde{F_{\phi}^{m}}^{(n)}=\widetilde{F_{\phi}^{m}}^{(n-1)}+n\rho^{n-1}f\left[\frac{1}{2}fg^{kl}\psi_{kl}+\upsilon(d+2m-2n)\right]+O(\rho^{n}).$

The determinant of the coefficient matrix of $g^{kl}\psi_{kl}$ and $\upsilon$
is

$(2n-d-m)(n-d-m).$

Therefore if $n\notin\\{\frac{d+m}{2},d+m\\}$, then there are unique
$g^{kl}\psi_{kl}$ and $\upsilon$ such that
$g^{ij}\widetilde{R}_{ij}^{(n)},(\widetilde{F_{\phi}^{m}})^{(n)}=O(\rho^{n})$.
However, if $n\in\\{\frac{d+m}{2},d+m\\}$, then we may not be able to
simultaneously solve $g^{ij}\widetilde{R}_{ij}^{(n)}=O(\rho^{n})$ and
$\widetilde{F_{\phi}^{m}}=O(\rho^{n})$.

Suppose first that $d+m\notin\mathbb{N}$. Combining the above discussion with
Borel’s lemma yields a straight and normal metric measure structure
$(\widetilde{g},\widetilde{f})$ such that
$\widetilde{R}_{ij},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$.

Suppose next that $d+m\in 2\mathbb{N}$. Set $n=\frac{d+m}{2}$. Equation 4.8
implies that

(4.9)
$\frac{m}{f^{2}}\widetilde{F_{\phi}^{m}}^{(\frac{d+m}{2})}-g^{ij}(\widetilde{\text{Ric}_{\phi}^{m}})_{ij}^{(\frac{d+m}{2})}=\frac{m}{f^{2}}\widetilde{F_{\phi}^{m}}^{(\frac{d+m}{2}-1)}-g^{ij}\widetilde{R}_{ij}^{(\frac{d+m}{2}-1)}\\\
+\frac{(d+m)^{2}}{2}\rho^{\frac{d+m}{2}-1}\big{(}\frac{1}{2}g^{kl}\psi_{kl}+\frac{m}{f}\upsilon\big{)}+O(\rho^{\frac{d+m}{2}}).$

In particular, there is a unique choice of
$\frac{1}{2}g^{kl}\psi_{kl}+\frac{m}{f}\upsilon$ such that the left hand side
of Equation 4.9 is $O(\rho^{\frac{d+m}{2}})$. Making this choice yields a
unique straight and normal metric measure structure
$(\widetilde{g},\widetilde{f})$ such that
$(\widetilde{R}_{ij},\widetilde{F_{\phi}^{m}})=O_{ij}^{2,+}(\rho^{\frac{d+m}{2}-1})$.
Moreover, if the weighted obstruction is zero, then
$\widetilde{R}_{ij}^{(n)}=O(\rho^{\frac{d+m}{2}})$. Hence, we may iteratively
solve LABEL:ricij and LABEL:rphi to obtain a straight and normal metric
measure structure $(\widetilde{g},\widetilde{f})$ such that
$\widetilde{R}_{ij},\widetilde{F_{\phi}^{m}}=O(\rho^{d+m-1})$. The possible
obstruction is the same as that discussed in the next case.

Suppose finally that $d+m\in\mathbb{N}\setminus 2\mathbb{N}$. Set $n=d+m$.
From the discussion of the first paragraph, we may choose the trace-free part
of $\psi_{ij}$ such that $\widetilde{R}_{ij}^{(d+m)}$ is pure trace.
Additionally, by Equation 4.8,

$\displaystyle g^{ij}\widetilde{R}_{ij}^{(d+m)}$
$\displaystyle=g^{ij}\widetilde{R}_{ij}^{(d+m-1)}+m(d+m)\rho^{d+m-1}\left(\frac{1}{2}g^{kl}\psi_{kl}-\frac{d}{f}\upsilon\right)+O(\rho^{d+m}),$
$\displaystyle\widetilde{F_{\phi}^{m}}^{(d+m)}$
$\displaystyle=\widetilde{F_{\phi}^{m}}^{(d+m-1)}+f^{2}(d+m)\rho^{d+m-1}\left(\frac{1}{2}g^{kl}\psi_{kl}-\frac{d}{f}\upsilon\right)+O(\rho^{d+m}).$

Therefore we may choose $g^{ij}\psi_{ij}$ and $\upsilon$ such that
$g^{ij}\widetilde{R}_{ij}^{(d+m)},\widetilde{F_{\phi}^{m}}^{(d+m)}=O(\rho^{d+m})$
if and only if Equation 4.7 holds. If Equation 4.7 holds, then we may continue
iteratively improving $(\widetilde{g},\widetilde{f})$ as in the first
paragraph. Hence, by Borel’s lemma, there is a straight and normal metric
measure structure $(\widetilde{g},\widetilde{f})$ such that
$\widetilde{R}_{ij},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$. ∎

###  4.2. Solving $\widetilde{R}_{I\infty}=0$

In this subsection, we show that if $(\widetilde{g},\widetilde{f})$ is as in
Theorem 4.5, then the components $\widetilde{R}_{I\infty}$ can be made to
vanish to the appropriate order and that Equation 4.7 automatically holds.
This proves Theorem 3.8.

###### Proof of Theorem 3.8.

Let $(\widetilde{g},\widetilde{f})$ be as in Theorem 4.5. Equation 2.1 implies
that

(4.10)
$\widetilde{g}^{JK}\widetilde{\partial}_{J}\widetilde{R}_{KI}-\widetilde{g}^{JK}\widetilde{\Gamma}_{JK}^{Q}\widetilde{R}_{QI}-\widetilde{g}^{JK}\widetilde{\Gamma}_{JI}^{Q}\widetilde{R}_{KQ}\\\
-\widetilde{g}^{JK}\widetilde{R}_{IJ}\partial_{K}\widetilde{\phi}-\frac{1}{2}\partial_{I}\widetilde{R_{\widetilde{\phi}}^{m}}-\frac{1}{f^{2}}\widetilde{F}_{\phi}^{m}\partial_{I}\widetilde{\phi}=0.$

Set

$n=\begin{cases}\infty,&\text{if }d+m\notin\mathbb{N},\\\
\frac{d+m}{2}-1,&\text{if }d+m\in 2\mathbb{N},\\\ d+m-1,&\text{if
}d+m\in\mathbb{N}\setminus 2\mathbb{N}.\end{cases}$

