# Turán Inequalities for Infinite Product Generating Functions

Bernhard Heim  Lehrstuhl A für Mathematik, RWTH Aachen University, 52056
Aachen, Germany<EMAIL_ADDRESS>and  Markus Neuhauser  Kutaisi
International University, 5/7, Youth Avenue, Kutaisi, 4600 Georgia  Lehrstuhl
A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
<EMAIL_ADDRESS>
###### Abstract.

In the $1970$s, Nicolas proved that the partition function $p(n)$ is log-
concave for $n>25$. In [HNT21], a precise conjecture on the log-concavity for
the plane partition function $\mathop{\rm pp}(n)$ for $n>11$ was stated. This
was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a
general picture. We associate to double sequences $\\{g_{d}(n)\\}_{d,n}$ with
$g_{d}(1)=1$ and

$0\leq g_{d}\left(n\right)-n^{d}\leq
g_{1}\left(n\right)\left(n-1\right)^{d-1}$

polynomials $\\{P_{n}^{g_{d}}(x)\\}_{d,n}$ given by

$\sum_{n=0}^{\infty}P_{n}^{g_{d}}(x)\,q^{n}:=\mathop{\rm
exp}\left(x\sum_{n=1}^{\infty}g_{d}(n)\frac{q^{n}}{n}\right)=\prod_{n=1}^{\infty}\left(1-q^{n}\right)^{-xf_{d}(n)}.$

We recover $p(n)=P_{n}^{\sigma_{1}}(1)$ and $\mathop{\rm
pp}\left(n\right)=P_{n}^{\sigma_{2}}(1)$, where $\sigma_{d}(n):=\sum_{\ell\mid
n}\ell^{d}$ and $f_{d}(n)=n^{d-1}$. Let $n\geq 6$. Then the sequence
$\\{P_{n}^{\sigma_{d}}(1)\\}_{d}$ is log-concave for almost all $d$ if and
only if $n$ is divisible by $3$. Let $\mathop{\rm id}(n)=n$. Then
$P_{n}^{\mathop{\rm id}}(x)=\frac{x}{n}L_{n-1}^{(1)}(-x)$, where
$L_{n}^{\left(\alpha\right)}\left(x\right)$ denotes the $\alpha$-associated
Laguerre polynomial. In this paper, we invest in Turán inequalities

$\Delta_{n}^{g_{d}}(x):=\left(P_{n}^{g_{d}}(x)\right)^{2}-P_{n-1}^{g_{d}}(x)\,P_{n+1}^{g_{d}}(x)\geq
0.$

Let $n\geq 6$ and $0\leq x<2-\frac{12}{n+4}$. Then $n$ is divisible by $3$ if
and only if $\Delta_{n}^{g_{d}}(x)\geq 0$ for almost all $d$. Let $n\geq 6$
and $n\not\equiv 2\pmod{3}$. Then the condition on $x$ can be reduced to
$x\geq 0$. We determine explicit bounds. As an analogue to Nicolas’ result, we
have for $g_{1}=\mathop{\rm id}$ that $\Delta_{n}^{\mathop{\rm id}}(x)\geq 0$
for all $x\geq 0$ and all $n$.

###### Key words and phrases:

Generating functions, Log-concavity, Turán inequalities, Partition numbers.

###### 2010 Mathematics Subject Classification:

Primary 05A17, 11P82; Secondary 05A20

##  1\. Introduction and main results

In this paper, we study Turán inequalities
$p_{n}(x)^{2}-p_{n-1}(x)\,p_{n+1}(x)\geq 0$ for families of polynomials
$\\{p_{n}(x)\\}_{n}$ attached to arithmetic functions.

Our work is motivated by a recent result by Ono, Pujahari, and Rolen [OPR22]
and [HN22]. Ono, Pujahari, and Rolen proved the log-concavity conjecture
([HNT21], Conjecture 1) for plane partitions $\mathop{\rm pp}\left(n\right)$
for $n>11$. Twenty-four years ago, Nicolas [Ni78] had already proved the log-
concavity property for the partition numbers $p(n)$ for $n>25$. This result
was reproved by DeSalvo and Pak [DP15]. For an introduction to partition
numbers and plane partition numbers, we refer to Andrews’ book [An98].
Further, to study the concept of log-concavity and related topics, Brenti
[Br89] and Stanley [St89, St99] are suitable references.

This paper is also a significant generalization of our previous result in
[HN22]. Let
$a_{d}(n):=\frac{1}{n}\sum_{k=1}^{n}\left(\sum_{\ell|k}\ell^{d}\right)\,a_{d}(n-k)$,
with $a_{d}(0)=1$. Let $n\geq 6$ be fixed. Then the sequence
$\left\\{a_{d}\left({n}\right)\right\\}_{d}$ is log-concave for almost all
$\in\mathbb{N}$ if and only if $n\equiv 0\pmod{3}$. Note that $p(n)=a_{1}(n)$
and $\mathop{\rm pp}(n)=a_{2}(n)$. The quantities $p(n)$ and $\mathop{\rm
pp}\left(n\right)$ are induced by certain arithmetic functions. This leads to
the following generalization.

######  Definition.

Let $\mathbb{D}$ be the set of all double sequences $\\{g_{d}(n)\\}_{d,n\geq
1}$ with normalization $g_{d}(1)=1$, such that
$\sum_{n=1}^{\infty}g_{d}(n)\,q^{n-1}$ is regular at $q=0$ with radius of
convergence $R$, and

$0\leq g_{d}\left(n\right)-n^{d}\leq
g_{1}\left(n\right)\,\left(n-1\right)^{d-1}$

for all $d$ and $n$.

We invest in sequences of polynomials $\\{P_{n}^{g_{d}}(x)\\}_{n}$, defined by
the recurrence relation:

$P_{n}^{g_{d}}(x):=\frac{x}{n}\,\sum_{k=1}^{n}g_{d}(k)\,P_{n-k}^{g_{d}}(x),\qquad\text{with
}P_{0}^{g_{d}}(x):=1.$

We have the generating series

$\sum_{n=0}^{\infty}P_{n}^{g_{d}}(x)\,q^{n}=\mathop{\rm
exp}\left(x\sum_{n=1}^{\infty}g_{d}(n)\frac{q^{n}}{n}\right)=\prod_{n=1}^{\infty}\left(1-q^{n}\right)^{-x\,f_{d}(n)},$

where $n\,f_{d}(n)=\sum_{\ell\mid n}\mu(\ell)\,g_{d}(n/\ell)$ with $\mu$ the
Moebius function. Examples for $g_{d}(n)$ are $\\{\sigma_{d}(n)\\}$ and
$\\{\psi_{d}(n)\\}$, where $\sigma_{d}(n)=\sum_{\ell\mid n}\ell^{d}$ and
$\psi_{d}(n)=n^{d}$. Turán’s inequality of $\\{P_{n}^{g_{d}}(x)\\}$ at $n$ for
a subset of $\mathbb{R}$ is defined by

$\Delta_{n}^{g_{d}}(x):=\left(P_{n}^{g_{d}}(x)\right)^{2}-P_{n-1}^{g_{d}}(x)\,P_{n+1}^{g_{d}}(x)\geq
0.$

Let $x_{0}$ be fixed, we call $P_{n}^{g_{d}}(x_{0})$ log-concave at $n$ if
$\Delta_{n}^{g_{d}}(x_{0})\geq 0$.

We note that the partition function and the plane partition function satisfy
$p(n)=P_{n}^{\sigma_{1}}(1)$ and $\mathop{\rm
pp}\left(n\right)=P_{n}^{\sigma_{2}}(1)$. Let $E^{g_{d}}$ be the set of all
$n\in\mathbb{N}$ with $\Delta_{n}^{g_{d}}(1)<0$, denoted as strictly log-
convex.

Nicolas [Ni78] proved that the partition function $p(n)$ is log-concave for
almost all $n$. The set of exceptions is given by
$E^{\sigma_{1}}=\left\\{2k+1\,:\,0\leq k\leq 12\right\\}$. Ono, Pujahari, and
Rolen [OPR22] proved that $E^{\sigma_{2}}=\\{1,3,5,7,9,11\\}$. Numerical
investigations [HN22] for $n\leq 10^{5}$ indicate that
$E^{\sigma_{3}}=\\{1,3,5,7\\}$, $E^{\sigma_{4}}=E^{\sigma_{5}}=\\{1,5\\}$.
Surprisingly, $E^{\sigma_{20}}$ has at least $10$ elements. We believe that
the general and clean patterns associated to double sequences in $\mathbb{D}$
are displayed by $g_{d}(n)=\psi_{d}(n)$ (see Table 1).

$\begin{array}[]{r|cccccccccccccccccccc}\hline\cr n\backslash
d&{1}&{2}&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\\ \hline\cr\hline\cr
1&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\\
2&&&&&&&&&&&&&&&&&&\\\ 3&&&&&&&&&&&&&&&&&&\\\
4&&&&&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\\
5&&&&&&&&&&&&&&&&&&\\\ 6&&&&&&&&&&&&&&&&&&\\\
7&&&&&&&&&&&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\\
8&&&&&&&&&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\\
9&&&&&&&&&&&&&&&&&&\\\ 10&&&&&&&&&&&&&&&&\bullet&\bullet&\bullet\\\
11&&&&&&&&&&&&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\\
12&&&&&&&&&&&&&&&&&&\\\ 13&&&&&&&&&&&&&&&&&&\\\
14&&&&&&&&&&&&&&&\bullet&\bullet&\bullet&\bullet\\\ \hline\cr\end{array}$
Table 1.  Exceptions for $g_{d}\left(n\right)=n^{d}$, $1\leq d\leq 18$ and
$1\leq n\leq 14$

In our main result we capture the impact of the residue of $n$ divided by $3$
and the range of the argument of the $\Delta_{n}^{g_{d}}(x)$.

######  Theorem 1.1.

Let $\\{g_{d}(n)\\}$ be a double sequence in $\mathbb{D}$. Let $n\geq 6$.
Moreover let

(1.1)
$\Delta_{n}^{g_{d}}(x)=\left(P_{n}^{g_{d}}(x)\right)^{2}-P_{n-1}^{g_{d}}(x)\,P_{n+1}^{g_{d}}(x)\geq
0$

be the Turán inequality.

  * a)

Let $0\leq x<2-\frac{12}{n+4}$. Then (1.1) holds true for almost all $d$ if
and only if $n$ is divisible by $3$.

  * b)

Let $n\not\equiv 2\pmod{3}$ and $x\geq 0$. Then (1.1) holds true for almost
all $d$ if and only if $n$ is divisible by $3$.

The case $g_{d}\left(n\right)=\sigma_{d}\left(n\right)$ and $x=1$ leads to the
results obtained in [HN22], Theorem 1.2 and Theorem 1.3. An explicit analysis
of the bounds obtained in the proof of Theorem 1.1 leads to the following:

######  Theorem 1.2.

Let $\left\\{g_{d}\left(n\right)\right\\}$ be a double sequence in
$\mathbb{D}$. Let $n\geq 3$ and $n\neq 5$. Let $R$ be the radius of
convergence of $\sum_{n=1}^{\infty}g_{1}(n)\,\frac{q^{n}}{n}$. For each $x$,
let $r(x)$ be chosen with $0<r(x)<R$ and $P_{n}^{g_{1}}(x)\leq r(x)^{-n}$ for
all $n$. Then we have the following properties.

  * (i)

Let $n\equiv 0\pmod{3}$ and $x>0$. Then $\Delta_{n}^{g_{d}}(x)\geq 0$ for
$d\geq d_{0}\left(n,x\right)$, where

$d_{0}\left(n,x\right)=1+\frac{2n}{3\ln\left(9/8\right)}\left(\ln\left(n/3\right)-\ln\left(x\right)-3\ln\left(r(x)\right)\right).$

  * (ii)

Let $n\equiv 1\pmod{3}$ and $x>0$. Then $\Delta_{n}^{g_{d}}(x)<0$ for $d\geq
d_{0}\left(n,x\right)$, where

$d_{0}\left(n,x\right)=1+\frac{2n}{3\ln\left(9/8\right)}\left(\ln\left(\frac{n-1}{3}\right)-\ln\left(x\right)-3\ln\left(r(x)\right)\right).$

  * (iii)

Let $n\equiv 2\pmod{3}$ and $0<x<2-\frac{12}{n+4}$. Then
$\Delta_{n}^{g_{d}}(x)<0$ for $d\geq d_{0}\left(n,x\right)$, where

$\displaystyle d_{0}\left(n,x\right)$ $\displaystyle=$ $\displaystyle
1+\frac{1}{\ln\left(9/8\right)}\left(-\ln\left(\frac{n-2}{3n+3}\left(\frac{1}{x}+\frac{1}{2}\right)-\frac{1}{3}\right)\right.$
$\displaystyle\left.+\,\frac{n-2}{3}\ln\left(\frac{n-2}{3}\right)-\frac{n+1}{3}\ln\left(x\right)-n\ln\left(r(x)\right)\right).$

######  Remarks.

a) The positive real number $r(x)$ always exists due to Cauchy–Hadamard’s
theorem.
b) Let $g_{d}(n)=n^{d}$. Then $\Delta_{4}^{g_{5}}(x)$ has sign changes for
positive real $x$, since there are two positive, real zeros
$\alpha_{1}<\alpha_{2}$.

##  2\. Records

Let $g_{d}(n)=\sigma_{d}(n)$. Then for $d=1$ and $d=2$ complete results for
the log-concavity $\Delta_{n}^{g_{d}}(1)\geq 0$, including the explicit
$E^{\sigma_{d}}$, are provided by Nicolas [Ni78] and Ono–Pujahari–Rolen
[OPR22]. For $n\leq 10^{5}$ and $d\leq 8$ further results have been obtained
by Heim–Neuhauser [HN22].

Let $g_{d}(n)=\psi_{d}(n)$. In Table 2 we have displayed the results for
$1\leq d\leq 9$. In this paper, we prove the analogue to Nicolas’ result and
give some numerical evidence for the case $d=2$, which is for
$\sigma_{2}\left(n\right)$ the log-concavity for plane partitions.

$d$ | log-concave | strictly log-convex | proof / verification
---|---|---|---
$1$ | $n>1$ | $n=1$ | Heim–Neuhauser 2022
$2$ | $n>1$ | $n=1$ | H–N 2022 ($n\leq 10^{4}$)
$3$ | $n>1$ | $n=1$ |
$4$ | $n>1$ | $n=1$ |  xxxxxx⋮
$5$ | $n>4,\\{2,3\\}$ | $\\{1,4\\}$ |  xxxxxx⋮
$6$ | $n>4,\\{2,3\\}$ | $\left\\{1,4\right\\}$ |  xxxxxx⋮
$7$ | $n>4,\\{2,3\\}$ | $\left\\{1,4\right\\}$ |  xxxxxx⋮
$8$ | $n>4,\\{2,3\\}$ | $\left\\{1,4\right\\}$ |  xxxxxx⋮
$9$ | $n>8,\\{2,3,5,6,7\\}$ | $\\{1,4,8\\}$ | H–N 2022 ($n\leq 10^{4}$)
Table 2.  Properties of $\\{P_{n}^{\psi_{d}}(1)\\}$

More generally, let $\\{g_{d}(n)\\}$ be a double sequence in $\mathbb{D}$.
Then $\Delta_{1}^{g_{d}}(1)$ is always negative, since

$\Delta_{1}^{g_{d}}(x)=\frac{x}{2}\,\left(x-g_{d}(2)\right)$

and $g_{d}(2)\geq 2$. This explains the results for $n=1$ at $x=1$.