Taking $I=l$ and $I=\infty$ in Equation 4.10 and
computing$\;\bmod\;O(\rho^{n})$ yields

(4.11a) $\bigg{[}d+m-2-2\rho\partial_{\rho}\bigg{]}\widetilde{R}_{\infty
l}-\rho g^{ks}g^{\prime}_{sl}\widetilde{R}_{\infty
k}+2\rho\widetilde{R}_{l\infty}\partial_{\rho}\phi+\rho\partial_{l}\widetilde{R}_{\infty\infty}=O(\rho^{n}),$
(4.11b)
$\bigg{[}d+m-2-\rho\partial_{\rho}\bigg{]}\widetilde{R}_{\infty\infty}-\frac{1}{2}\partial_{\rho}\big{(}g^{ij}\widetilde{R}_{ij}-\frac{m}{f^{2}}\widetilde{F_{\phi}^{m}}\big{)}\\\
+g^{ij}(\widetilde{\nabla}_{\phi})_{i}\widetilde{R}_{j\infty}+\rho(2\partial_{\rho}\phi-g^{ij}g^{\prime}_{ij})\widetilde{R}_{\infty\infty}=O(\rho^{n}).$

First, suppose that $d+m\notin\mathbb{N}$. We know that
$\widetilde{R}_{ij},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$. From Equation
4.11a and Equation 4.11b we conclude that $\widetilde{R}_{\infty
l},\widetilde{R}_{\infty\infty}=O(\rho^{\infty})$.

Second, suppose that $d+m\in\mathbb{N}\setminus 2\mathbb{N}$. Equation 4.11
implies that $\widetilde{R}_{\infty
l},\widetilde{R}_{\infty\infty}=O(\rho^{n})$, and that
$g^{ij}\widetilde{R}_{ij}-mf^{-2}\widetilde{F_{\phi}^{m}}=O(\rho^{n+1})$. This
verifies Equation 4.7. Equation 4.6c gives us the unique value of
$\frac{1}{2}g^{kl}\psi_{kl}+\frac{m}{f}\upsilon$ such that
$\widetilde{R}_{\infty\infty}=O(\rho^{n})$. Hence, we can now uniquely
determine the values of $g^{kl}\psi_{kl}$ and $\upsilon$, and solve
$\widetilde{\text{Ric}}_{\phi}^{m},\widetilde{F_{\phi}^{m}}=O(\rho^{\infty})$.

Third, suppose that $d+m\in 2\mathbb{N}$. Equation 4.11a tells us that
$\widetilde{R}_{\infty l}=O(\rho^{n})$ and that
$\widetilde{R}_{\infty\infty}=O(\rho^{n-1})$. Now recall that
$g^{ij}\widetilde{R}_{ij}-mf^{-2}\widetilde{F_{\phi}^{m}}=O(\rho^{\frac{d+m}{2}})$.
Hence, Equation 4.11b tells us that
$\widetilde{R}_{\infty\infty}=O(\rho^{n})$.

Finally, since the terms of $(\widetilde{g},\widetilde{f})$ are uniquely
determined using the process described above, any two straight and normal
weighted ambient structures will be ambient-equivalent to the orders stated in
the theorem.∎

###  4.3. The weighted obstruction tensor

We conclude this section by establishing some properties of the weighted
obstruction tensor. To that end, we first introduce notation for the
corresponding term in the asymptotic expansion of $\widetilde{F_{\phi}^{m}}$.

######  Definition 4.7.

We define

$\mathcal{F}:=c_{d+m}{\left.\kern-1.2pt\partial_{\rho}^{\frac{d+m}{2}-1}\vphantom{\big{|}}\right|_{{\rho=0}}}\widetilde{F_{\phi}^{m}},$

where $c_{d+m}$ is defined in Equation 1.2.

Note that we can use the recursive construction of
$(\widetilde{g},\widetilde{f})$ to express $\mathcal{O}_{ij}$ and
$\mathcal{F}$ as polynomials in $g,g^{-1},f,\operatorname{Rm}$ and $\nabla$.

Both $\mathcal{O}_{ij}$ and $\mathcal{F}$ are weighted conformal invariants.
Moreover, there is a simple relationship between $\mathcal{F}$ and the trace
and divergence of $\mathcal{O}_{ij}$ (cf. [Case2014s]).

######  Theorem 4.8.

Let $(M^{d},g,f,m,\mu),d\geq 3$, be a smooth metric measure space with $d+m\in
2\mathbb{N}$. Then

  1. (i)

$\mathcal{O}_{i}^{i}=\frac{m}{f^{2}}\mathcal{F};$

  2. (ii)

$\delta_{\phi}\mathcal{O}_{i}=\frac{1}{f^{2}}\mathcal{F}(\partial_{l}\phi);$
and

  3. (iii)

$\mathcal{O}_{ij}$ and $\mathcal{F}$ are weighted conformal invariants of
weight $2-d-m$.

###### Proof.

Since
$g^{ij}\widetilde{R}_{ij}-mf^{-2}\widetilde{F_{\phi}^{m}}=O(\rho^{\frac{d+m}{2}})$,
we see that $\mathcal{O}^{i}_{i}=\frac{m}{f^{2}}\mathcal{F}$.

Writing Equation 4.11a modulo $O(\rho^{\frac{d+m}{2}})$ and recalling that
$\widetilde{R}_{\infty l}=O(\rho^{\frac{d+m}{2}-1})$, we get

$\frac{1}{t^{2}}\widetilde{\nabla}^{j}\widetilde{R}_{jl}-\frac{1}{t^{2}}g^{ij}\widetilde{R}_{li}\partial_{j}\widetilde{\phi}-\frac{1}{f^{2}}\widetilde{F_{\phi}^{m}}\partial_{l}\widetilde{\phi}=O(\rho^{\frac{d+m}{2}}).$

Hence,
$\delta_{\phi}\mathcal{O}_{i}=\frac{1}{f^{2}}\mathcal{F}\partial_{l}\phi$.

The conformal invariance of the weighted ambient space implies that
$\mathcal{O}_{ij}$ is conformally invariant of weight $2-d-m$. The conformal
invariance of $\mathcal{F}$ follows from the identity
$\mathcal{O}_{i}^{i}=\frac{m}{f^{2}}\mathcal{F}$. ∎

##  5\. Weighted Poincaré spaces

In this section, we construct a weighted Poincaré space for a smooth metric
measure space.