For $x>0$ we have $\Delta_{1}^{\psi_{d}}\left(x\right)\geq 0$ if and only if
$x\geq 2^{d}$. Let $n\geq 2$. Table 3 records our results for Turán
inequalities for small $d$.

$d$ | Turán inequality | proof / verification
---|---|---
$1$ |  $\Delta_{n}^{\mathop{\rm id}}(x)\geq 0$ for $n>1$ and $x\in\mathbb{R}$ | Heim–Neuhauser 2022
$2,3,4$ |  $\Delta_{n}^{\psi_{d}}(x)\geq 0$ for $n>1$ and $x\in\mathbb{R}$ | H–N 2022 ($n\leq 10^{2}$)
Table 3.  Turán inequalities in the $d$ aspect

In the case $d=5$ a new feature appears (see Figure 2). Let $3\leq n\leq 100$
then there are exactly two simple positive zeros. Their position implies
$\Delta_{3}^{\psi_{5}}(1)>0$, $\Delta_{4}^{\psi_{5}}(1)<0$ and
$\Delta_{n}^{\psi_{5}}(1)>0$ for $5\leq n\leq 100$. We expect that this is
true for all $n\geq 5$.  Figure 1.  The zeros of $\Delta_{n}^{\psi_{2}}(x)$
with the largest positive real part for $1\leq n\leq 40$. Blue labels the real
zeros and red the imaginary zeros.

Figure 2.  The zeros of $\Delta_{n}^{\psi_{5}}(x)$ with the largest positive
real part for $1\leq n\leq 40$. Blue labels the real zeros and red the
imaginary zeros.

##  3\. Basic formulas

Let $g$ be a normalized arithmetic function. Let

$P_{n}^{g}(x):=\frac{x}{n}\sum_{k=1}^{n}g(k)\,P_{n-k}^{g}(x),\qquad\text{with
}P_{0}^{g}\left(x\right)=1.$

Then $P_{n}^{g}(x)$ are polynomials of degree $n$. We refer to [HN20] for a
detailed study of these polynomials. For example $P_{1}^{g}(x)=x$ and
$P_{2}^{g}(x)=x/2\,(x+g(2))$.

###  3.1. Coefficients of $P_{n}^{g}(x)$

Let

$P_{n}^{g}(x)=\sum_{k=0}^{n}A_{n,k}^{g}\,\,x^{k}.$

Then $A_{0,0}^{g}=1$. Let $n\geq 1$ then $A_{n,0}^{g}=0$,
$A_{n,1}^{g}=g(n)\,/\,n$ and $A_{n,n}^{g}=1\,/\,n!$. We also have [HN20] for
$1\leq m<n$ and $n-m=1,2,3$:

$\displaystyle A_{n,n-1}^{g}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\,\,g\left(2\right)\binom{n}{2},$ $\displaystyle
A_{n,n-2}^{g}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\,\,\left[3\left(g\left(2\right)\right)^{2}\binom{n}{4}+2g\left(3\right)\binom{n}{3}\right],$
$\displaystyle A_{n,n-3}^{g}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\,\,\left[15\left(g\left(2\right)\right)^{3}\binom{n}{6}+20g\left(2\right)g\left(3\right)\binom{n}{5}+6g\left(4\right)\binom{n}{4}\right].$

######  Lemma 3.1.

Let $g$ be a normalized arithmetic function and

$\Delta_{n}^{g}(x):=P_{n}^{g}(x)^{2}-P_{n-1}^{g}(x)\,P_{n+1}^{g}(x).$

Then $\Delta_{1}^{g}(x)=\frac{x}{2}\left(x-g(2)\right)$ and
$\Delta_{2}^{g}(x)=\frac{x^{2}}{12}\left(x^{2}+3\,g(2)^{2}-4\,g(3)\right)$.
Further,

$\displaystyle\Delta_{3}^{g}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{144}x^{6}+\frac{1}{48}g\left(2\right)x^{5}+\left(\frac{1}{16}\left(g\left(2\right)\right)^{2}-\frac{1}{18}g\left(3\right)\right)x^{4}+(-\frac{1}{16}\left(g\left(2\right)\right)^{3}$
$\displaystyle{}+\frac{1}{6}g\left(3\right)g\left(2\right)-\frac{1}{8}g\left(4\right))x^{3}+\left(-\frac{1}{8}g\left(4\right)g\left(2\right)+\frac{1}{9}\left(g\left(3\right)\right)^{2}\right)x^{2}.$

This follows from the explicit form of the polynomials. We have

$\displaystyle P_{3}^{g}(x)$ $\displaystyle=$
$\displaystyle\frac{x}{6}\left(x^{2}+3\,g(2)\,x+2\,g(3)\right),$
$\displaystyle P_{4}^{g}(x)$ $\displaystyle=$
$\displaystyle\frac{x}{24}\left(x^{3}+6\,g(2)\,x^{2}+\left(8\,g(3)+3\,g(2)^{2}\right)\,x+6\,g(4)\right).$

###  3.2. Properties of $\Delta_{n}^{g}(x)$

Let us establish the following notation:

$\Delta_{n}^{g}(x)=\sum_{k=0}^{2n}D_{n,k}^{g}\,\,x^{k}.$

In contrast to $P_{n}^{g}(x)$, the coefficients of $\Delta_{n}^{g}(x)$ are not
always non-negative in general. Nevertheless, we have $\Delta_{n}^{g}(0)=0$
and the important asymptotic property

$\lim_{x\to\infty}\Delta_{n}^{g}(x)=\infty.$

This follows from $D_{n,2n}=\frac{1}{(n!)^{2}\,(n+1)}$. Let $n\geq 2$. We can
always factor out $x^{2}$ and still have polynomials, since
$D_{n,0}^{g}=D_{n,1}^{g}=0$. The new constant term is given by $D_{n,2}^{g}$,
which does not need to be non-negative:

$\displaystyle D_{n,2}^{g}$ $\displaystyle=$
$\displaystyle\frac{1}{n^{2}}\,\left[g(n)^{2}-\frac{n^{2}}{n^{2}-1}\,g(n-1)\,g(n+1)\right],$
$\displaystyle D_{n,3}^{g}$ $\displaystyle=$ $\displaystyle
2\frac{g\left(n\right)}{n^{2}}\sum_{k=1}^{n-1}\frac{g\left(k\right)g\left(n-k\right)}{k}-\frac{g\left(n-1\right)}{n-1}\sum_{k=1}^{n}\frac{g\left(n+1-k\right)g\left(k\right)}{2\left(n+1-k\right)k}$
$\displaystyle{}-\frac{g\left(n+1\right)}{n+1}\sum_{k=1}^{n-2}\frac{g\left(n-1-k\right)g\left(k\right)}{2\left(n-1-k\right)k}.$

###  3.3. Special cases

Let $\\{g_{d}(n)\\}$ be a double sequence in $\mathbb{D}$. We have
$\Delta_{1}^{g_{d}}(x)=x\,(x-g_{d}(2))/2$. Thus, $\Delta_{1}^{g_{d}}(x)=0$ if
$x=0$ or $x=g_{d}(2)$. Thus, $\Delta_{1}^{g_{d}}(x)>0$ if and only if
$x\not\in[0,g_{d}(2)]$. This implies that $\Delta_{1}^{g_{d}}(x)<0$ for
$x\in\left(0,g_{d}\left(2\right)\right)$ and all $d\in\mathbb{N}$. The case
$n=2$ is still directly accessible. We have $\Delta_{2}^{g_{d}}(x)=0$ if $x=0$
or $x^{2}=4\,g_{d}(3)-3\,\left(g_{d}\left(2\right)\right)^{2}$. We consider
$4\,g_{d}(3)-3\,\left(g_{d}\left(2\right)\right)^{2}\geq 0$ and $x\neq 0$. Let
$g_{d}=\psi_{d}$ or $g_{d}=\sigma_{d}$. Then $\Delta_{2}^{\psi_{d}}(x)>0$ for
$d\in\mathbb{N}$, especially $\Delta_{2}^{\psi_{1}}(x)=x^{4}/12$.

##  4\. Proof of Theorem 1.1

Our strategy is to utilize the well-known formula ([Ko04], section 4.7):

$P_{n}^{g_{d}}\left(x\right)=\sum_{k\leq
n}\sum_{\begin{subarray}{c}m_{1},\ldots,m_{k}\geq 1\\\
m_{1}+\ldots+m_{k}=n\end{subarray}}\frac{1}{k!}\,\frac{g_{d}\left(m_{1}\right)\cdots
g_{d}\left(m_{k}\right)}{m_{1}\cdots m_{k}}\,\,x^{k}.$

###  4.1. Lower and upper bounds

In Section 3 [HN22] we have obtained lower and upper bounds for
$P_{n}^{\sigma_{d}}(1)$. The invented proof method can be generalized in a
straightforward manner to obtain the following result for all double sequences
in $\mathbb{D}$ and the associated polynomials for $x>0$.

######  Proposition 4.1.

Let the double sequence $\\{g_{d}(n)\\}_{d,n\in\mathbb{N}}$ be an element of
$\mathbb{D}$. Let $n\geq 3$ and $x>0$. Then we have for all $d\geq 1$ the
following upper and lower bounds.

Let $n\equiv 0\pmod{3}$ and $n^{\prime}:=n/3$. Then

$\frac{3^{\left(d-1\right)n^{\prime}}}{\left(n^{\prime}\right)!}\,\,x^{n^{\prime}}\,<\,P_{n}^{g_{d}}(x)\,\leq\,3^{\left(d-1\right)n^{\prime}}P_{n}^{g_{1}}(x).$

Further, let $n\equiv 1\pmod{3}$ and $n^{\prime}:=(n-4)/3$. Then

$\frac{\left(4\cdot
3^{n^{\prime}}\right)^{d-1}}{\left(n^{\prime}\right)!}\,\left(x^{n^{\prime}+1}+\frac{x^{n^{\prime}+2}}{2}\right)\,<\,P_{n}^{g_{d}}(x)\,\leq\,\left(4\cdot
3^{n^{\prime}}\right)^{d-1}\,\,P_{n}^{g_{1}}(x).$

Further, let $n\equiv 2\pmod{3}$ and $n^{\prime}:=(n-2)/3$. Then

$\frac{\left(2\cdot
3^{n^{\prime}}\right)^{d-1}}{\left(n^{\prime}\right)!}\,\,x^{n^{\prime}+1}\,<\,P_{n}^{g_{d}}(x)\,\leq\,\left(2\cdot
3^{n^{\prime}}\right)^{d-1}P_{n}^{g_{1}}(x).$

Additionally, let $n\equiv 2\pmod{3}$ and $n\geq 8$. Let
$n^{\prime}:=(n-2)/3$. Then

$P_{n}^{g_{d}}(x)\leq\frac{\left(2\cdot
3^{n^{\prime}}\right)^{d-1}}{\left(n^{\prime}\right)!}\,x^{n^{\prime}+1}+\left(16\cdot
3^{n^{\prime}-2}\right)^{d-1}\,P_{n}^{g_{1}}(x).$

###  4.2. Proof of Theorem 1.1

We apply Proposition 4.1.

####  4.2.1. The case $n\equiv 0\pmod{3}$

In the first step we apply Proposition 4.1. This leads to

$\frac{\left(P_{n}^{g_{d}}\left(x\right)\right)^{2}}{P_{n-1}^{g_{d}}\left(x\right)P_{n+1}^{g_{d}}\left(x\right)}\geq\frac{x^{2n/3}}{\left(\left(n/3\right)!\right)^{2}P_{n-1}^{g_{1}}\left(x\right)P_{n+1}^{g_{1}}\left(x\right)}\left(\frac{9}{8}\right)^{d-1}.$

We choose $r(x)>0$, such that $P_{n}^{g_{1}}(x)\leq r(x)^{-n}$ for all $n$.
Let

$d_{0}=d_{0}\left(n,x\right)=1+\frac{2n}{3\ln\left(9/8\right)}\left(\ln\left(n/3\right)-\ln\left(x\right)-3\ln\left(r\left({x}\right)\right)\right).$

Then $\left\\{P_{n}^{g_{d}}\left(x\right)\right\\}_{n}$ is strictly log-
concave at $n$ for $d\geq d_{0}$, since

$\frac{\left(P_{n}^{g_{d}}\left(x\right)\right)^{2}}{P_{n-1}^{g_{d}}\left(x\right)P_{n+1}^{g_{d}}\left(x\right)}\geq\frac{x^{2n/3}}{\left(n/3\right)^{2n/3}\left(r\left({x}\right)\right)^{-2n}}\left(\frac{9}{8}\right)^{d-1}>1.$

####  4.2.2. The case $n\equiv 1\pmod{3}$

In the first step we apply Proposition 4.1. This leads to

$\frac{\left(P_{n}^{g_{d}}\left(x\right)\right)^{2}}{P_{n-1}^{g_{d}}\left(x\right)P_{n+1}^{g_{d}}\left(x\right)}\,\leq\,\left(\left(\frac{n-1}{3}\right)!\right)^{2}\,\,P_{n}^{g_{1}}(x)^{2}\,\,x^{-2\left(n-1\right)/3}\,\,\left(\frac{8}{9}\right)^{d-1}.$

We choose $r(x)>0$, such that $P_{n}^{g_{1}}(x)\leq r(x)^{-n}$ for all $n$.
Let

$d_{0}=d_{0}\left(n,x\right)=1+\frac{2n}{3\ln\left(9/8\right)}\left(\ln\left(\frac{n-1}{3}\right)-3\ln\left(r\left({x}\right)\right)-\ln\left(x\right)\right).$

Then the sequence $\left\\{P_{n}^{g_{d}}\left(x\right)\right\\}_{n}$ is
strictly log-convex at $n$ for $d\geq d_{0}$, since

$\frac{\left(P_{n}^{g_{d}}\left(x\right)\right)^{2}}{P_{n-1}^{g_{d}}\left(x\right)P_{n+1}^{g_{d}}\left(x\right)}\leq\left(\frac{n-1}{3}\right)^{2n/3}\left(r\left({x}\right)\right)^{-2n}x^{2n/3}\left(\frac{8}{9}\right)^{d-1}<1.$

####  4.2.3. The case $n\equiv 2\pmod{3}$

This final case involves some additional considerations. Again we first apply
Proposition 4.1 and obtain