Let $M_{+}$ be a smooth manifold with boundary $M$. Denote by $M^{0}_{+}$ the
interior of $M_{+}$. A smooth metric measure space
$(M^{0}_{+},g_{+},r_{+},m,\mu)$ is _conformally compact_ if there is a
defining function $r$ for $M$ such that

  1. (i)

$(r^{2}g_{+},rf_{+})$ extends smoothly to $M_{+}$; and

  2. (ii)

$(M,{\left.\kern-1.2ptr^{2}g_{+}\vphantom{\big{|}}\right|_{M}},{\left.\kern-1.2ptrf_{+}\vphantom{\big{|}}\right|_{M}},m,\mu)$
is a smooth metric measure space.

We call
$(M,[{\left.\kern-1.2ptr^{2}g_{+}\vphantom{\big{|}}\right|_{M}},{\left.\kern-1.2ptrf_{+}\vphantom{\big{|}}\right|_{M}}],m,\mu)$
the conformal infinity of $(M^{0}_{+},g_{+},r_{+},m,\mu)$.

In the following, we identify $M_{+}$ with an open neighborhood of
$M\times\\{0\\}$ in $M\times[0,\infty)$ and choose $r$ to be the standard
coordinate on $[0,\infty)$.

######  Definition 5.1.

A _weighted Poincaré space_ for $(M^{d},[g,f],m,\mu)$ is a metric measure
structure $(g_{+},f_{+})$ on $M\times[0,\epsilon)$ such that

  1. (i)

$g_{+}$ has signature $(d+1,0)$;

  2. (ii)

$(g_{+},f_{+})$ has $(M^{d},[g,f],m,\mu)$ as conformal infinity; and

  3. (iii)

    1. (a)

if $d+m\notin 2\mathbb{N}$, then

(5.1)
$\displaystyle\big{(}\text{Ric}_{\phi}^{m}(g_{+})+(d+m)g_{+},F_{\phi}^{m}(g_{+})-(d+m)f_{+}^{2}\big{)}=O(r^{\infty});$

    2. (b)

if $d+m\in 2\mathbb{N}$, then

(5.2)
$\displaystyle\big{(}\text{Ric}_{\phi}^{m}(g_{+})+(d+m)g_{+},F_{\phi}^{m}(g_{+})-(d+m)f_{+}^{2}\big{)}=O^{2,+}_{\alpha\beta}(r^{d+m-2}).$

If $(M_{+},g_{+},f_{+},m,\mu)$ is conformally compact, then
$|dr/r|_{g_{+}}=|dr|_{r^{2}g_{+}}$ extends smoothly to the boundary. We call
$(M_{+},g_{+},f_{+},m,\mu)$ _asymptotically hyperbolic_ if $|dr/r|_{g_{+}}=1$
on $M$. In this case, all the sectional curvatures of $g_{+}$ approach $-1$ at
a boundary point [MazzeoMelrose1987].

######  Definition 5.2.

A weighted Poincaré space $(M_{+},g_{+},f_{+},m,\mu)$ is in _normal form
relative to $(g,f)$_ if $g_{+}=r^{-2}(dr^{2}+g_{r})$ and $f_{+}=r^{-1}f_{r}$.
Here $g_{r}$ is a one-parameter family of metrics on $M$ such that $g_{0}=g$,
and $f_{r}$ is a one-parameter family of functions such that $f_{0}=f$.

######  Proposition 5.3.

Let $(M_{+},g_{+},f_{+},m,\mu)$ be asymptotically hyperbolic. Then there
exists a neighborhood $U$ of $M\times\\{0\\}$ in $M\times[0,\infty)$ on which
there is a unique diffeomorphism $\phi$ from $U$ into $M_{+}$ such that
${\left.\kern-1.2pt\phi\vphantom{\big{|}}\right|_{M}}$ is the identity map and
$(\phi^{*}g_{+},\phi^{*}f_{+})$ is in normal form relative to $(g,f)$ on $U$.

###### Proof.

This follows as in the conformal case [FeffermanGraham2012]*Proposition 4.3. ∎

######  Definition 5.4.

An asymptotically hyperbolic smooth metric measure space, denoted as
$(M_{+},g_{+},f_{+},m,\mu)$, is _even_ if $(r^{2}g_{+},rf_{+})$ is the
restriction to $M_{+}$ of a metric measure structure
$(g_{+}^{\prime},f_{+}^{\prime})$ on an open set $V\subset
M\times(-\infty,\infty)$ containing $M_{+}$ such that $V$ and
$(g_{+}^{\prime},f_{+}^{\prime})$ are invariant under $r\mapsto-r$.

A diffeomorphism $\psi:M_{+}\mapsto M\times[0,\infty)$ satisfying
${\left.\kern-1.2pt\psi\vphantom{\big{|}}\right|_{{M\times\\{0\\}}}}=\mathrm{Id}$
is _even_ if $\psi$ is the restriction of a diffeomorphism on an open set $V$
as above which commutes with $r\mapsto-r$.

It is easily seen that if $\psi$ if an even diffeomorphism and
$(M_{+},g_{+},f_{+},m,\mu)$ is an even asymptotically hyperbolic smooth metric
measure, then $\psi^{*}g_{+}$ is also even.

######  Theorem 5.5.

Let $(M^{d},[g,f],m,\mu),d\geq 3$, be a conformal class of smooth metric
measure spaces. Then there exists an even weighted Poincaré space
corresponding to it. Moreover, if $(g_{+}^{1},f_{+}^{1})$ and
$(g_{+}^{2},f_{+}^{2})$ are two even weighted Poincaré structures defined on
$(M_{+}^{1})^{\circ}$ and $(M_{+}^{2})^{\circ}$ respectively, then there exist
open sets $U_{1}\subset M_{+}^{1}$ and $U_{2}\subset M_{+}^{2}$ containing
$M\times\\{0\\}$, and an even diffeomorphism $\phi:U_{1}\mapsto U_{2}$, such
that ${\left.\kern-1.2pt\phi\vphantom{\big{|}}\right|_{{M\times\\{0\\}}}}$ is
the identity map and

  1. (i)

if $d+m\notin 2\mathbb{N}$, then
$(g_{+}^{1}-\phi^{*}{g_{+}^{2}},-2(f_{+}^{1}-\phi^{*}f_{+}^{2}))=O(\rho^{\infty})$;

  2. (ii)

if $d+m\in 2\mathbb{N}$, then
$(g_{+}^{1}-\phi^{*}{g_{+}^{2}},-2(f_{+}^{1}-\phi^{*}f_{+}^{2}))=O_{\alpha\beta}^{+}(r^{d+m-2})$.

Theorem 5.5 will be proved at the end of this section.

Let $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ be a
straight weighted pre-ambient space for $(M,g,f,m,\mu)$. Then
$\widetilde{g}(T,T)$ vanishes to first order on
$\mathcal{G}\times\\{0\\}\subset\widetilde{\mathcal{G}}$. Therefore, shrinking
$\widetilde{\mathcal{G}}$ if necessary, the hypersurface
$H:=\widetilde{\mathcal{G}}\cap\\{\widetilde{g}(T,T)=-1\\}$ lies on one side
of $\mathcal{G}\times\\{0\\}$. Also, since $\widetilde{g}(T,T)$ is homogeneous
of degree $2$, each $\delta_{s}$-orbit on this side of
$\mathcal{G}\times\\{0\\}$ intersects $H$ exactly once. We extend the
projection $\pi:\mathcal{G}\mapsto M$ to
$\widetilde{\mathcal{G}}\subset\mathcal{G}\times\mathbb{R}\mapsto
M\times\mathbb{R}$ by projecting only the first factor. Define
$\chi:M\times\mathbb{R}\mapsto M\times[0,\infty)$ by
$(x,\rho)\mapsto(x,\sqrt{2|\rho|})$. Then there is an open set $M_{+}\subset
M\times[0,\infty)$ containing $M\times\\{0\\}$ such that
${\left.\kern-1.2pt\chi\circ\pi\vphantom{\big{|}}\right|_{H}}:H\mapsto
M_{+}^{\circ}$ is a diffeomorphism.

######  Proposition 5.6.

If $(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ is a straight
and normal weighted pre-ambient space of the form of Equation 1.1, and if $H$
and $M_{+}^{\circ}$ are as above, then

(5.3) $\displaystyle
g_{+}:=(({\left.\kern-1.2pt\chi\circ\pi\vphantom{\big{|}}\right|_{{H}}})^{-1})^{*}\widetilde{g},$
$\displaystyle
f_{+}:=(({\left.\kern-1.2pt\chi\circ\pi\vphantom{\big{|}}\right|_{{H}}})^{-1})^{*}\widetilde{f},$

defines an even, asymptotically hyperbolic, normal smooth metric measure space
$(M_{+},g_{+},f_{+},m,\mu)$ with conformal infinity $(M^{d},g,f,m,\mu)$.

###### Proof.

We have $H=\\{2\rho t^{2}=-1\\}$. Introduce new variables $r>0$ and $s>0$ by
$-2\rho=r^{2}$ and $s=rt$. Then

$\displaystyle\widetilde{g}$
$\displaystyle=\frac{s^{2}}{r^{2}}(g_{-\frac{1}{2}r^{2}}+dr^{2})-ds^{2},$
$\displaystyle\widetilde{f}$ $\displaystyle=\frac{s}{r}f_{-\frac{1}{2}r^{2}},$

Note that $H=s^{-1}(\\{1\\})$. Thus the restriction of $\widetilde{g}$ to $TH$
is $r^{-2}(g_{-\frac{1}{2}r^{2}}+dr^{2})$. Similarly, the restriction of
$\widetilde{f}$ to $H$ is $r^{-1}f_{-\frac{1}{2}r^{2}}$. By the definition of
$\chi$, we see that $r$ is the coordinate in the second factor of
$M\times[0,\infty)$. Therefore $(g_{+},f_{+})$ is an even asymptotically
hyperbolic smooth metric measure space with conformal infinity
$(M^{d},[g,f],m,\mu)$ in normal form relative to $(g,f)$. ∎

We now show that weighted Poincaré spaces and weighted ambient spaces are
closely related.

######  Proposition 5.7.

Let $(M_{+}^{d+1},g_{+},f_{+},m,\mu)$ be a smooth metric measure space.
Consider $(M_{+}\times\mathbb{R}_{+},\widetilde{g},\widetilde{f},m,\mu)$,
where $\widetilde{g}=s^{2}g_{+}-ds^{2}$ and $\widetilde{f}=sf_{+}$. Then

(5.4) $\displaystyle\text{Ric}_{\phi}^{m}(\widetilde{g})$
$\displaystyle=\text{Ric}(g_{+})+(d+m)g_{+},$
$\displaystyle\widetilde{F_{\phi}^{m}}$
$\displaystyle=F_{\phi}^{m}(g_{+})-(d+m)f_{+}^{2}.$

###### Proof.

The Christoffel symbols of the metric $\widetilde{g}=-ds^{2}+s^{2}g_{+}$ are

(5.5) $\displaystyle\widetilde{\Gamma}_{IJ}^{0}$
$\displaystyle=\begin{pmatrix}0&0\\\ 0&s(g_{+})_{ij}\end{pmatrix},$
$\displaystyle\widetilde{\Gamma}_{IJ}^{k}$
$\displaystyle=\begin{pmatrix}0&\frac{1}{s}\delta_{i}^{k}\\\
\frac{1}{s}\delta_{i}^{k}&(\Gamma_{g_{+}})_{ij}^{k}\end{pmatrix}.$

Using the formula

$R_{ij}=\partial_{L}\Gamma_{ij}^{L}-\partial_{i}\Gamma_{Lj}^{L}+\Gamma_{ij}^{P}\Gamma_{LP}^{L}-\Gamma_{Lj}^{P}\Gamma_{iP}^{L}$

and Equation 5.5, we readily compute that
$\operatorname{Ric}_{\phi}^{m}(\widetilde{g})=\operatorname{Ric}(g_{+})+(d+m)g_{+}$.
Using the formula

$\widetilde{F_{\phi}^{m}}=sf_{+}\left(-\frac{d+1}{s}f_{+}+\frac{1}{s^{2}}\Delta_{g_{+}}sf_{+}\right)+(m-1)\left(-f_{+}^{2}+|\nabla_{g_{+}}f_{+}|^{2}-\mu\right).$

we readily compute that
$\widetilde{F_{\phi}^{m}}=F_{\phi}^{m}(g_{+})-(d+m)f_{+}^{2}.$ ∎

###### Proof of Theorem 5.5.

Pick a representative $(M^{d},g,f,m,\mu)$. Theorem 3.8 implies that there is a
weighted ambient metric in straight and normal form relative to $(g,f)$. By
Proposition 5.6, the metric measure structure $(g_{+},f_{+})$ is even and in
normal form relative to $(g,f)$. Proposition 5.7 implies that conditions (3)
and (4) of Definition 5.1 are satisfied, so that
$(M_{+}^{\circ},g_{+},f_{+},m,\mu)$ is a weighted Poincaré metric space for
$(M^{d},g,f,m,\mu)$. This proves the first part of the theorem. The second
part follows similarly from the argument in Theorem 3.8. ∎

##  6\. On the expansion of the weighted ambient structure

Although it is difficult in general to compute the terms of
$(\widetilde{g},\widetilde{f})$, we are able to compute the first few terms by
hand.