$\displaystyle\left(P_{n}^{g_{d}}\left(x\right)\right)^{2}-P_{n-1}^{g_{d}}\left(x\right)P_{n+1}^{g_{d}}\left(x\right)$
$\displaystyle\leq$ $\displaystyle\left(\frac{\left(2\cdot
3^{\left(n-2\right)/3}\right)^{d-1}x^{\left(n+1\right)/3}}{\left(\left(n-2\right)/3\right)!}+\left(16\cdot
3^{\left(n-8\right)/3}\right)^{d-1}P_{n}^{g_{1}}\left(x\right)\right)^{2}$
$\displaystyle{}-\frac{\left(4\cdot
3^{\left(n-5\right)/3}\right)^{d-1}\left(x^{\left(n-2\right)/3}+x^{\left(n+1\right)/3}/2\right)}{\left(\left(n-5\right)/3\right)!}\frac{3^{\left(d-1\right)\left(n+1\right)/3}x^{\left(n+1\right)/3}}{\left(\left(n+1\right)/3\right)!}$
$\displaystyle\leq$ $\displaystyle\left(\frac{\left(2\cdot
3^{\left(n-2\right)/3}\right)^{d-1}x^{\left(n+1\right)/3}}{\left(\left(n-2\right)/3\right)!}\right)^{2}\left(1-\frac{\left(n-2\right)/3}{\left(n+1\right)/3}\left(\frac{1}{x}+\frac{1}{2}\right)\right.$
$\displaystyle{}+2\left(8\cdot
3^{-2}\right)^{d-1}x^{-\left(n+1\right)/3}P_{n}^{g_{1}}\left(x\right)\left(\left(n-2\right)/3\right)!$
$\displaystyle{}\left.+\left(\left(\left(n-2\right)/3\right)!\left(8\cdot
3^{-2}\right)^{d-1}x^{-\left(n+1\right)/3}P_{n}^{g_{1}}\left(x\right)\right)^{2}\right).$

The last inequality can only be not larger than zero if
$0<x<2-\frac{12}{n+4}$. We choose $r(x)>0$ such that $P_{n}^{g_{1}}(x)\leq
r(x)^{-n}$ for all $n$. Then

$\left(\frac{n-2}{3}\right)!\left(\frac{8}{9}\right)^{d-1}x^{-\left(n+1\right)/3}P_{n}^{g_{1}}\left(x\right)\leq\left(\frac{n-2}{3}\right)^{\left(n-2\right)/3}\left(\frac{8}{9}\right)^{d-1}x^{-\left(n+1\right)/3}\left(r\left({x}\right)\right)^{-n}<1$

for

$d>1+\frac{1}{\ln\left(9/8\right)}\left(\frac{n-2}{3}\ln\left(\frac{n-2}{3}\right)-\frac{n+1}{3}\ln\left(x\right)-n\ln\left(r\left({x}\right)\right)\right).$

Let $d_{0}=d_{0}\left(n,x\right)$ be defined by

$\displaystyle
1+\frac{1}{\ln\left(9/8\right)}\left(-\ln\left(\frac{n-2}{3n+3}\left(\frac{1}{x}+\frac{1}{2}\right)-\frac{1}{3}\right)\right.$
$\displaystyle\phantom{1+\frac{1}{\ln\left(9/8\right)}\left(\right.}\left.{}+\frac{n-2}{3}\ln\left(\frac{n-2}{3}\right)-\frac{n+1}{3}\ln\left(x\right)-n\ln\left(r\left({x}\right)\right)\right).$

Then the sequence $\\{P_{n}^{g_{d}}(x)\\}_{d}$ is strictly log-convex for all
$d\geq d_{0}$.

##  5\. Turán inequalities

Let $\\{g_{d}(n)\\}$ be a double sequence in $\mathbb{D}$. We are interested
in finding the set of positive real numbers, such that
$\Delta_{n}^{g_{d}}(x)\geq 0$, with special emphasis on the behavior at $x=1$.

In [HN21] a conjecture for $\Delta_{n}^{\sigma_{1}}(x)$ was stated, which
generalized a conjecture of Chern–Fu–Tang [CFT18] related to integers $x\geq
2$. The Chern–Fu–Tang conjecture was proven by Bringmann, Kane, Rolen, and
Tripp [BKRT21]. Recently, a second conjecture [HNT21] was proposed for
$\Delta_{n}^{\sigma_{2}}(x)$. We have shown for $x=1$, the case of plane
partitions, that $\Delta_{n}^{\sigma_{2}}(1)>0$ for almost all $n$. Finally,
Ono, Pujahari, and Rolen [OPR22] have proven that
$\Delta_{n}^{\sigma_{2}}(1)>0$ for all $n\geq 12$. We now show that
$\Delta_{n}^{\psi_{1}}\left(x\right)\geq 0$ for all $x\in\mathbb{R}$. This is
the first case where a full result on Turán inequalities is obtained for a
double sequence in $\mathbb{D}$ with $d$ fixed.

###  5.1. $\Delta_{n}^{\psi_{1}}(x)\geq 0$

We have $\psi_{1}(n)=1$. The polynomials $P_{n}^{\psi_{1}}(x)$ had been
studied in [HLN19] and identified with the $\alpha$-associated Laguerre
polynomials. We have $P_{n}^{\psi_{1}}(x)=\frac{x}{n}L_{n-1}^{(1)}(-x)$, where

$\sum_{n=0}^{\infty}L_{n}^{(\alpha)}(x)\,t^{n}=\frac{1}{(1-t)^{\alpha+1}}\,\,\mathrm{e}^{-x\frac{t}{1-t}},\qquad\alpha>-1.$

The Laguerre polynomials of degree $n$ are given by $L_{n}(x)=L_{n}^{(0)}(x)$.
It is known that $\alpha$-associated Laguerre polynomials for $\alpha\geq 0$
satisfy:

$\left(L_{n}^{(\alpha)}(x)\right)^{2}-L_{n-1}^{(\alpha)}(x)\,L_{n+1}^{(\alpha)}(x)=\sum_{k=0}^{n-1}\frac{\binom{\alpha+n-1}{n-k}}{n\,\binom{n}{k}}\left(L_{k}^{(\alpha-1)}(x)\right)^{2}>0.$

These Turán inequalities are not sufficient to prove
$\Delta_{n}^{\psi_{1}}(x)\geq 0$. We have to show for all $x\in\mathbb{R}$
that

$\Delta_{n}^{\psi_{1}}\left(x\right)=\frac{1}{n^{2}}\left(L_{n-1}^{(1)}(-x)\right)^{2}-\frac{1}{n^{2}-1}L_{n-2}^{(1)}(-x)\,\,L_{n}^{(1)}(-x)\geq
0.$

Szegő [Sz48] proved in 1948, that $L_{n}^{(\alpha)}(x)/L_{n}^{(\alpha)}(0)$
satisfies Turán inequalities, where $L_{n}^{(1)}(0)=n+1$. This proves our
claim.

###  5.2. Challenges

We propose three open questions.

####  5.2.1. Log-concavity of partition and plane partition numbers

Reprove the results of Nicolas [Ni78] and Ono et al. [OPR22] on the log-
concavity of the partition numbers and the plane partition numbers utilizing
the zero distribution of the polynomials
$\left\\{P_{n}^{\sigma_{d}}\left(x\right)\right\\}$ for $d=1$ and $d=2$.

####  5.2.2. Turán inequalities $\Delta_{n}^{\psi_{d}}(x)\geq 0$

Based on our numerical investigations on the zeros of
$\Delta_{n}^{\psi_{d}}(x)$ we believe that it is very likely for $2\leq d\leq
4$ that $\Delta_{n}^{\psi_{d}}(x)\geq 0$ for $n\geq 2$ and $x\in\mathbb{R}$.
Prove this observation.

####  5.2.3. The case $n\equiv 0\pmod{3}$

The following problem was presented at the Conference: $100$ Years of Mock
Theta Functions in 2022 at Vanderbilt University (organized by Rolen and
Wagner). Prove that $\Delta_{n}^{\psi_{d}}\left(x\right)\geq 0$ for all
$n\equiv 0\pmod{3}$ and all $d\in\mathbb{N}$.

## References

  * [An98] G. E. Andrews: _The theory of partitions._ Cambridge Univ. Press, Cambridge (1998).
  * [Br89] F. Brenti: _Unimodal, log-concave and Pólya frequency sequences in combinatorics._ Mem. Am. Math. Soc. 413 (1989).
  * [BKRT21] K. Bringmann, B. Kane, L. Rolen, Z. Tripp: _Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser_. Trans. Am. Math. Soc. Ser. B 8 No. 21 (2021), 615–634.
  * [CFT18] S. Chern, S. Fu, D. Tang: _Some inequalities for $k$-colored partition functions._ Ramanujan J. 46 (2018), 713–725.
  * [DP15] S. DeSalvo, I. Pak: _Log-concavity of the partition function._ Ramanujan J. 38 (2015), 61–73.
  * [HLN19] B. Heim, F. Luca, M. Neuhauser: _Recurrence relations for polynomials obtained by arithmetic functions._ Int. Journal of Number Theory 15 No. 6 (2019), 1291–1303.
  * [HN20] B. Heim, M. Neuhauser: _Formulas for coefficients of polynomials assigned to arithmetic functions_ ,  arXiv:2010.07890v2 [math.NT].
  * [HN21] B. Heim, M. Neuhauser: _Polynomization of the Chern–Fu–Tang conjecture._ Res. Numb. Theory 7 No. 26 (2021).
  * [HN22] B. Heim, M. Neuhauser: _Log-concavity of infinite product generating functions_ ,  arXiv:2202.00627v1 [math.CO]. To appear in Research in Number Theory.
  * [HNT21] B. Heim, M. Neuhauser, R. Tröger: _Inequalities for plane partitions_ , arXiv:2109.15145v1 [math.CO].
  * [Ko04] B. Kostant: _Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra._ Invent. Math. 158 (2004), 181–226.
  * [Ni78] J.-L. Nicolas: _Sur les entiers $N$ pour lesquels il y a beaucoup des groupes abéliens d’ordre $N$._ Ann. Inst. Fourier 28 No. 4 (1978), 1–16.
  * [OPR22] K. Ono, S. Pujahari, L. Rolen: _Turán inequalities for the plane partition function_ , arXiv:2201.01352v1 [math.NT].
  * [St89] R. Stanley: _Log-concave and unimodal sequences in algebra, combinatorics, and geometry._ In: M. F. Capobianco, M. G. Guan, D. F. Hsu, F. Tian (eds.) Graph theory and its applications: East and West. Proceedings of the first China–USA international conference, held in Jinan, China, June 9–20, 1986. Ann. New York Acad. Sci. 576, New York Acad. Sci., New York (1989), 500–535.
  * [St99] R. Stanley: _Enumerative combinatorics._ Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin, Cambridge Studies in Advanced Mathematics 62 Cambridge University Press, Cambridge (1999).
  * [Sz48] G. Szegő: _On an inequality of P. Turán concerning Legendre polynomials._ Bulletin of the American Mathematical Society 54 (1948), 401–405.


# Ricci flow and volume renormalizability

Eric Bahuaud  Department of Mathematics, Seattle University, 901 12th Ave,
Seattle, WA 98122  bahuaude(at)seattleu.edu ,  Rafe Mazzeo  Department of
Mathematics, Stanford University, Stanford, CA 94305
mazzeo(at)math.stanford.edu and  Eric Woolgar  Department of Mathematical and
Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1,
Canada  ewoolgar(at)ualberta.ca

###### Abstract.

Relative to a special boundary defining function, conformally compact
asymptotically hyperbolic metrics have an expansion in a collar neighborhood
of conformal infinity. When this expansion is even to a certain order and
satisfies an additional auxiliary condition, a renormalized volume is defined.
We prove such expansions are preserved under the normalized Ricci flow. We
study the variation of curvature functionals in this setting. As an
application we obtain the following variation formula.

$\frac{d\,}{dt}\mathrm{RenV}(M^{n},g(t))=-\mathop{\vphantom{T}}^{R}\\!\\!\\!\int_{M^{n}}\left(S(g(t))+n(n-1)\right)\,dV_{g(t)},$

where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and
$\mathop{\vphantom{T}}^{R}\\!\\!\\!\int\cdot dV_{g}$ is the Riesz
renormalization. This extends our earlier work to a broader class of metrics.

###### Key words and phrases:

Ricci flow, conformally compact metrics, asymptotically hyperbolic metrics,
renormalized volume

A complete, conformally compactifiable Riemannian manifold $(M,g)$ is called
asymptotically hyperbolic if it is conformally isometric to the interior of a
smooth compact manifold $\overline{M}$ with boundary $\partial M$ with
curvatures tending to $-1$ at $\partial M$. Equivalently, $g$ can be written
as $g=\psi^{*}\rho^{-2}\overline{g}$, where $\overline{g}$ is smooth and
nondegenerate up to the boundary $\partial M$ defined by $\rho=0$, and
$|d\rho/\rho|^{2}_{g}\to 1$ at this boundary. We call $\rho$ a _boundary
defining function_ for $\partial M$ and call $(\partial M,[h_{0}])$ the
_boundary-at-infinity_ of $(M,g)$, where $[h_{0}]$ is the conformal class of
the metric $h_{0}$ induced on $\partial M$ by $\overline{g}$. Such metrics
provide the setting for many interesting problems and investigations in
geometric analysis and physics. It is known [11] that if $g$ is AH, then there
exists another boundary defining function $x$ such that, in a collar
neighborhood of the boundary,

$g=\frac{dx^{2}+h(x)}{x^{2}},$ (0.1)

where $h\sim\sum x^{j}h_{j}$ is a smooth family of metrics on $\partial M$.
Then $x$ is called a _special defining function_ for $\partial M$. The
‘special metrics’ in this class are called Poincaré-Einstein (PE); a more
general class consists of the asymptotically Poincaré-Einstein (APE) metrics,
which are conformally compact and have an Einstein tensor

$E(g)=\operatorname{Rc}(g)+(n-1)g$ (0.2)

that vanishes near $\partial M$ up to corrections of order $x^{n}$ ($n=\dim
M$). There are slight differences in this description depending on whether $n$
is even or odd. To simplify the discussion here, and also because it is our
primary focus in this paper, we assume henceforth that $n$ is even.

The normalized Ricci flow in this setting is the system of equations

$\partial_{t}\,g=-2E(g)\ ,\ t\in[0,T),\qquad g(0)={g^{0}}.$ (0.3)

The normalization factor fixes the conformal infinity in time. PE metrics are
the stationary points of this normalized Ricci flow, but this flow also
preserves the class of APE metrics [3].

AH metrics have exponential volume growth, but using the regularity of the
metric at infinity, i.e., near $\partial M$, it is sometimes possible to
define an Hadamard renormalized volume $\mathrm{RenV}(M,g)$. This is done by
computing the volume of $(M,g)$ in the compact set $\\{x\geq\epsilon\\}$,
expanding the result in powers of $\epsilon$ (there can also be $\log\epsilon$
terms when the bulk dimension $n$ is odd), and taking the limit at
$\epsilon=0$ after discarding the singular terms in this expansion. While this
may be done for any AH metric with a fixed special defining function $x$, it
turns out [10] that this has an invariant meaning which is independent of the
choice of $x$ provided $g$ satisfies an auxiliary condition concerning the
omission of a certain number of odd terms in the expansion of the term $h(x)$.
More precisely, (i) $h(x)$ must have an even expansion to order $n-2$, and
(ii) the first odd term in this expansion must have vanishing trace. We call
AH metrics which satisfy condition (i) _partially even_. Partially even
metrics that satisfy condition (ii) are called _volume renormalizable_. Thus
we are asserting that (when $n$ is even) the renormalized volume of an AH
metric $g$ has a well-defined, invariant meaning if and only if these
conditions are satisfied. Significantly, every even dimensional PE metric, as
well as every even dimensional APE metric, satisfies these conditions.

AH metrics with various assumptions on the evenness of the expansion have been
considered frequently before. Perhaps most important is the appearance of this
condition in geometric scattering theory. The paper [14] by the second author
and Melrose established a general construction in the framework of what is now
called geometric microlocal analysis for the resolvent of the scalar Laplacian
of an AH metric. More specifically, it is proved there that the family of
operators $(\Delta+\zeta(n-1-\zeta))^{-1}$, which depends holomorphically on
$\zeta$ in the right half-plane $\operatorname{Re}\zeta>(n-1)/2$, extends to a
meromorphic function in the punctured plane
$\mathbb{C}\setminus\\{(n-1)/2-j,j\in\mathbb{N}\\}$. That analysis left open
the possibility that there could be an accumulation of poles at this sequence
of points along the (shifted) negative real axis. An important paper by
Guillarmou [13] clarified the situation by showing that this accumulation does
not happen precisely when the metric $g$ is even to all orders. This has many
significant consequences for the scattering theory. This evenness condition
has also appeared in the more recent and quite different approach by Vasy [16]
for studying the meromorphic continuation of this resolvent family.

In our earlier paper [3], we studied the behaviour of $\mathrm{RenV}$ under
Ricci flow for an evolving family of APE metrics. Our goal in this paper is to
extend this analysis by studying this Ricci evolution and its effect on the
renormalized volume for the entire class of partially even metrics. The key
difference between this larger class of metrics and the APEs is that the
coefficient tensors $h_{2j}$, $1\leq j\leq(n-2)/2$, and
$\mathrm{tr}^{h_{0}}h_{n-1}$ in the expansion for $h$ are formally determined
by the leading coefficient $h_{0}$ when $g$ is APE, whereas they are
independent of one another when $g$ is partially even. The interrelationships
between these tensors when $g$ is APE is precisely what forces the Einstein
tensor to vanish to order $n$ in that case, while for partially even metrics,
the Einstein tensor need only vanish to second order.

We now state our main theorems.

######  Theorem A.

Suppose that $(M,{g^{0}})$, $\dim M=:n=2m$, is partially even, and let $g(t)$
be a solution of (0.3) with $g(0)={g^{0}}$ with maximal interval of existence
$[0,T_{0})$. Then $(M,g(t))$ remains partially even for $t<T_{0}$.

This is proved in several steps. The first is to show that the Ricci-DeTurck
flow preserves the property of being partially even relative to a fixed
defining function. This depends crucially on the earlier result by the first
author [2] that the Ricci-DeTurck and Ricci flows preserve the property of
being conformally compact (or polyhomogeneous). We then argue that the
transformation which carries this to a solution of the Ricci flow preserves
the form of the expansion. We then show that the re-expression of $g(t)$
relative to the appropriate special boundary defining function $x$ (which
depends on $t$) still preserves this form of the expansion.

Our second main result computes the variation of the renormalized volume along
this flow. We cast this slightly more generally by proving that the variation
through volume renormalizable metrics of any curvature functional is just the
renormalized first variation of the same form that, on compact manifolds,
yields the Euler-Lagrange equations (usually after integration by parts), see
Theorem 4.4 below. An immediate application is the

######  Theorem B.

Let $(M,{g^{0}})$, $n=2m$, be volume renormalizable, and $g(t)$ the solution
of (0.3) with $g(0)={g^{0}}$. Then

$\frac{d\,}{dt}\mathrm{RenV}=-\mathop{\vphantom{T}}^{R}\\!\\!\\!\int_{M}S(g(t))+n(n-1)\,dV_{g_{t}}\
.$ (0.4)

Here $\mathop{\vphantom{T}}^{R}\\!\\!\\!\int(\cdot)dV_{g}$ indicates that the
integral is not classically convergent, but must be regularized in a way to be
made precise below. The analogue of formula (0.4) in [3] appears almost
exactly the same except that when $g(t)$ is APE, the integral is convergent
and there is no need to renormalize the integral.

There is recent work on the Kähler-Ricci flow in the setting of asymptotically
complex hyperbolic metrics. It is proved in [15] that polyhomogeneity is
preserved under this flow. There is an analogue of renormalized volume in this
setting and it would be interesting to understand how our results could be
carried over.

This paper is organized as follows. In Section 1 we review the families of
asymptotically hyperbolic metrics considered here and establish some of basic
properties involving even expansions. In Section 2 we prove Theorem A that the
normalized Ricci flow of a partially even metric remains partially even, and
in Section 3 we show that the volume renormalizability condition also persists
under the Ricci flow. We defer several long computations needed in this
section to the Appendix. Finally in Section 4 we review the Riesz
renormalization and apply it to the variation of curvature functionals. We
then prove Theorem B.

### Acknowledgements

EB and RM are grateful to Robin Graham for discussions related to this work.
The work of EW was supported by an NSERC Discovery Grant RGPIN 203614. RM was
supported by the NSF grants DMS-1105050 and DMS-1608223.

##  1\. Fefferman-Graham expansions of AH metrics

###  1.1. Taxonomy of conformally compactifiable metrics

As in the Introduction, let $\overline{M}$ be a compact manifold with boundary
and $M$ its interior.

A metric $g$ on $M$ is called conformally compact if

$g=\rho^{-2}\overline{g}\ ,$ (1.1)

where both $\rho$ and $\overline{g}$ are $\mathcal{C}^{\infty}$ up to
$\partial M$, and $\overline{g}$ is a metric on $\overline{M}$.

Any such metric has a sort of boundary value on $\partial M$,

$\mathfrak{c}(g)=\left[\left.\rho^{2}g\right|_{T\partial M}\right],$ (1.2)

called its conformal infinity. Furthermore, a conformally compact metric is
called _asymptotically hyperbolic_ , abbreviated as AH, if
$|d\rho|^{2}_{\overline{g}}=1$ at $\partial M$. Note that $g$ is unchanged if
we replace $\rho$ and $\overline{g}$ by $a\rho$ and $a^{2}\overline{g}$ for
any smooth positive function $a$, but nonetheless both $\mathfrak{c}(g)$ and
the AH condition are well-defined.

Now, if $g$ is AH and we select a representative metric
$h_{0}\in\mathfrak{c}(g)$, then there exists a uniquely determined ‘special’
boundary defining function $x$ which satisfies $|dx|^{2}_{x^{2}g}\equiv 1$ in
a neighborhood of the boundary. With this choice of $x$, the flow lines of the
gradient $\nabla^{\overline{g}}x$ identify a collar neighborhood of $\partial
M$ with $[0,1)\times\partial M$, and by Gauss’ Lemma

$g=\frac{dx^{2}+h(x)}{x^{2}},$ (1.3)

where $h(x)$ is a smooth family of metrics on $\partial M$. This is the
Graham-Lee normal form [11]. We shall be primarily concerned with the terms in
the asymptotic expansion

$h(x)=h_{0}+h_{1}x+h_{2}x^{2}+\cdots.$ (1.4)

For some of the computations below we will also need a full coordinate system
near the boundary of $M$. Let $\\{y^{\alpha}\\}$ be any coordinates on
$\partial M$ extended to be constant along the integral curves of
$\nabla^{\overline{g}}x$. We use greek indices to index these tangential
directions, $x$ to index $\frac{\partial}{\partial x}$, and latin indices to
include both tangential and normal directions. In the expansion above, the
coefficient tensors are functions of the $y$-coordinates alone.

######  Definition 1.1.

An AH metric $g$ is called _even to order $2\ell$_ if, in Graham-Lee normal
form (1.2), the expansion for $h(x)$ contains no terms $h_{2j+1}$ with
$j\leq\ell-1$, i.e.,

$h(x)\sim
h_{0}+x^{2}h_{2}+\dots+x^{2\ell}h_{2\ell}+x^{2\ell+1}h_{2\ell+1}+....,$ (1.5)

Our main focus will be on AH metrics $g$ which are even to order $n-2$, and we
call these simply _partially even_.

######  Definition 1.2.

We say that $g$ is _volume renormalizable_ (VR) if it is partially even and in
addition satisfies

$\mathrm{tr}^{h_{0}}h_{n-1}=0.$ (1.6)

We now recall one of the most important classes of AH metrics

######  Definition 1.3.

An AH metric $g$ is called _Poincaré-Einstein_ (PE) if its Einstein tensor
$E(g)$, defined in (0.2), vanishes identically.

Since our primary concern is with asymptotic expansions at the boundary, it is
also natural to introduce the

######  Definition 1.4.

An AH metric $g$ is called _asymptotically Poincaré-Einstein_ (APE) if
$|E(g)|_{g}=\mathcal{O}(x^{n})$.

If $g$ is APE, or in particular PE, then it is known, see [9], that every odd
coefficient $h_{2j+1}$, $0\leq j\leq(n-4)/2$, must vanish while the even
coefficients $h_{2j}$, $1\leq j\leq(n-2)/2$, can be expressed as universal
differential operators applied to $h_{0}$, see [9], and finally,
$\mathrm{tr}^{h_{0}}h_{n-1}=0$. This implies that when $n$ is even, PE
$\implies$ APE $\implies$ VR $\implies$ partially even $\implies$ AH.

We have not included the slightly different results when $n$ is odd; the key
difference in that case is that when $g$ is PE, the expansion for $h(x)$ may
also include the terms $x^{n-1+j}(\log x)^{\ell}\tilde{h}_{j\ell}$, $j\geq 0$.
When $n=2m$, if $g$ is merely $\mathcal{C}^{2}$ conformally compact and
Einstein, then a regularity theorem due to Chruściel, Delay, Lee and Skinner
[6] shows that if $h_{0}\in\mathcal{C}^{\infty}$, then $h(x)$ is automatically
smooth up to $x=0$, i.e., it is not necessary to assume a priori that $g$ is
smoothly conformally compact, so long as its conformal infinity contains a
smooth representative.

###  1.2. Partially even metrics

As before, we assume that $n=\dim M$ is even, and write $n=2m$. To make
Definition 1.1 sensible, we must establish the following

######  Proposition 1.5.

If an AH metric $g$ is even to order $2\ell$ in the normal form corresponding
to one choice of metric $h_{0}\in\mathfrak{c}(g)$, then it is even to the same
order with respect to any other metric $h_{0}^{\prime}\in\mathfrak{c}(g)$.

The proof appears in [13, Lemma 2.1]. We give a proof by an alternative
method.

###### Proof.

Write $h_{0}^{\prime}=e^{2\omega_{0}}h_{0}$, where
$\omega_{0}\in\mathcal{C}^{\infty}(\partial M)$. The special boundary defining
function $x^{\prime}$ associated to $h_{0}^{\prime}$ is then given by
$x^{\prime}=e^{\omega}x$, where $|d(e^{\omega}x)/(e^{\omega}x)|^{2}_{g}\equiv
1$. Expand this, writing $\overline{g}=x^{2}g$, and using that
$|dx/x|^{2}_{g}\equiv 1$, to get

$2x\langle
d\omega,dx\rangle_{\overline{g}}+x^{2}|d\omega|^{2}_{\overline{g}}=0,\qquad\left.\omega\right|_{\partial
M}=\omega_{0}.$ (1.7)

As in [11], divide by $x$ to obtain a nondegenerate Hamilton-Jacobi equation,
which means that there is a unique solution $\omega\in\mathcal{C}^{\infty}$
for $0\leq x<\epsilon$. In the normal form for $\overline{g}$, this equation
becomes

$\partial_{x}\omega+\frac{1}{2}x\left((\partial_{x}\omega)^{2}+h^{\alpha\beta}(x)\partial_{y^{\alpha}}\omega\partial_{y^{\beta}}\omega\right)=0.$
(1.8)

There are two different ways to see from this that $\omega$ must be even to
order $2\ell+2$. We now explain both approaches.

The first involves simply computing the successive coefficients. Thus expand
$\omega\sim\sum\omega_{j}(y)x^{j}$, substitute into the equation, and collect
like powers of $x$, to get the sequence of equations

$2j\omega_{j}+\sum_{2p+q+r+2=j}h^{\alpha\beta}_{2p}(\partial_{y^{\alpha}}\omega_{q})(\partial_{y^{\beta}}\omega_{r})=0,\qquad
j\leq 2\ell.$ (1.9)

If $j=1$, then the second summand is empty, so $\omega_{1}=0$. Next, consider
any $\omega_{2j+1}$, $j\leq\ell$. The terms in the accompanying sum have
$2p+q+r+2=2j+1$, hence $p\leq j-1$, $q,r\leq 2j-1$, and precisely one of $q$
or $r$ must be odd. By induction, either $\omega_{q}=0$ or $\omega_{r}=0$. In
any case, each summand has a vanishing factor, so $\omega_{2j+1}=0$. This
means that the first possible odd term is $x^{2\ell+3}$.

For the other approach we need a bit of notation. Let $\widetilde{M}$ denote
the double of $M$ across its boundary. The atlas of $\mathcal{C}^{\infty}$
functions on $\widetilde{M}$ is determined uniquely once we specify an
identification of a collar neighborhood $\mathcal{U}$ of $\partial M$ with the
product $\partial M\times[0,\epsilon)$; the double of this neighborhood,
$\widetilde{\mathcal{U}}$, is then endowed with the atlas of
$\mathcal{C}^{\infty}$ functions on the product $\partial
M\times(-\epsilon,\epsilon)$. This product decomposition is determined by
following the gradient flow lines of $\nabla^{\overline{g}}x$; in other words,
the choice of $h_{0}\in\mathfrak{c}(g)$ determines $x$ and $\overline{g}$ and
hence the $\mathcal{C}^{\infty}$ structure on $\widetilde{M}$.

Now fix the atlas on $\widetilde{M}$ associated to $h_{0}$; it is clear that
$\omega$ is even to order $2\ell+2$ if and only if its even extension
$\tilde{\omega}$ to $\widetilde{M}$ is $\mathcal{C}^{2\ell+2,\mu}$ for every
$0<\mu<1$. (Of course, $\tilde{\omega}$ is also $\mathcal{C}^{\infty}$ on each
‘side’ of $\partial M$ in $\widetilde{M}$.) However, this extended function is
a solution of the nondegenerate extended Hamilton-Jacobi equation obtained by
extending the coefficients $h^{\alpha\beta}$ of (1.8) as even functions and
$x$ as an odd function. The extended coefficients $h^{\alpha\beta}(x)$ are in
the Hölder class $\mathcal{C}^{2\ell,\mu}$ (but not in $\mathcal{C}^{2\ell+1}$
if the coefficients of $|x|^{2\ell+1}$ in their expansions do not vanish), so
standard theory shows that the solution lies in $\mathcal{C}^{2\ell+1,\mu}$.
This shows that $\tilde{\omega}$ is even to order $2\ell+2$ as claimed.