Let $(\widetilde{g},\widetilde{f})$ be as in Equation 1.1. Then

(6.1) $\displaystyle(\widetilde{\text{Ric}_{\phi}^{m}})_{ij}$
$\displaystyle=\rho g^{\prime\prime}_{ij}-\rho
g^{kl}g^{\prime}_{ik}g^{\prime}_{jl}+\frac{1}{2}\rho
g^{kl}g^{\prime}_{kl}g^{\prime}_{ij}+\rho\frac{m}{\widetilde{f}}g_{ij}^{\prime}\widetilde{f}^{\prime}$
$\displaystyle\quad\quad\quad\quad-\left(\frac{d+m}{2}-1\right)g^{\prime}_{ij}-\frac{1}{2}g^{kl}g^{\prime}_{kl}g_{ij}-\frac{m}{\widetilde{f}}g_{ij}\widetilde{f}^{\prime}+(\text{Ric}_{\phi}^{m})_{ij},$
$\displaystyle\widetilde{F_{\phi}^{m}}$ $\displaystyle=-2\rho
ff^{\prime\prime}-\rho
ff^{\prime}g^{ij}g^{\prime}_{ij}-2(m-1)\rho(f^{\prime})^{2}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\frac{1}{2}f^{2}g^{ij}g^{\prime}_{ij}+(2m+d-2)ff^{\prime}+F_{\phi}^{m}.$

Using these, we readily compute that

$\displaystyle g_{ij}^{\prime}(\cdot,0)=2(P_{\phi}^{m})_{ij},\quad
f^{\prime}(\cdot,0)=\frac{f}{m}Y_{\phi}^{m}.$

Moreover, differentiating Equation 6.1 once with respect to $\rho$ and
evaluating at $\rho=0$ yields

$\displaystyle(d+m-4)g_{ij}^{\prime\prime}$
$\displaystyle=-2(B_{\phi}^{m})_{ij}+2(d+m-4)(P_{\phi}^{m})_{i}^{k}(P_{\phi}^{m})_{jk},$
$\displaystyle(d+m-4)f^{\prime\prime}$
$\displaystyle=\frac{f}{m}g^{ij}(B_{\phi}^{m})_{ij}.$

In particular, if $d+m=4$, then $\mathcal{O}_{ij}=(B_{\phi}^{m})_{ij}$.

##  7\. Explicit examples of weighted ambient structures

We conclude this article by proving Theorem 1.2. The following three
subsections handle the the three separate cases mentioned in the theorem.

###  7.1. Quasi-Einstein space

Suppose that $(M^{d},g,f,m,\mu)$ is quasi-Einstein. Then

(7.1) $\text{Ric}_{\phi}^{m}=\lambda g,\quad F_{\phi}^{m}=-\lambda f^{2}.$

Set

$g_{\rho}:=(1+\lambda\rho)^{2}g,\quad f_{\rho}:=(1+\lambda\rho)f.$

Direct computation yields $\widetilde{\text{Ric}_{\phi}^{m}}=0$ and
$\widetilde{F_{\phi}^{m}}=0$. Thus
$(\mathcal{G}\times(-\epsilon,\epsilon),\widetilde{g},\widetilde{f},m,\mu)$ is
a weighted ambient metric for $(M^{d},g,f,m,\mu)$. This verifies Theorem
1.2(i).

###  7.2. Weighted locally conformally flat space

Recall from Section 2 that the locally conformally flat condition is
equivalent to $A_{\phi}^{m}=0$ and $dP_{\phi}^{m}=0$. Therefore
$(P_{\phi}^{m})_{ij;k}=(P_{\phi}^{m})_{(ij;k)}$. We use this fact repeatedly
in the rest of the section.

Let $(g_{\rho},f_{\rho})$ be as in the statement of Theorem 1.2. Note that

$(g_{\rho})_{ij}=g_{il}(U_{\phi}^{m})_{k}^{l}(U_{\phi}^{m})_{j}^{k}=(U_{\phi}^{m})_{ik}(U_{\phi}^{m})_{j}^{k},$

where

$(U_{\phi}^{m})_{j}^{i}:=\delta_{j}^{i}+\rho(P_{\phi}^{m})_{j}^{i}.$

Set $V_{\phi}^{m}:=(U_{\phi}^{m})^{-1}$. Then
$(V_{\phi}^{m})_{k}^{i}(U_{\phi}^{m})_{j}^{k}=\delta_{j}^{i}$. Note that
$(U_{\phi}^{m})_{ij}$ and $(V_{\phi}^{m})_{ij}$ are both symmetric.
Additionally,

(7.2) $\displaystyle(V_{\phi}^{m})_{i}^{k}(g_{\rho})_{kj}$
$\displaystyle=(U_{\phi}^{m})_{ij},$ (7.3)
$\displaystyle(g_{\rho})_{ij}^{\prime}$
$\displaystyle=2(P_{\phi}^{m})_{ik}(U_{\phi}^{m})_{j}^{k}.$

We now relate the Levi-Civita connections ${}^{g_{\rho}}\nabla$ and
${}^{g}\nabla$, and the curvature tensors ${}^{g_{\rho}}R_{ijkl}$ and
${}^{g}R_{ijkl}$.

######  Lemma 7.1.

Let $(g_{\rho},f_{\rho})$ and $(g,f)$ be defined as in Theorem 1.2. The Levi-
Civita connections of $g_{\rho}$ and $g$ are related by

(7.4)
${}^{g_{\rho}}\nabla_{i}\eta_{j}={}^{g}\nabla_{i}\eta_{j}-\rho(V_{\phi}^{m})_{l}^{k}(P_{\phi}^{m})_{i;j}^{l}\eta_{k},$

and the curvature tensors by

(7.5)
${}^{g_{\rho}}R_{ijkl}={}^{g}R_{abkl}(U_{\phi}^{m})_{i}^{a}(U_{\phi}^{m})_{j}^{b}.$

###### Proof.

First note that, since $dP_{\phi}^{m}=0$, the right hand side of Equation 7.4
defines a torsion-free connection.