To finish the argument, if $\hat{h}_{0}=e^{2\omega_{0}}h_{0}$ and
$\hat{x}=e^{2\omega}x$ is the new special boundary defining function, then the
metric $\hat{g}=e^{2\omega}\overline{g}=(\hat{x})^{2}g$ is
$\mathcal{C}^{2\ell,\mu}$ on $\widetilde{M}$, so the metric induced by
pullback under the map $(\hat{x},y)\to\exp^{\hat{g}}_{y}(\hat{x}\hat{\nu})$ is
$\mathcal{C}^{2\ell-1,\mu}$ (here $\hat{\nu}=e^{-\omega_{0}(y)}\bar{\nu}$ is
the $\hat{g}$ unit normal to $\partial M$). But this metric is in normal form
and hence is even to order $2\ell$. ∎

It follows from this proof that if $g$ is an AH metric which is even to order
$2\ell$, then it defines an equivalence class of defining functions $[x]$,
where $x\sim x^{\prime}$ if $x^{\prime}/x$ is even to order $2\ell+2$.

We now extend this discussion by considering how to recognize partially even
AH metrics which are not written in normal form. Thus fix an AH metric $g$ and
suppose that $x$ is some boundary defining function. As before we set
$\overline{g}=x^{2}g$ and use the gradient lines of $\nabla^{\overline{g}}x$
to define a smooth structure on the double $\widetilde{M}$, and finally,
consider the extended metric $\tilde{g}$ which restricts to $\overline{g}$ on
$M$ and satisfies $I^{*}\tilde{g}=\tilde{g}$, where
$I:\widetilde{M}\to\widetilde{M}$ is reflection across the submanifold
$\partial M\subset\widetilde{M}$.

######  Definition 1.6.

We say that $g$ is even to order $j$ relative to an arbitrary boundary
defining function $x$ if $\tilde{g}$ is $\mathcal{C}^{j}$ on $\widetilde{M}$
(still recalling that it is $\mathcal{C}^{\infty}$ on each side of $\partial
M$).

If $j=2\ell$, it is straightforward to see that if $g$ satisfies this
condition, then relative to any choice of coordinates $y$ on $\partial M$, the
components $\overline{g}_{xx}$ and $\overline{g}_{\alpha\beta}$ are even to
order $2\ell$ while $\overline{g}_{x\alpha}$ is odd to order $2\ell+1$ (i.e.,
it is of the form $xm_{x\alpha}$ where $m_{x\alpha}$ is even to order
$2\ell$). To explain this last condition, note that a term of order $2\ell$ in
the expansion for $\overline{g}_{x\alpha}$ with the correct parity must be of
the form $a(y)(\mathrm{sgn}\,x)|x|^{2\ell}$, and hence is not
$\mathcal{C}^{2\ell}$. On the other hand, if $j=2\ell+1$, then evenness to
order $j$ implies that $\overline{g}_{xx}$ and $\overline{g}_{\alpha\beta}$
are even to order $j+1=2\ell+2$ while $\overline{g}_{x\alpha}$ is still just
odd to order $2\ell+1$. If the components of $\overline{g}$ satisfy these
parity conditions, then by Cramer’s formula, the components
$(\overline{g})^{\,ij}$ of the inverse matrix are also $\mathcal{C}^{2\ell}$
across $x=0$, hence these too satisfy the same parity conditions. Clearly, if
$g$ is even to order $2\ell$ in the sense of Definition 1.2, then $g$ is even
to order $2\ell$ in the sense of Definition 1.6 relative to a special defining
function $x$. Conversely, we have the following result.

######  Lemma 1.7.

If $g$ is even to order $2\ell$ relative to any fixed boundary defining
function in the sense of Definition 1.6, then it is even to order $2\ell$ in
the sense of Definition 1.6, i.e., it is even to order $2\ell$ with respect to
any special boundary defining function.

###### Proof.

If $x$ denotes the initial defining function, let $h_{0}=x^{2}g$ restricted to
$T\partial M$. We seek $x^{\prime}=e^{\omega}x$ such that
$|dx^{\prime}/x^{\prime}|^{2}_{g}=1$, $\omega_{0}(y)=\omega(0,y)\equiv 0$, or
equivalently

$2\langle
dx,d\omega\rangle_{\overline{g}}+x|d\omega|^{2}_{\overline{g}}=\frac{1-|dx|^{2}_{\overline{g}}}{x},\qquad\left.\omega\right|_{\partial
M}=0.$ (1.10)

In contrast to (1.8), the right hand side of this equation is nonvanishing,
but there still exists a unique solution $\omega$ which is
$\mathcal{C}^{\infty}$ for $0\leq x<\epsilon$, and the only issue is the
degree of smoothness of its even extension across $x=0$. However,
$x^{-1}(|dx|^{2}_{\overline{g}}-1)=x^{-1}(\overline{g}^{xx}-1)$ is odd to
order $2\ell-1$, hence lies in $\mathcal{C}^{2\ell-1}$, and the solution of
this equation is one order smoother, i.e. lies in $\mathcal{C}^{2\ell}$. This
proves that $\omega$ is even to order $2\ell$.

Now set $g^{\prime}=(x^{\prime})^{2}g=e^{2\omega}\overline{g}$ and write
$\tilde{g}^{\prime}$ for its $I$-invariant extension. From the paragraph
above, $\tilde{g}^{\prime}\in\mathcal{C}^{2\ell}$. In fact, the metric is
$\mathcal{C}^{\infty}$ in the tangential direction and its irregularity is
only in the direction normal to the boundary (technically, it is
polyhomogeneous at the boundary). It is standard to show that the exponential
mapping $\Phi:NM\to M$ from the normal bundle of $\partial M$ to $M$,
$\Phi(x,y)=\exp_{y}(x\nu(y))$ is smooth in the ‘base’ variable $y\in\partial
M$ and $\mathcal{C}^{2\ell+1}$ in $x$. Hence
$\Phi^{*}\tilde{g}^{\prime}=d(x^{\prime})^{2}+\tilde{h}^{\prime}_{\alpha\beta}(x^{\prime},y)dy^{\alpha}dy^{\beta}$,
where $\tilde{h}^{\prime}_{\alpha\beta}\in\mathcal{C}^{2\ell}$, or finally,
$g^{\prime}$ is even to order $2\ell$. ∎

##  2\. Partially even metrics and Ricci flow

Our aim in this section is to prove the following.

######  Proposition 2.1.

Suppose that ${g^{0}}$ is an AH metric which is even to order $2\ell$. Let
$g(t)$ be the unique solution to the Ricci flow equation (0.3) with initial
condition ${g^{0}}$. Then $g(t)$ remains even to order $2\ell$ throughout its
time of existence.

Notice that we do not use the gauged Ricci flow equation here since we are
taking the existence of the solution as given. Indeed, we take from [2] the
fact that there is a unique solution to (0.3) and this remains smoothly
conformally compact throughout its maximal time interval of existence.

###### Proof.

First appeal to the transformation formula for the Ricci tensor under a
conformal change of metric, see [4]: if $g=x^{-2}\overline{g}$, then
$E(g)=\operatorname{Rc}(g)+(n-1)g=x^{-2}\overline{E}(\overline{g})$, where

$\overline{E}(\overline{g})=-(n-1)\,(|dx|^{2}_{\overline{g}}-1)\,\overline{g}+\left((n-2)\operatorname{Hess}_{\overline{g}}(x)+(\Delta_{\overline{g}}x)\,\overline{g}\right)x+\operatorname{Rc}(\overline{g})\,x^{2}\
.$ (2.1)

In the following computations, we use a fixed defining function $x$, which is
a special boundary defining function for the initial metric ${g^{0}}$, but of
course no longer has this property for $g(t)$, $t>0$.

We shall analyze the equation

$\partial_{t}\,\overline{g}=-2\overline{E}(\overline{g})\ ,$ (2.2)

which is obtained if we multiply (0.3) by $x^{2}$.

As a first step, consider the restriction of this equation to $x=0$:

$\left.\partial_{t}\left(\overline{g}\right|_{x=0}\right)=2(n-1)(|dx|_{\overline{g}}^{2}-1)\left.\left(\overline{g}\right|_{x=0}\right)\
.$ (2.3)

Since $g$ is AH, the leading coefficient $|dx|^{2}_{\overline{g}}-1$ vanishes
at $t=0$, which shows that the restriction (not just the pullback) of
$\overline{g}$ to the boundary is invariant under the flow. It is thus
reasonable to write the evolving solution in the form $g(t)={g^{0}}+k(t)$,
where $|k(t)|_{{g^{0}}}=\mathcal{O}(x)$. This was the ansatz in (0.3) in [2],
and a uniqueness theorem in this setting [5] shows that any solution of (2.2)
in this quasi-isometry class must agree with the solution in [2]. The proof in
[2] also shows that $k(t)\in\mathcal{C}^{\infty}(\overline{M})$ for every
$t\geq 0$. Consequently, $\overline{E}(x^{2}g(t))=0$ at $\partial M$ for all
$t\geq 0$.

Now rewrite (2.2) as an equation for the evolving tensor
$v(t)=\overline{g}(t)-\overline{{g^{0}}}$:

$\begin{split}\partial_{t}\overline{k}(t)=&\,2(n-1)\,(|dx|^{2}_{\overline{g}(t)}-1)\,\overline{g}(t)-2\left((n-2)\operatorname{Hess}_{\overline{g}(t)}(x)+(\Delta_{\overline{g}(t)}x)\,\overline{g}(t)\right)x\\\
&\,-2\operatorname{Rc}(\overline{g}(t))\,x^{2}.\end{split}$ (2.4)

Define the double $\widetilde{M}$ in terms of the fixed defining function $x$
and consider (2.4) on this space. The key point is that $\overline{E}$
commutes with $I^{*}$, i.e. if $\hat{g}$ is any extension of $\overline{g}$ to
$\widetilde{M}$, then

$I^{*}\overline{E}(\hat{g})=\overline{E}(I^{*}\hat{g})\ .$ (2.5)

We shall work with the even extension $\tilde{g}$, where
$I^{*}\tilde{g}=\tilde{g}$, so that
$I^{*}\overline{E}(\tilde{g})=\overline{E}(\tilde{g})$.

We now argue through a sequence of claims:

Claim 1: Let $g$ be even to order $j$. Then
$\overline{E}(\tilde{g})\in\mathcal{C}^{j}$, or equivalently,
$\overline{E}(\overline{g})$ is even to order $j$.

To prove this, note that $\overline{E}$ is a “$0$-differential operator”,
which means that every derivative $\partial_{x}$ or $\partial_{y^{\alpha}}$ is
accompanied by a factor of $x$. When $j$ is even, then the first term in
$\tilde{g}$, the even extension of $x^{2}g$, which is not
$\mathcal{C}^{\infty}$, occurs with the factor $|x|^{j+1}$; furthermore, the
normal derivatives which act on this are either $x\partial_{x}$ or
$x^{2}\partial_{x}^{2}$, neither of which decrease the order of vanishing.
This means that $\overline{E}(\tilde{g})\in\mathcal{C}^{j}$ across $x=0$.
Similarly, when $j$ is odd, the leading nonsmooth term for the reflected
metric occurs at order $j+1$ again.

To state the next claim, fix any $\overline{g}$ which is even to order $j\leq
2\ell$, and let $v$ be smooth up to $x=0$ and denote by $\Pi^{e,j}(x^{j}v)$
and $\Pi^{o,j}(x^{j}v)$ the even and odd components of $x^{j}v$ at order $j$.
Also, write

$\overline{E}(\overline{g}+x^{j}v)=\overline{E}(\overline{g})+\left.D\overline{E}\right|_{\overline{g}}(x^{j}v)+\mathcal{O}(x^{2j}),$
(2.6)

and for simplicity, set
$\left.D\overline{E}\right|_{\overline{g}}=\mathcal{L}$.

Claim 2: If $v$ is smooth for $x\geq 0$, then

$\Pi^{e,j}(\mathcal{L}x^{j}v)=\mathcal{L}(\Pi^{e,j}(x^{j}v)).$ (2.7)

Suppose that $x^{j}v$ is even to order $j$ and define
$\hat{g}=\overline{g}+\Pi^{e,j}(x^{j}v)$. By Claim 1,
$\overline{E}(\overline{g})\in\mathcal{C}^{j}$. Furthermore,
$\overline{E}(\hat{g})\in\mathcal{C}^{j}$ and the quadratic error term
vanishes like $x^{2j}$. Hence $\mathcal{L}(x^{j}v)\in\mathcal{C}^{j}$, which
means that it is even to order $j$ as well. This proves (2.7).

Claim 3: Suppose that $\overline{g}$ is even to order $k$ and
$v\in\mathcal{C}^{\infty}(\overline{M})$; then for $j\leq k$,

$\Pi^{o,j}(\mathcal{L}x^{j}v)=\mathcal{L}(\Pi^{o,j}(x^{j}v))=0.$ (2.8)

Unlike the even case, this seems to require direct calculation. As before, set
$\hat{g}=\overline{g}+x^{j}v$. We shall calculate the leading coefficients of
the linearizations of each of the terms $(|dx|_{\hat{g}}^{2}-1)\hat{g}$,
$x\mathrm{Hess}_{\hat{g}}(x)$, $x\Delta_{\hat{g}}x\,\hat{g}$ and
$x^{2}\operatorname{Rc}(\hat{g})$ separately (for simplicity we do not bother
to assemble these with the correct coefficients into the operator
$\mathcal{L}$). This will show immediately that both (2.8) and (2.7) must
hold.

In these calculations it is convenient to work in a coordinate system $(x,y)$
around $p\in\partial M$ for which
$\overline{g}=dx^{2}+\overline{g}_{\alpha\beta}dy^{\alpha}dy^{\beta}$ (thus
$x$ is the $\overline{g}$-distance to $\partial M$), and so that in addition
$\overline{g}_{\alpha\beta}(p)=\delta_{\alpha\beta}$ and all Christoffel
symbols of $\overline{g}$ vanish at $p$. The following computations are all
carried out at the point $p$ and modulo $x^{j+1}$, and we use $\equiv$ in this
sense. Thus, for example, $\hat{g}^{-1}\equiv\delta-v$, etc.

The first term is

$(|dx|^{2}_{\hat{g}}-1)\hat{g}=(\hat{g}^{00}-1)\hat{g}\equiv(\delta^{00}-1-x^{j}v_{00})(\overline{g}+v)=-x^{j}v_{00}\overline{g}.$
(2.9)

Next,

$x\mathrm{Hess}_{\hat{g}}(x)_{ij}=-x\widehat{\Gamma}^{x}_{ij},\quad\mbox{and}\
\qquad x\Delta_{\hat{g}}x=-x\hat{g}^{ij}\widehat{\Gamma}^{x}_{ij}.$ (2.10)

A computation in the appendix shows that

$x\widehat{\Gamma}^{x}_{xx}\equiv(j/2)x^{j}v_{00},\quad
x\widehat{\Gamma}^{x}_{x\alpha}\equiv 0,\quad\mbox{and}\qquad
x\widehat{\Gamma}^{x}_{\alpha\beta}\equiv-(j/2)x^{j}v_{\alpha\beta},$ (2.11)

hence in addition,

$x\hat{g}^{ij}\widehat{\Gamma}^{x}_{ij}\equiv(j/2)x^{j}(v_{00}-\overline{g}^{\alpha\beta}v_{\alpha\beta}).$
(2.12)

As for the final set of terms,

$\begin{split}x^{2}\operatorname{Rc}(\hat{g})_{xx}\equiv&\,-x^{2}\partial_{x}\widehat{\Gamma}^{\alpha}_{\alpha
x}\equiv-\frac{1}{2}j(j-1)\delta^{\alpha\beta}x^{j}v_{\alpha\beta}\\\
x^{2}\operatorname{Rc}(\hat{g})_{x\alpha}\equiv&\,0\\\
x^{2}\operatorname{Rc}(\hat{g})_{\alpha\beta}\equiv&\,-\frac{1}{2}j(j-1)/2x^{j}v_{\alpha\beta}.\end{split}$
(2.13)

It is clear from these calculations that for any $j$,
$\mathcal{L}\circ\Pi^{o,j}=0$. On the other hand, since any
$v\equiv\Pi^{e,j}v+\Pi^{o,j}v$ and $\mathcal{L}\Pi^{e,j}v$ is even at order
$j$, we deduce that $\Pi^{o,j}(\mathcal{L}v)=0$ as well. This proves the
claim.

We are now in a position to complete the proof. Trivially, $\overline{g}(t)$
is even to order $0$. Suppose we have shown that $\overline{g}(t)$ is even to
order $j$ for any $j$ with $0\leq j\leq 2\ell-1$. Write
$\overline{g}(t)=\overline{{g^{0}}}+v(t)$, and assume by induction we have
shown that $v(t)$ is even to order $j$. Applying $\Pi^{o,j}$ to the equation
(2.4) and noting that

$\Pi^{o,j}\overline{E}(\overline{g}+v)=\mathcal{L}(\Pi^{o,j}(v))=0,$ (2.14)

we obtain

$\partial_{t}\Pi^{o,j}(v(t))=0,\quad\left.\Pi^{o,j}v\right|_{t=0}\equiv 0.$
(2.15)

This is a well-defined evolution equation acting on $\Pi^{o,j}v$, the first
odd term in the expansion for $v$. This implies of course that
$\Pi^{o,j}v(t)\equiv 0$, hence $v$ is in fact even to order $j+1$. This
argument works for every $j\leq 2\ell-1$, so $\overline{g}(t)$ remains even to
order $2\ell$, as claimed. ∎

##  3\. The volume renormalizability condition

We have now proved that if ${g^{0}}$ is even to order $2m-2$ (where $2m=\dim
M$), then $g(t)$ remains even to order $2m-2$ throughout the interval of
existence. In this section we study the evolution of the first odd term
$h_{2m-1}$. Recall that ${g^{0}}$ is volume renormalizable when
$\mathrm{tr}^{h_{0}}h_{2m-1}=0$.

######  Proposition 3.1.

If ${g^{0}}$ is volume renormalizable, then the solution $g(t)$ to the Ricci
flow with initial condition ${g^{0}}$ remains volume renormalizable so long as
this solution is defined.

The proof proceeds very much as before. We show that some version of the
scalar function $\mathrm{tr}^{h_{0}}h_{2m-1}$ satisfies a homogeneous ordinary
differential equation in $t$ with initial condition $0$, and hence vanishes
for $t\geq 0$. The actual quantity we study is slightly more complicated, and
is defined below.

Having fixed the initial metric ${g^{0}}$ and representative
$h_{0}\in\mathfrak{c}({g^{0}})$, let $x$ be the corresponding special boundary
defining function. We define the $t$-independent metric

$G=\frac{dx^{2}+h_{0}}{x^{2}}$ (3.1)

by truncating the expansion of the tangential metric $h(x)$ in the normal form
for ${g^{0}}$. We also set $\overline{G}=x^{2}G$.

Next define the function

$F:=\mathrm{tr}^{\overline{G}}\overline{g}(t)=\mathrm{tr}^{G}g(t).$ (3.2)

Clearly $F$ is smooth up to $x=0$ for every $t\geq 0$. Moreover, since $G$ and
$g$ are even, $F$ is even to order $2m-2$. Our interest is in the first odd
term in the expansion of $F$.

Let us introduce some notation: if $f\in\mathcal{C}^{\infty}(\overline{M})$
then the coefficient of the $i^{\rm th}$ term its expansion with respect to
the fixed defining function $x$ will be denoted $\\{f\\}_{i}$.

######  Lemma 3.2.

Let $\mu=\mu(t,y)=\\{F\\}_{2m-1}$. Then

$\mu=\\{\overline{g}_{xx}\\}_{2m-1}+(h_{0})^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}.$
(3.3)

###### Proof.

Note that since $\overline{G}^{x\alpha}\equiv 0$,

$F=\overline{G}^{xx}\overline{g}_{xx}+\overline{G}^{\alpha\beta}\overline{g}_{\alpha\beta},$
(3.4)

where all indices are raised with respect to $\overline{G}$. The result is now
straightforward. ∎

######  Proposition 3.3.

Setting $\nu=\\{\overline{g}_{xx}\\}_{2m-1}$ and
$\mu=\\{\overline{g}_{xx}\\}_{2m-1}+(h_{0})^{\alpha\beta}\\{\overline{g}(t)_{\alpha\beta}\\}_{2m-1}$
as above, and denoting by ′ the time-derivative $\partial_{t}$, then

$\begin{split}\mu^{\prime}=&\,-2(2m-1)\mu\\\
\nu^{\prime}=&\,(2m-3)(2m-1)\mu.\end{split}$ (3.5)

###### Proof.

We must compute the first odd term in the expansion of

$\begin{split}F^{\prime}=&\,G^{ij}g_{ij}^{\prime}=-2G^{ij}E_{ij}=-2\overline{G}^{ij}\overline{E}_{ij}\\\
=&\,-2(2m-1)\,(1-|dx|^{2}_{\overline{g}})\overline{G}^{ij}\overline{g}_{ij}-2x\overline{G}^{ij}\left[(n-2)\operatorname{Hess}_{\overline{g}}(x)_{ij}+(\Delta_{\overline{g}}x)\,\overline{g}_{ij}\right]\\\
&\,-2x^{2}\overline{G}^{ij}\operatorname{Rc}(\overline{g})_{ij}.\end{split}$
(3.6)

Let us decompose the expression on the right as the sum of the three terms

$\begin{split}\mathcal{A}=&\,-2(2m-1)(1-|dx|^{2}_{\overline{g}})\overline{G}^{ij}\overline{g}_{ij},\\\
\mathcal{B}=&\,-2x(2m-2)\overline{G}^{ij}\operatorname{Hess}_{\overline{g}}(x)_{ij}-2x\overline{g}^{kl}\operatorname{Hess}_{\overline{g}}(x)_{kl}\cdot\overline{G}^{ij}\overline{g}_{ij},\\\
\mathcal{C}=&\,-2\overline{G}^{ij}(x^{2}\operatorname{Rc}(\overline{g})_{ij}).\end{split}$
(3.7)

Powers of $x$ have been distributed so that these may be expressed in terms of
$0$-derivatives $x\partial_{x}$ and $x\partial_{y_{\alpha}}$, which assists
with the parity bookkeeping.

First consider $\mathcal{A}$. Write $|dx|^{2}_{\overline{g}}\sim
1+\sum_{j=1}^{m-1}A_{2j}x^{2j}+A_{2m-1}x^{2m-1}$. Since
$F=\overline{G}^{ij}\overline{g}_{ij}$ has leading coefficient
$(\overline{G}^{ij}\overline{g}_{ij})|_{x=0}=2m$ and is even to order $2m-2$,
we have

$\begin{split}\\{\mathcal{A}\\}_{2m-1}=&\,-2(2m-1)\left\\{(1-|dx|^{2}_{\overline{g}})\overline{G}^{ij}\overline{g}_{ij}\right\\}_{2m-1}\\\
=&\,-2(2m-1)\left(\left\\{1-|dx|^{2}_{\overline{g}}\right\\}_{0}\left\\{\overline{G}^{ij}\overline{g}_{ij}\right\\}_{2m-1}+\left\\{1-|dx|^{2}_{\overline{g}}\right\\}_{2m-1}\left\\{\overline{G}^{ij}\overline{g}_{ij}\right\\}_{0}\right)\\\
=&\,4m(2m-1)A_{2m-1}=-4m(2m-1)\nu,\end{split}$ (3.8)

where the last equality follows from Lemma A.1 in the appendix.

The calculation for $\mathcal{B}$ is more difficult, and we relegate certain
long parity calculations to the appendix. We begin by studying
$x\operatorname{Hess}_{\overline{g}}(x)$, and record the parity of each
component, the leading coefficient and the coefficient of the first odd term.

First,

$[x\operatorname{Hess}_{\overline{g}}(x)]_{\alpha\beta}=x\left(\partial_{\alpha\beta}^{2}x-\overline{\Gamma}^{k}_{\alpha\beta}\partial_{k}x\right)=-x\overline{\Gamma}^{x}_{\alpha\beta}.$
(3.9)

From the appendix, this expression is even to order $2m-2$ with leading order
$2$ and first odd coefficient

$\left\\{[x\operatorname{Hess}_{\overline{g}}(x)]_{\alpha\beta}\right\\}_{2m-1}=\frac{1}{2}(2m-1)\\{\overline{g}_{\alpha\beta}\\}_{2m-1}.$
(3.10)

Next,

$[x\operatorname{Hess}_{\overline{g}}(x)]_{xx}=x\left(\partial_{xx}^{2}x-\overline{\Gamma}^{k}_{xx}\partial_{k}x\right)=-x\overline{\Gamma}^{x}_{xx};$
(3.11)

this term is also even to order $2m-2$ with leading order $2$ and first odd
coefficient

$\left\\{[x\operatorname{Hess}_{\overline{g}}(x)]_{xx}\right\\}_{2m-1}=-\frac{1}{2}(2m-1)j.$
(3.12)

Finally,

$\displaystyle[x\operatorname{Hess}_{\overline{g}}(x)]_{x\alpha}$
$\displaystyle=x\left(\partial_{x\alpha}^{2}x-\overline{\Gamma}^{k}_{x\alpha}\partial_{k}x\right)=-x\overline{\Gamma}^{x}_{x\alpha}.$
(3.13)

From the appendix, this term has leading order $1$ and is odd to order $2m-1$,
and we do not need more information about it.

Since $G^{0\alpha}\equiv 0$, the first term of $\mathcal{B}$,

$\begin{split}-2(2m-2)\overline{G}^{ij}\cdot&x\operatorname{Hess}_{\overline{g}}(x)_{ij}\\\
&=-2(2m-2)(\overline{G}^{xx}\cdot
x\operatorname{Hess}_{\overline{g}}(x)_{xx}+\overline{G}^{\alpha\beta}\cdot
x\operatorname{Hess}_{\overline{g}}(x)_{\alpha\beta}),\end{split}$ (3.14)

equals

$\begin{split}&\,\left\\{-2(2m-2)\overline{G}^{ij}\cdot
x\operatorname{Hess}_{\overline{g}}(x)_{ij}\right\\}_{2m-1}\\\
=&\,-2(2m-2)\left(\\{x\operatorname{Hess}_{\overline{g}}(x)_{xx}\\}_{2m-1}+h_{0}^{\alpha\beta}\\{x\operatorname{Hess}_{\overline{g}}(x)_{\alpha\beta})\\}_{2m-1}\right)\\\
=&\,-2(2m-2)\left(-\frac{1}{2}(2m-1)\nu+h_{0}^{\alpha\beta}\frac{1}{2}(2m-1)(h_{2m-1})_{\alpha\beta}\right)\\\
=&\,-(2m-1)(2m-2)\left(-\nu+h_{0}^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\right)\\\
=&\,-(2m-1)(2m-2)\left(-\nu+(\mu-\nu)\right)\\\
=&\,-(2m-1)(2m-2)\left(\mu-2\nu\right).\end{split}$ (3.15)

We consider separately the individual summands in the second term of
$\mathcal{B}$,

$\begin{split}&\,-2\overline{g}^{kl}\cdot
x\operatorname{Hess}_{\overline{g}}(x)_{kl}\cdot\overline{G}^{ij}\overline{g}_{ij}\\\
=&\,-2F\left(\overline{g}^{xx}x\operatorname{Hess}_{\overline{g}}(x)_{xx}+2\overline{g}^{x\alpha}x\operatorname{Hess}_{\overline{g}}(x)_{x\alpha}+\overline{g}^{\alpha\beta}x\operatorname{Hess}_{\overline{g}}(x)_{\alpha\beta}\right)\end{split}$
(3.16)

The first of these contributes

$\begin{split}\left\\{-2F\overline{g}^{xx}x\operatorname{Hess}_{\overline{g}}(x)_{xx}\right\\}_{2m-1}&=-2\left\\{F\overline{g}^{xx}\right\\}_{0}\left\\{x\operatorname{Hess}_{\overline{g}}(x)_{xx}\right\\}_{2m-1}\\\
&=2(2m)\frac{1}{2}(2m-1)\nu=2m(2m-1)\nu.\end{split}$ (3.17)

Using the analysis of (3.13) and the fact that $\overline{g}^{x\alpha}$ has
leading order $x$ and is odd to order $2m-1$, the middle term,
$-4F\overline{g}^{x\alpha}x\operatorname{Hess}_{\overline{g}}(x)_{x\alpha}$ is
even to order $2m$, hence contributes nothing at order $2m-1$. As for the
final term,

$\begin{split}\left\\{-2F\overline{g}^{\alpha\beta}x\operatorname{Hess}_{\overline{g}}(x)_{\alpha\beta}\right\\}_{2m-1}&=-2\\{F\overline{g}^{\alpha\beta}\\}_{0}\\{x\operatorname{Hess}_{\overline{g}}(x)_{\alpha\beta}\\}_{2m-1}\\\
&=-2(2mh_{0}^{\alpha\beta})\frac{1}{2}(2m-1)\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\\\
&=-2m(2m-1)h_{0}^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\\\
&=-2m(2m-1)(\mu-\nu)\\\ &=\ 2m(2m-1)(\nu-\mu).\end{split}$ (3.18)

Altogether, then,

$\begin{split}\\{\mathcal{B}\\}_{2m-1}&=-(2m-1)(2m-2)(\mu-2\nu)+2m(2m-1)(2\nu-\mu)\\\
&=-2(2m-1)(2m-1)\mu+4(2m-1)(2m-1)\nu\\\ &=2(2m-1)^{2}(2\nu-\mu)\ .\end{split}$
(3.19)

We turn finally to $\mathcal{C}$. The computations are outlined in the
appendix. Since $G^{x\alpha}=0$ we find