We now show that $g_{\rho}$ is parallel with respect to the connection
determined by the right side of Equation 7.4. On differentiating the metric,
we get

(7.6)
$^{g}\nabla_{k}(g_{\rho})_{ij}=2\rho(P_{\phi}^{m})_{ij;k}+2\rho^{2}(P_{\phi}^{m})_{(i}^{l}(P_{\phi}^{m})_{j)l;k}.$

Using the definitions of $U_{\phi}^{m}$ and $V_{\phi}^{m}$ and the symmetry of
$(P_{\phi}^{m})_{ij;k}$, we get

$\displaystyle(V_{\phi}^{m})_{l}^{m}(P_{\phi}^{m})_{k;(i}^{l}(g_{\rho})_{j)m}=(P_{\phi}^{m})_{k;(i}^{l}(U_{\phi}^{m})_{j)l}=(P_{\phi}^{m})_{ij;k}+\rho(P_{\phi}^{m})_{k;(i}^{l}(P_{\phi}^{m})_{j)l}.$

Therefore

${}^{g}\nabla_{k}(g_{\rho})_{ij}-2\rho(V_{\phi}^{m})_{l}^{m}(P_{\phi}^{m})_{k;(i}^{l}(g_{\rho})_{j)m}=0.$

Combining the previous two paragraphs verifies Equation 7.4.

Now set

$(D_{\phi}^{m})_{jk}^{i}:=\rho(V_{\phi}^{m})_{l}^{i}(P_{\phi}^{m})_{j;k}^{l},$

so that $(D_{\phi}^{m})_{jk}^{i}$ is the difference of ${}^{g}\nabla$ and
${}^{g_{\rho}}\nabla$. The difference of the curvature tensors of the
connections is given in terms of $D_{\phi}^{m}$ by

${}^{g_{\rho}}R_{mjkl}(g_{\rho})^{im}-{}^{g}R_{jkl}^{i}=2(D_{\phi}^{m})_{j[l;k]}^{i}+2(D_{\phi}^{m})_{j[l}^{c}(D_{\phi}^{m})_{k]c}^{i};$

see [FeffermanGraham2012]. We compute that

$\displaystyle(D_{\phi}^{m})_{jl;k}^{i}$
$\displaystyle=\rho\left((V_{\phi}^{m})_{a;k}^{i}(P_{\phi}^{m})_{j;l}^{a}+(V_{\phi}^{m})_{a}^{i}(P_{\phi}^{m})_{j;lk}^{a}\right)$
$\displaystyle=-\rho(V_{\phi}^{m})_{b}^{i}(U_{\phi}^{m})_{c;k}^{b}(V_{\phi}^{m})_{a}^{c}(P_{\phi}^{m})_{j;l}^{a}+\rho(V_{\phi}^{m})_{a}^{i}(P_{\phi}^{m})_{j;lk}^{a}$
$\displaystyle=-\rho^{2}(V_{\phi}^{m})_{b}^{i}(P_{\phi}^{m})_{c;k}^{b}(V_{\phi}^{m})_{a}^{c}(P_{\phi}^{m})_{j;l}^{a}+\rho(V_{\phi}^{m})_{a}^{i}(P_{\phi}^{m})_{j;lk}^{a}$
$\displaystyle=-(D_{\phi}^{m})_{ck}^{i}(D_{\phi}^{m})_{jl}^{c}+\rho(V_{\phi}^{m})_{a}^{i}(P_{\phi}^{m})_{j;lk}^{a}.$

Therefore,

$\displaystyle{}^{g_{\rho}}R_{mjkl}(g_{\rho})^{im}$
$\displaystyle={}^{g}R_{jkl}^{i}+2\rho(V_{\phi}^{m})^{ia}(P_{\phi}^{m})_{aj;[lk]}$
$\displaystyle={}^{g}R_{jkl}^{i}+\rho(V_{\phi}^{m})^{ia}\left({}^{g}R_{alk}^{b}(P_{\phi}^{m})_{bj}+{}^{g}R_{jlk}^{b}(P_{\phi}^{m})_{ab}\right).$

Since $(V_{\phi}^{m})_{i}^{k}(g_{\rho})_{kj}=(U_{\phi}^{m})_{ij}$, we conclude
that

$\displaystyle{}^{g_{\rho}}R_{ijkl}$
$\displaystyle={}^{g}R_{jkl}^{b}(g_{\rho})_{bi}+\rho(U_{\phi}^{m})_{i}^{a}\left({}^{g}R_{alk}^{b}(P_{\phi}^{m})_{bj}+{}^{g}R_{jlk}^{b}(P_{\phi}^{m})_{ab}\right)$
$\displaystyle={}^{g}R_{jkl}^{b}(U_{\phi}^{m})_{ba}(U_{\phi}^{m})_{i}^{a}+\rho(U_{\phi}^{m})^{a}_{i}\left({}^{g}R_{alk}^{b}(P_{\phi}^{m})_{bj}+{}^{g}R_{jlk}^{b}(P_{\phi}^{m})_{ab}\right)$
$\displaystyle=\left[{}^{g}R_{abkl}\left(\delta_{j}^{b}+\rho(P_{\phi}^{m})_{j}^{b}\right)\right](U_{\phi}^{m})_{i}^{a}$
$\displaystyle={}^{g}R_{abkl}(U_{\phi}^{m})_{j}^{b}(U_{\phi}^{m})^{a}{}_{i}.\qed$

The metric $\widetilde{g}$ in Theorem 1.2(ii) is in fact flat.

######  Proposition 7.2.

Let $(g_{\rho},f_{\rho})$ and $(g,f)$ be as in Theorem 1.2. Then
$\widetilde{g}$ is flat.