$\overline{G}^{ij}\operatorname{Rc}(\overline{g})_{ij}=\overline{G}^{xx}\operatorname{Rc}(\overline{g})_{xx}+\overline{G}^{\alpha\beta}\operatorname{Rc}(\overline{g})_{\alpha\beta},$
(3.20)

and so from the appendix

$\begin{split}\\{\mathcal{C}\\}_{2m-1}=&\
\\{-2x^{2}\overline{G}^{ij}\operatorname{Rc}(\overline{g})_{ij}\\}_{2m-1}\\\
=&\
-2\\{x^{2}\operatorname{Rc}(\overline{g})_{xx}\\}_{2m-1}-2h_{0}^{\alpha\beta}\\{x^{2}\operatorname{Rc}(\overline{g})_{\alpha\beta}\\}_{2m-1}\\\
=&\
-2(-(2m-1)(m-1)h_{0}^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1})\\\
&\
-2h_{0}^{\alpha\beta}(-(2m-1)(m-1)\\{\overline{g}_{\alpha\beta}\\}_{2m-1})\\\
=&\ 4(m-1)(2m-1)h_{0}^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\\\
=&\ 4(m-1)(2m-1)(\mu-\nu).\end{split}$ (3.21)

Collecting all of this information, we obtain

$\begin{split}\mu^{\prime}=&\
\\{\mathcal{A}\\}_{2m-1}+\\{\mathcal{B}\\}_{2m-1}+\\{\mathcal{C}\\}_{2m-1}\\\
=&\ -4m(2m-1)\nu-(4m-2)(2m-1)\mu+2(2m-1)(4m-2)\nu\\\ &\
+4(m-1)(2m-1)(\mu-\nu)\\\ =&\ -2(2m-1)\mu.\end{split}$ (3.22)

To obtain the evolution equation for $\nu=\\{\overline{g}_{xx}\\}_{2m-1}$,
recall that

$\partial_{t}\overline{g}_{xx}=x^{2}\partial_{t}g_{xx}=-2\overline{E}(g)_{xx}.$
(3.23)

Now

$\begin{split}-2\overline{E}(g)_{xx}=&\
-2(2m-1)\,(1-|dx|^{2}_{\overline{g}})\,\overline{g}_{xx}-2x\left[(n-2)\operatorname{Hess}_{\overline{g}}(x)_{xx}+(\Delta_{\overline{g}}x)\,\overline{g}_{xx}\right]\\\
&\ -2x^{2}\operatorname{Rc}(\overline{g})_{xx}.\end{split}$ (3.24)

Computing as above gives

$\nu^{\prime}=(2m-3)(2m-1)\mu.$ (3.25)

∎

Applying the result of Proposition 3.3 we immediately obtain

######  Corollary 3.4.

If $(\mu(0),\nu(0))=(0,0)$ then $(\mu(t),\nu(t))=(0,0)$ along the flow. In
particular, $h_{0}^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}=0$ for
$0\leq t<T_{0}$.

We now prove Proposition 3.1.

###### Proof of Proposition 3.1.

Suppose that ${g^{0}}$ is a volume renormalizable metric to order $2m-2$. Let
$h_{0}\in\mathfrak{c}({g^{0}})$ and $x$ the corresponding special boundary
defining function, so that

${g^{0}}=\frac{dx^{2}+h_{x}}{x^{2}},$ (3.26)

where $h_{x}|_{x=0}=h_{0}$. As usual, fix tangential coordinates $y^{\alpha}$
extended to be constant along the integral curves of $\nabla^{\overline{g}}x$.

Let $g(t)$ satisfy the Ricci flow equation. We have already shown that it
remains even to order $2m-2$. We write $x_{t}$ for the evolving special
boundary defining function corresponding to $g(t)$ and recall that
$\mathfrak{c}(g(t))$ is constant in $t$, so that

$g(t)=\frac{dx_{t}^{2}+h_{0}+h_{2}x_{t}^{2}+\cdots+h_{2m-2}(t)x_{t}^{2m-2}+h_{2m-1}(t)x_{t}^{2m-1}+\cdots}{x_{t}^{2}}\
.$ (3.27)

We wish to prove that $\mathrm{tr}^{h_{0}}h_{2m-1}(t)=0$.

Denoting by $x$ the special bdf for ${g^{0}}$, we have

$x_{t}=e^{\omega_{t}(x,y)}x\ ,$ (3.28)

where $\omega_{t}$ vanishes to second order since the conformal infinity is
fixed and is even to order $2m$. Computing further shows that

$dx_{t}=(x\partial_{x}\omega_{t}+1)e^{\omega_{t}}dx+(x\partial_{y^{\alpha}}\omega_{t})e^{\omega_{t}}dy^{\alpha}\
,$ (3.29)

and

$\begin{split}dx_{t}^{2}=&\,(x\partial_{x}\omega_{t}+1)^{2}e^{2\omega_{t}}dx^{2}+2(x\partial_{x}\omega_{t}+1)(x\partial_{y^{\alpha}}\omega_{t})e^{2\omega_{t}}dy^{\alpha}dx\\\
&\,+(x\partial_{y^{\alpha}}\omega_{t})(x\partial_{y^{\beta}}\omega_{t})e^{2\omega_{t}}dy^{\alpha}dy^{\beta}.\end{split}$
(3.30)

Inserting this into the expression for $g(t)$ yields

$\begin{split}g(t)=&\,\left(x\partial_{x}\omega_{t}+1\right)^{2}\frac{dx^{2}}{x^{2}}+2\left(x\partial_{x}\omega_{t}+1\right)\left(x\partial_{y^{\alpha}}\omega_{t}\right)\frac{dy^{\alpha}dx}{x^{2}}\\\
&\,+\left[e^{-2\omega}(h_{0})_{\alpha\beta}+(h_{2})_{\alpha\beta}x^{2}+\cdots+(h_{2m-2})_{\alpha\beta}e^{(2m-4)\omega}x^{2m-2}\right.\\\
&\,\left.+(h_{2m-1})_{\alpha\beta}e^{(2m-3)\omega}x^{2m-1}+\cdots+(x\partial_{y^{\alpha}}\omega_{t})(x\partial_{y^{\beta}}\omega_{t})e^{2\omega}\right]\frac{dy^{\alpha}dy^{\beta}}{x^{2}}\
.\end{split}$ (3.31)

Now observe the following. By the proof of Lemma 1.7,

the coefficient of $dx^{2}$ is even to order $2m-2$, and its first odd
coefficient is $2(2m-1)\\{\omega\\}_{2m-1}$, which vanishes since $\omega$ is
even to order $2m$. The coefficient of $dxdy^{\alpha}$ is odd to order $2m-1$.
The entire coefficient of $dy^{\alpha}dy^{\beta}$ in square brackets of is
even to order $2m-2$. Using the expansion for $\omega$, its leading term is
$h_{0}$ while the term at order $2m-1$ equals
$(h_{2m-1})_{\alpha\beta}-2\\{\omega\\}_{2m-1}(h_{0})_{\alpha\beta}$. Thus
Corollary 3.4 shows that $\mathrm{tr}^{h_{0}}h_{2m-1}(t)=0$ along the flow. ∎

##  4\. Variation of renormalized curvature functionals

We now take up the proof of Theorem B. We begin with a quick review of Riesz
renormalization, referring to [1] and [7] for more details, and then study the
variation of renormalized curvature integrals.

###  4.1. Regularized and renormalized integrals

Fix a product decomposition $[0,\epsilon_{0})\times\partial M$ of some
neighborhood of $\partial M$, with projection on the first factor a fixed
defining function. Recall that a function $u$ on $M$ is said to be
polyhomogeneous if, in terms of some (and hence any) boundary defining
function $x$, there is an expansion

$u\sim\sum_{j}\sum_{\ell=0}^{N_{j}}u_{j\ell}(y)x^{\gamma_{j}}(\log x)^{\ell};$
(4.1)

here $\gamma_{j}$ is a sequence of complex numbers with
$\operatorname{Re}\gamma_{j}\to\infty$ and the coefficient functions
$u_{j\ell}$ are $\mathcal{C}^{\infty}$ on $\partial M$. For simplicity we
assume that $\operatorname{Re}\gamma_{0}\leq\operatorname{Re}\gamma_{j}$ for
all $j$. This expansion and the differentiated expansion to any order hold in
the traditional sense.

######  Definition 4.1.

If $u$ is polyhomogeneous at $\partial M$, then

$z\mapsto I(z):=\int_{M}x^{z}udV_{g}$ (4.2)

extends to a meromorphic function in the entire complex plane. By definition,
the _Riesz regularized_ integral of $u$ is the finite part

$\mathop{\vphantom{T}}^{R}\\!\\!\\!\int_{M}u\,dV_{g}:=\operatorname{\underset{z=0}{FP}}I(z).$
(4.3)

When $I(z)$ has a simple pole at $z=0$,
$\operatorname{\underset{z=0}{FP}}I(z)=\lim_{z\to
0}\left(I(z)-\operatorname{\underset{z=0}{Res}}I(z)\right)$.

To write this out more carefully, recall first that
$dV_{g}=x^{-n}\sqrt{\frac{\det h_{x}}{\det
h_{0}}}dxdV_{h_{0}}=:x^{-n}J(x,y)\,dxdV_{h_{0}}$, where the Jacobian factor
$J(x,y)$ has the expansion $\sum_{k\geq 0}J_{k}x^{k}$. Writing the expansion
of $u$ as above, we see that $I(z)$ is holomorphic on
$\\{z\in\mathbb{C}:\operatorname{Re}(z)>n-1-\operatorname{Re}\gamma_{0}\\}$.
Now consider each term

$\begin{split}&\int_{M}x^{z-n+\gamma_{j}+i}(\log
x)^{\ell}u_{j\ell}(y)J_{i}(y)\,dxdy\\\ =&\,\int_{0}^{\epsilon}\int_{\partial
M}x^{z-n+\gamma_{j}+i}(\log x)^{\ell}u_{j\ell}(y)J_{i}(y)\,dxdy\\\
&\,+\int_{x\geq\epsilon}\int_{\partial M}x^{z-n+\gamma_{j}+i}(\log
x)^{\ell}u_{j\ell}(y)J_{i}(y)\,dxdy,\end{split}$ (4.4)

where $J(x,y)\sim\sum J_{i}(y)x^{i}$. The second term on the right is entire
in $z$, while the first extends meromorphically with a pole of order $\ell+1$
at $z=n-\gamma_{j}-i-1$.

As an application, suppose that $x$ and $\tilde{x}=xe^{\omega}$ are two
boundary defining functions, where
$\omega\in\mathcal{C}^{\infty}(\overline{M})$, then
$\tilde{x}^{z}-x^{z}=(e^{z\omega}-1)x^{z}=:A_{j}(y,z)x^{j+z}$. Suppose also
that $u\sim\sum_{j\geq 0}u_{j}(y)x^{j}$, i.e., all $\gamma_{j}$ are
nonnegative integers and each $N_{j}=0$. A short calculation now shows that if
$I(z)$ and $\tilde{I}(z)$ are the regularizations with respect to these two
defining functions, then near $z=0$,

$\tilde{I}(z)-I(z)=\int_{\partial M}C(y,0)\,dV_{h_{0}}+{\mathcal{O}}(z)\ ,$
(4.5)

where $C(y,z)=\sum_{i+j+k=n-1}u_{i}(y)A_{j}(y,z)J_{k}(y)$. The following is an
immediate consequence:

######  Proposition 4.2.

Suppose that $g$ and $u$ are both even to order $n-2=2m-2$. Then
$C(y,0)=\frac{1}{2}u_{0}A_{0}\mathrm{tr}^{h_{0}}h_{n-1}$. If, on the other
hand, the expansion for $u$ starts with the term $x^{p}$ ($p$ even), then
$C(y,0)=\frac{1}{2}u_{p}A_{0}\mathrm{tr}^{h_{0}}h_{n-1-p}$.

###### Proof.

By the parity assumptions, $C(y,0)=u_{0}(y)A_{0}(y,0)J_{n-1}(y)$; all other
terms vanish. It remains to show that

$J_{n-1}=\frac{1}{2}\mathrm{tr}^{h_{0}}h_{n-1}.$ (4.6)

However, by Proposition 1.5, if $g$ is even to order $2m-2$ then $\omega$ is
even to order $2m$, and by simple calculation $\sqrt{h}/\sqrt{h_{0}}$ is even
to order $2m$ as well. Thus $(e^{z\omega-1})\sqrt{h}/\sqrt{h_{0}}$ is even to
order $2m-2$ and the coefficient of $x^{2m-1}$ is
$\frac{1}{2}\mathrm{tr}^{h_{0}}h_{n-1}$.

∎

The key example is $u\equiv 1$, in which case the renormalized integral is the
renormalized volume. If $g$ is volume renormalizable, then
$\mathrm{tr}^{h_{0}}h_{n-1}=0$, which shows that the renormalized volume is
well-defined with these hypotheses. A very similar argument shows that

######  Corollary 4.3.

Suppose the conditions of Proposition 4.2 hold and
$\mathrm{tr}^{h_{0}}h_{n-1}=0$. Then the renormalized integral of $u$ is
independent of the choice of conformal representative of the boundary metric.

###  4.2. Variations of renormalized integrals

Let $g_{t}$ be a family of metrics on a compact manifold $Z$, and consider the
Riemannian curvature functional

$I_{Z}(g):=\int_{Z}u(g)\,dV_{g},$ (4.7)

where $u(g)$ is some scalar quantity defined from the curvature tensor and its
covariant derivatives. One is often interested in critical points of this
action, which are solutions of the Euler-Lagrange equation. The first step in
computing these critical points is the variation of the integrand

$\left.\frac{\partial\,}{\partial
t}\right|_{t=0}\int_{Z}u_{t}\,dV_{g_{t}}=\int_{Z}\left.\frac{\partial\,}{\partial
t}\right|_{t=0}(u_{t}\,dV_{g_{t}}),$ (4.8)

followed by integration by parts. We are interested in the corresponding
calculation on a conformally compact manifold. In particular, we shall only
take the action of partially even volume renormalizable metrics and consider
variations amongst such metrics. Thus suppose that $g(t)$ is volume
renormalizable for each $t$ and ${g^{0}}=:g$; suppose also that the conformal
infinity of $g(t)$ is independent of $t$, with fixed representative $h_{0}$.
Let $x_{t}$ be the corresponding family of special boundary defining
functions, and write $x_{t}=e^{\omega_{t}}x$. Let $u_{t}=u(g_{t})$ be the
scalar quantity associated to $g_{t}$. Set

${\cal L}(g)=\left(\frac{\partial}{\partial
t}\bigg{|}_{t=0}u_{t}+\frac{u_{0}}{2}\mathrm{tr}^{g}\frac{\partial}{\partial
t}\bigg{|}_{t=0}g_{t}\right)$ (4.9)

Writing $x_{0}=x$ and $u_{0}=u$, then we have

######  Theorem 4.4.

With the notation above,

$\frac{\partial}{\partial
t}\bigg{|}_{t=0}\mathop{\vphantom{T}}^{R}\\!\\!\\!\int
u_{t}\,dV_{g_{t}}=\frac{\partial}{\partial
t}\bigg{|}_{t=0}\operatorname{\underset{z=0}{FP}}\;\int_{M}x_{t}^{z}u_{t}dV_{g_{t}}=\operatorname{\underset{z=0}{FP}}\int_{M}x^{z}{\cal
L}(g)dV_{g}\ ,$ (4.10)

where the local expression for $\mathcal{L}$ is exactly the same as in the
compact case.