###### Proof.

The curvature tensor of $\widetilde{g}=2\rho d\rho dt+t^{2}g_{\rho}+2td\rho
dt$ is [FeffermanGraham2012]

$\displaystyle\begin{split}&\widetilde{R}_{IJK0}=0,\end{split}$
$\displaystyle\begin{split}&\widetilde{R}_{ijkl}=t^{2}\left[{}^{g_{\rho}}R_{ijkl}+(g_{\rho})_{i[l}(g_{\rho})_{k]j}^{\prime}+(g_{\rho})_{j[k}(g_{\rho})_{l]i}^{\prime}-\rho(g_{\rho})_{i[l}^{\prime}(g_{\rho})_{k]j}^{\prime}\right],\end{split}$
$\displaystyle\begin{split}&\widetilde{R}_{\infty
jkl}=\frac{1}{2}t^{2}\left[\nabla_{l}(g_{\rho})_{jk}^{\prime}-\nabla_{k}(g_{\rho})_{jl}^{\prime}\right],\end{split}$
$\displaystyle\begin{split}&\widetilde{R}_{\infty
jk\infty}=\frac{1}{2}t^{2}\left[(g_{\rho})_{jk}^{\prime\prime}-\frac{1}{2}(g_{\rho})^{pq}(g_{\rho})_{jp}^{\prime}(g_{\rho})_{kq}^{\prime}\right].\end{split}$

Hence, $\widetilde{g}$ is flat if and only if

(7.8a)
$\displaystyle\begin{split}&0={}^{g_{\rho}}R_{ijkl}+(g_{\rho})_{i[l}(g_{\rho})_{k]j}^{\prime}+(g_{\rho})_{j[k}(g_{\rho})_{l]i}^{\prime}-\rho(g_{\rho})_{i[l}^{\prime}(g_{\rho})_{k]j}^{\prime},\end{split}$
(7.8b)
$\displaystyle\begin{split}&0={}^{g_{\rho}}\nabla_{k}(g_{\rho})_{ij}^{\prime}-{}^{g_{\rho}}\nabla_{j}(g_{\rho})_{ik}^{\prime},\end{split}$
(7.8c)
$\displaystyle\begin{split}&0=(g_{\rho})_{ij}^{\prime\prime}-\frac{1}{2}(g_{\rho})^{pq}(g_{\rho})_{ip}^{\prime}(g_{\rho})_{jq}^{\prime}.\end{split}$

LABEL:curvature3 follows directly from the definition of $g_{\rho}$.

We next verify LABEL:curvature2. Differentiating Equation 7.6 with respect to
$\rho$ yields

(7.9)
$^{g}\nabla_{k}(g_{\rho})_{ij}^{\prime}=2(P_{\phi}^{m})_{ij;k}+4\rho(P_{\phi}^{m})_{(i}^{l}(P_{\phi}^{m})_{j)l;k}.$

Using Equation 7.3 gives

(7.10)
$\displaystyle(V_{\phi}^{m})_{l}^{s}(P_{\phi}^{m})_{k;(i}^{l}(g_{\rho})_{j)s}^{\prime}=2(P_{\phi}^{m})_{k;(i}^{l}(P_{\phi}^{m})_{j)l}.$

Combining Equations 7.4, 7.9 and 7.10 yields

${}^{g_{\rho}}\nabla_{k}(g_{\rho})_{ij}^{\prime}=2(P_{\phi}^{m})_{ij;k}.$

Since $dP_{\phi}^{m}=0$, we conclude that LABEL:curvature2 holds.

We now verify Equation 7.8a. Note that

$g_{\rho}-\frac{1}{2}\rho g_{\rho}^{\prime}=U_{\phi}^{m}.$

Thus

$\displaystyle(g_{\rho})_{i[l}(g_{\rho})_{k]j}^{\prime}+(g_{\rho})_{j[k}(g_{\rho})_{l]i}^{\prime}-\rho(g_{\rho})_{i[l}^{\prime}(g_{\rho})_{k]j}^{\prime}=(U_{\phi}^{m})_{i[l}(g_{\rho})_{k]j}^{\prime}+(U_{\phi}^{m})_{j[k}(g_{\rho})_{l]i}^{\prime}.$

Using Equation 7.3, we deduce that

(7.11)
$(g_{\rho})_{i[l}(g_{\rho})_{k]j}^{\prime}+(g_{\rho})_{j[k}(g_{\rho})_{l]i}^{\prime}-\rho(g_{\rho})_{i[l}^{\prime}(g_{\rho})_{k]j}^{\prime}\\\
=2(U_{\phi}^{m})_{i}^{a}(U_{\phi}^{m})_{j}{}^{b}\left(g_{a[l}(P_{\phi}^{m})_{k]b}+g_{b[k}(P_{\phi}^{m})_{l]a}\right).$

Since $A_{\phi}^{m}=0$, it holds that

(7.12)
${}^{g}R_{abkl}=-2\left(g_{a[l}(P_{\phi}^{m})_{k]b}+g_{b[k}(P_{\phi}^{m})_{l]a}\right).$

Combining Equations 7.5, 7.13 and 7.12 yields Equation 7.8a.∎

###### Proof of Theorem 1.2(ii).

Proposition 7.2 implies that
$\widetilde{\text{Ric}_{\phi}^{m}}=-\frac{m}{\widetilde{f}}\widetilde{\nabla}^{2}\widetilde{f}$.

We now prove that $\widetilde{\nabla}^{2}\widetilde{f}=0$. Recall
[FeffermanGraham2012] that the Christoffel symbols of $\widetilde{g}=2\rho
dt^{2}+t^{2}g_{\rho}+2\rho dt^{2}$ are

(7.13)
$\displaystyle\begin{split}\widetilde{\Gamma}_{IJ}^{0}&=\left(\begin{array}[]{ccc}0&0&0\\\
0&-\frac{1}{2}t(g_{\rho})_{ij}^{\prime}&0\\\ 0&0&0\end{array}\right),\\\
\widetilde{\Gamma}_{IJ}^{k}&=\left(\begin{array}[]{ccc}0&t^{-1}\delta_{j}^{k}&0\\\
t^{-1}\delta_{i}^{k}&\Gamma_{ij}^{k}&\frac{1}{2}(g_{\rho})^{kl}(g_{\rho})_{il}^{\prime}\\\
0&\frac{1}{2}(g_{\rho})^{kl}(g_{\rho})_{jl}^{\prime}&0\end{array}\right),\\\
\widetilde{\Gamma}_{IJ}^{\infty}&=\left(\begin{array}[]{ccc}0&0&t^{-1}\\\
0&-(g_{\rho})_{ij}+\rho(g_{\rho})_{ij}^{\prime}&0\\\
t^{-1}&0&0\end{array}\right).\end{split}$

Direct computation immediately yields

$(\widetilde{\nabla}^{2}\widetilde{f})_{0I}=(\widetilde{\nabla}^{2}\widetilde{f})_{\infty\infty}=0.$

We next prove that $(\widetilde{\nabla}^{2}\widetilde{f})_{i\infty}=0$.
Observe that by Lemma 2.1,

(7.14)
$\frac{1}{m}\partial_{i}(fY_{\phi}^{m})=(P_{\phi}^{m})_{i}^{k}\partial_{k}f,$

and by Equation 7.3,

$\frac{1}{2}g^{kl}g^{\prime}_{li}=(V_{\phi}^{m})^{k}_{s}(P_{\phi}^{m})^{s}_{i}.$

We then compute that

$\displaystyle\frac{1}{t}(\widetilde{\nabla}^{2}\widetilde{f})_{i\infty}$
$\displaystyle=\partial_{i}\left(\frac{Y_{\phi}^{m}}{m}f\right)-\widetilde{\Gamma}_{i\infty}^{k}\partial_{k}\left(f+\rho\frac{Y_{\phi}^{m}}{m}f\right)$
$\displaystyle=(P_{\phi}^{m})^{k}_{i}\partial_{k}f-(V_{\phi}^{m})^{k}_{s}(P_{\phi}^{m})^{s}_{i}\partial_{k}f-\rho(V_{\phi}^{m})^{k}_{s}(P_{\phi}^{m})^{s}_{i}(P_{\phi}^{m})^{l}_{k}\partial_{l}f.$

The conclusion follows from the identity
$-\rho(P_{\phi}^{m})^{l}_{k}=\delta^{l}_{k}-(U_{\phi}^{m})^{l}_{k}.$

We now prove that $(\widetilde{\nabla}^{2}\widetilde{f})_{ij}=0$. Note that

$\frac{1}{t}\widetilde{\nabla}_{ij}^{2}f={}^{g_{\rho}}\nabla_{ij}^{2}(f+\frac{f}{m}\rho
Y_{\phi}^{m})-\frac{1}{t}\Gamma_{ij}^{0}(f+\frac{f}{m}\rho
Y_{\phi}^{m})-\frac{1}{m}\Gamma_{ij}^{\infty}fY_{\phi}^{m}.$

Now applying Equations 7.3 and 7.4 yields

(7.15)
$\frac{1}{t}(\widetilde{\nabla}^{2}\widetilde{f})_{ij}={}^{g}\nabla^{2}_{ij}(f+\frac{f}{m}Y_{\phi}^{m}\rho)-\rho(V_{\phi}^{m})_{l}^{k}(P_{\phi}^{m})^{l}_{i;j}\partial_{k}(f+\frac{f}{m}Y_{\phi}^{m}\rho)\\\
+f((P_{\phi}^{m})_{ik}+\frac{1}{m}Y_{\phi}^{m}g_{ik})(U_{\phi}^{m})_{j}^{k}.$

Equation 7.14 implies that

(7.16) $\displaystyle{}^{g}\nabla^{2}_{ij}(fY_{\phi}^{m})$
$\displaystyle=m\big{[}(P_{\phi}^{m})^{k}_{i;j}\partial_{k}f+(P_{\phi}^{m})_{i}^{k}\nabla^{2}_{jk}f].$

Also, Lemma 2.1 implies that

(7.17) $\displaystyle{}^{g}\nabla^{2}_{ij}f$
$\displaystyle=-f(P_{\phi}^{m})_{ij}-\frac{f}{m}Y_{\phi}^{m}g_{ij}.$

Equations 7.14, 7.15, 7.16 and 7.17, put together, yield
$(\widetilde{\nabla}^{2}\widetilde{f})_{ij}=0$. Hence
$(\widetilde{\text{Ric}_{\phi}^{m}})_{IJ}=0$.

We now prove that $\widetilde{F_{\phi}^{m}}=0$. Since
$(\widetilde{\nabla}^{2}\widetilde{f})_{IJ}=0$, it holds that
$\widetilde{\Delta}\widetilde{f}=0$. For $m=1$, we have
$\widetilde{F_{\phi}^{m}}=0$. Suppose now that $m\neq 1$. Since
$(\widetilde{\nabla}^{2}\widetilde{f})_{I\infty}=0$, it holds that
$\partial_{\rho}(|\widetilde{\nabla}\widetilde{f}|^{2}-\mu)=0$. Hence, it
suffices to show that
${\left.\kern-1.2pt(|\widetilde{\nabla}\widetilde{f}|^{2}-\mu)\vphantom{\big{|}}\right|_{\rho=0}}=0$.
A direct computation yields

$({\left.\kern-1.2pt|\widetilde{\nabla}\widetilde{f}|^{2}-\mu)\vphantom{\big{|}}\right|_{\rho=0}}=2\frac{f^{2}}{m}Y_{\phi}^{m}+|\nabla
f|^{2}-\mu.$

On the one hand, it holds that [Case2014s]

(7.18) $\frac{f^{2}}{m}Y_{\phi}^{m}=-\frac{1}{d+m-2}\left[(m-1)(|\nabla
f|^{2}-\mu)+f\Delta f+f^{2}J_{\phi}^{m}\right].$

On the other hand, the trace of Equation 2.2b gives

(7.19) $f\Delta f+f^{2}J_{\phi}^{m}+\frac{d-m}{m}f^{2}Y_{\phi}^{m}=0.$

Combining Equation 7.18 with Equation 7.19 yields
$\widetilde{F_{\phi}^{m}}=0$.

As both $\widetilde{\text{Ric}_{\phi}^{m}}_{IJ}=0$ and
$\widetilde{F_{\phi}^{m}}=0$, we conclude that
$(\widetilde{\mathcal{G}},\widetilde{g},\widetilde{f},m,\mu)$ is a weighted
ambient metric for $(M^{d},g,f,m,\mu)$. ∎

###  7.3. The Gover–Leitner conditions

Let $(M^{d},g,1,m,\mu),d\geq 3,$ be a smooth metric measure space such that

(7.20) $\text{Ric}_{\phi}^{m}=-(d-1)\mu g.$

Then $F_{\phi}^{m}=-(m-1)\mu$.

Set $\lambda=-\mu/2$ and $g_{\rho}=(1+\lambda\rho)^{2}g$ and
$f_{\rho}=1-\lambda\rho$. Define $(\widetilde{g},\widetilde{f})$ as in
Equation 1.1. It follows immediately from Equation 6.1 that
$\widetilde{\text{Ric}}_{\phi}^{m}=0$ and $\widetilde{F}_{\phi}^{m}=0$. Hence,
$(\widetilde{G},\widetilde{g},\widetilde{f},m,\mu)$ is the weighted ambient
metric of $(M,g,1,m,\mu)$. Moreover, Equation 7.13 and Equation 7.20 imply
that

(7.21) $\displaystyle\begin{split}P_{\phi}^{m}&=\lambda g,\\\
Y_{\phi}^{m}&=-m\lambda.\end{split}$

This verifies Theorem 1.2(iii).

## Acknowledgements

JSC was supported by the Simons Foundation (Grant #524601).

## References