###### Proof.

We begin by computing

$\begin{split}\frac{\partial}{\partial
t}\bigg{|}_{t=0}\operatorname{\underset{z=0}{FP}}\;\int_{M}x_{t}^{z}u_{t}\,dV_{g_{t}}=&\,\operatorname{\underset{z=0}{FP}}\int_{M}zx^{z-1}_{0}\dot{x_{t}}u_{0}\,dV_{{g^{0}}}+\operatorname{\underset{z=0}{FP}}\int_{M}x^{z}_{0}\frac{\partial}{\partial
t}\bigg{|}_{t=0}\left(u_{t}\,dV_{g_{t}}\right)\\\
=&\,\operatorname{\underset{z=0}{FP}}\left(z\int_{M}x^{z}\dot{\omega_{t}}u\,dV_{g}\right)+\operatorname{\underset{z=0}{FP}}\int_{M}x^{z}\frac{\partial}{\partial
t}\bigg{|}_{t=0}\left(u_{t}\,dV_{g_{t}}\right)\\\
=&\,\operatorname{\underset{z=0}{Res}}\int_{M}x^{z}\dot{\omega_{t}}u\,dV_{g}+\operatorname{\underset{z=0}{FP}}\int_{M}x^{z}{\cal
L}(g)\,dV_{g}\ ,\end{split}$ (4.11)

where $\dot{x_{t}}:=\frac{\partial}{\partial t}\big{|}_{t=0}x_{t}$, and
similarly for $\dot{\omega}_{t}$.

Since $\omega$ is even to order $2m$, so is ${\dot{\omega}}$. In addition,
${\dot{\omega}}\in{\cal O}(x^{2})$ since the conformal infinity is fixed. This
produces a shift by $2$ in the terms in the expansion of
${\dot{\omega}}u\,dV_{g}$. This means that poles arising from integrating this
expression appear at odd integers. In particular, there is no pole at $z=0$,
which means that

$\frac{\partial}{\partial
t}\bigg{|}_{t=0}\operatorname{\underset{z=0}{FP}}\;\int_{M}x_{t}^{z}u_{t}dV_{g_{t}}=\operatorname{\underset{z=0}{FP}}\int_{M}x^{z}{\cal
L}(g)dV_{g}\ .$ (4.12)

∎

######  Theorem 4.5 (Theorem B).

Suppose that $(M^{n},{g^{0}})$, $n=2m$, is volume renormalizable, and let
$g(t)$ be a solution of (0.3) with $g(0)={g^{0}}$. Then along the flow,

$\frac{d}{dt}\mathrm{RenV}=-\mathop{\vphantom{T}}^{R}\\!\\!\\!\int_{M}S(g(t))+n(n-1)dV_{g_{t}}.$
(4.13)

###### Proof.

Proposition 3.1 asserts that $g(t)$ remains volume renormalizable along the
normalized Ricci flow. Consequently the renormalized volume is defined along
the flow.

Recall that on a compact manifold the variation of the volume form $g\mapsto
dV_{g}$ equals $\frac{1}{2}\mathrm{tr}^{g(t)}\dot{g}(t)$, and for the
normalized Ricci flow,

$\frac{1}{2}\mathrm{tr}^{g(t)}\partial_{t}g=-\mathrm{tr}^{g(t)}E_{ij}=-\left(S(g(t))+n(n-1)\right).$
(4.14)

Finally, by Theorem 4.4,

$\frac{d}{dt}\mathrm{RenV}(M,g(t))=-\mathop{\vphantom{T}}^{R}\\!\\!\\!\int_{M}S(g(t))+n(n-1)dV_{g_{t}},$
(4.15)

as asserted. ∎

##  Appendix A Parity computations

In this appendix we record a number of parity computations used above.

######  Lemma A.1.

Let $g$ be a metric which is even to order $2m-2$. Suppose that $x$ is a
defining function for which $|dx|^{2}_{\overline{g}}=\overline{g}^{xx}\sim
1+\sum_{j=1}^{m-1}A_{2j}x^{2j}+A_{2m-1}x^{2m-1}$; then completing $x$ to a
coordinate system $(x,y)$, we have that

$\overline{g}_{xx}=1+\sum_{j=1}^{m-1}a_{2j}x^{2j}+a_{2m-1}x^{2m-1}\ ,$ (A.1)

where $a_{2m-1}=-A_{2m-1}$.

###### Proof.

We use the formula

$\overline{g}^{xx}\overline{g}_{xx}+\overline{g}^{x\alpha}\overline{g}_{x\alpha}=1,$
(A.2)

recalling that since $\overline{g}$ is even to order $2m-2$, the coefficients
$\overline{g}_{\alpha x}$ and $\overline{g}^{\alpha x}$ are odd to order
$2m-1$, hence may be ignored since their product do not contribute to an odd
term before order $2m+1$. Writing $\overline{g}_{xx}\sim
1+\sum_{j=1}^{m-1}a_{2j}x^{2j}+a_{2m-1}x^{2m-1}$, we see that the coefficient
at order $2m-1$ of the product $\overline{g}^{xx}\overline{g}_{xx}$ equals
$a_{2m-1}+A_{2m-1}$. However, this coefficient vanishes, which shows that the
term in $g_{xx}$ at order $2m-1$ is $-A_{2m-1}$. ∎

Suppose that $f(x,y)\sim a_{0}(y)+a_{1}(y)x+\ldots$ is smooth. To simplify
notation below, we denote by $\\{f\\}_{n}$ the $n^{\mathrm{th}}$ coefficient
function $a_{n}(y)$. The following is obvious:

######  Lemma A.2.

If $f$ and $g$ are smooth functions which are even as functions of $x$ to
order $2k$, then the product $fg$ is even to order $2k$, with
$\\{fg\\}_{0}=\\{f\\}_{0}\\{g\\}_{0}$ and
$\\{fg\\}_{2k+1}=\\{f\\}_{2k+1}\\{g\\}_{0}+\\{f\\}_{0}\\{g\\}_{2k+1}.$

Let $\overline{g}$ be a metric that is even to order $2m-2$, i.e.,

$\overline{g}=\overline{g}_{xx}dx^{2}+2\overline{g}_{x\alpha}dxdy^{\alpha}+\overline{g}_{\alpha\beta}dy^{\alpha}dy^{\beta}\
,$ (A.3)

where
$\overline{g}_{xx}=1+\\{\overline{g}_{xx}\\}_{2}x^{2}+\cdots,\overline{g}_{\alpha\beta}=(h_{0})_{\alpha\beta}+\\{\overline{g}_{\alpha\beta}\\}_{2}x^{2}+\cdots$
are even functions to order $2m-2$, and $\overline{g}_{x\alpha}$ has leading
order $x$ and is odd to order $2m-1$.

In the first table we compute the expansions of $0$-derivatives (i.e.,
derivatives with respect to $x\partial_{x}$, $x\partial_{y^{\alpha}}$) of the
various metric components. Note that because of the prefactor $x$ in the
derivative, all of these terms vanish at $x=0$; this will be important in view
of Lemma A.2. An asterisk in this table denotes a coefficient we do not need
to calculate.

Component | Parity to order | Final coefficient
---|---|---
x ∂_x ¯g_xx |  Even to order 2m-2 | { x ∂_x ¯g_xx }_2m-1 = (2m-1) {¯g_xx}_2m-1
x ∂_α ¯g_xx |  Odd to order 2m-1 | *
x ∂_x ¯g_xμ |  Odd to order 2m-1 | *
x ∂_ν ¯g_xμ |  Even to order 2m | *
x ∂_x ¯g_αβ |  Even to order 2m-2 | { x ∂_x ¯g_αβ }_2m-1 = (2m-1) { ¯g_αβ}_2m-1
x ∂_ν ¯g_αβ |  Odd to order 2m-1 | *

As for the Christoffel symbols, once again each of these vanishes at $x=0$,
but It emerges from this computation that any such symbol with an even number
of $x$ components is odd to order $2m-1$. In subsequent computations for the
Ricci tensor such symbols plays no role in the first odd term in the expansion
of quantities of interest.

Component | Parity to order | Final coefficient
---|---|---
x ¯Γ_xx^x |  Even to order 2m-2 | {x ¯Γ_xx^x}_2m-1 = 12 (2m-1) {¯g_xx}_2m-1
x ¯Γ^x_αβ |  Even to order 2m-2 | {x ¯Γ^x_αβ }_2m-1 = -12 (2m-1) { ¯g_αβ}_2m-1
x ¯Γ_αx^x |  Odd to order 2m-1 | *
x ¯Γ_αx^γ |  Even to order 2m-2 | {x ¯Γ_αx^γ }_2m-1 = 12(2m-1) h_0^γβ { ¯g_αβ}_2m-1
x ¯Γ_x x^γ |  Odd to order 2m-1 | *
x ¯Γ_αβ^γ |  Odd to order 2m-1 | *

These results are proved by straightforward calculations.

We now perform a similar analysis on the components of the Ricci curvature. We
write this out a bit more carefully. First, we have

$\begin{split}x^{2}\overline{Rc}_{xx}=&\,x^{2}{\overline{Rm}_{\alpha
xx}}^{\alpha}\\\
=&\,x^{2}\partial_{\alpha}\overline{\Gamma}_{xx}^{\alpha}-x^{2}\partial_{x}\overline{\Gamma}_{\alpha
x}^{\alpha}+x\overline{\Gamma}_{xx}^{s}x\overline{\Gamma}_{\alpha
s}^{\alpha}-x\overline{\Gamma}_{\alpha
x}^{s}x\overline{\Gamma}_{xs}^{\alpha}\\\
=&\,x^{2}\partial_{\alpha}\overline{\Gamma}_{xx}^{\alpha}-x^{2}\partial_{x}\overline{\Gamma}_{\alpha
x}^{\alpha}+x\overline{\Gamma}_{xx}^{x}x\overline{\Gamma}_{\alpha
x}^{\alpha}+x\overline{\Gamma}_{xx}^{\mu}x\overline{\Gamma}_{\alpha\mu}^{\alpha}-x\overline{\Gamma}_{\alpha
x}^{x}x\overline{\Gamma}_{xx}^{\alpha}\\\ &\,-x\overline{\Gamma}_{\alpha
x}^{\mu}x\overline{\Gamma}_{x\mu}^{\alpha}\\\
=&\,x\partial_{\alpha}\left(x\overline{\Gamma}_{xx}^{\alpha}\right)-x^{2}\partial_{x}\overline{\Gamma}_{\alpha
x}^{\alpha}+x\overline{\Gamma}_{xx}^{x}x\overline{\Gamma}_{\alpha
x}^{\alpha}+x\overline{\Gamma}_{xx}^{\mu}x\overline{\Gamma}_{\alpha\mu}^{\alpha}-x\overline{\Gamma}_{\alpha
x}^{x}x\overline{\Gamma}_{xx}^{\alpha}\\\ &\,-x\overline{\Gamma}_{\alpha
x}^{\mu}x\overline{\Gamma}_{x\mu}^{\alpha}\\\
=&\,x\partial_{\alpha}\left(x\overline{\Gamma}_{xx}^{\alpha}\right)-x\partial_{x}\left(x\overline{\Gamma}_{\alpha
x}^{\alpha}\right)+x\overline{\Gamma}_{\alpha
x}^{\alpha}+x\overline{\Gamma}_{xx}^{x}x\overline{\Gamma}_{\alpha
x}^{\alpha}+x\overline{\Gamma}_{xx}^{\mu}x\overline{\Gamma}_{\alpha\mu}^{\alpha}\\\
&\,-x\overline{\Gamma}_{\alpha
x}^{x}x\overline{\Gamma}_{xx}^{\alpha}-x\overline{\Gamma}_{\alpha
x}^{\mu}x\overline{\Gamma}_{x\mu}^{\alpha}\ .\end{split}$ (A.4)

The first and the final four terms in this last expression are even to order
$2m$ and thus do not contribute to the term at order $2m-1$; this uses that
each $x\overline{\Gamma}$ vanishes at $x=0$. Thus

$\begin{split}\\{x^{2}\overline{Rc}_{xx}\\}_{2m-1}=&\,\\{-x\partial_{x}\left(x\overline{\Gamma}_{\alpha
x}^{\alpha}\right)+x\overline{\Gamma}_{\alpha x}^{\alpha}\\}_{2m-1}\\\
=&\,\frac{1}{2}\left(-(2m-1)^{2}+2m-1\right)h_{0}^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\\\
=&\,-(2m-1)(m-1)h_{0}^{\alpha\beta}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\
.\end{split}$ (A.5)

The other relevant set of components is

$\begin{split}x^{2}\overline{Rc}_{\alpha\beta}=&\,x^{2}{\overline{Rm}_{s\alpha\beta}}^{s}\\\
=&\,x^{2}\left(\partial_{s}\overline{\Gamma}_{\alpha\beta}^{s}-\partial_{\alpha}\overline{\Gamma}_{s\beta}^{s}+\overline{\Gamma}_{\alpha\beta}^{r}\overline{\Gamma}_{sr}^{s}-\overline{\Gamma}_{s\beta}^{r}\overline{\Gamma}_{\alpha
r}^{s}\right)\\\
=&\,x^{2}\left(\partial_{x}\overline{\Gamma}_{\alpha\beta}^{x}+\partial_{\mu}\overline{\Gamma}_{\alpha\beta}^{\mu}-\partial_{\alpha}\overline{\Gamma}_{x\beta}^{x}-\partial_{\alpha}\overline{\Gamma}_{\mu\beta}^{\mu}+\overline{\Gamma}_{\alpha\beta}^{x}\overline{\Gamma}_{xx}^{x}+\overline{\Gamma}_{\alpha\beta}^{x}\overline{\Gamma}_{\mu
x}^{\mu}\right.\\\
&\,\left.+\overline{\Gamma}_{\alpha\beta}^{\mu}\overline{\Gamma}_{x\mu}^{x}+\overline{\Gamma}_{\alpha\beta}^{\mu}\overline{\Gamma}_{\lambda\mu}^{\lambda}-\overline{\Gamma}_{x\beta}^{x}\overline{\Gamma}_{\alpha
x}^{x}-\overline{\Gamma}_{\mu\beta}^{x}\overline{\Gamma}_{\alpha
x}^{\mu}-\overline{\Gamma}_{x\beta}^{\mu}\overline{\Gamma}_{\alpha\mu}^{x}-\overline{\Gamma}_{\lambda\beta}^{\mu}\overline{\Gamma}_{\alpha\mu}^{\lambda}\right)\
.\end{split}$ (A.6)

This entire expression is even to order $2m-2$, and moreover all terms of the
form $(x\overline{\Gamma})(x\overline{\Gamma})$ are even to order $2m$, as are
the derivative terms involving tangential partial derivatives. We thus find
that

$\begin{split}\\{x^{2}\overline{Rc}_{\alpha\beta}\\}_{2m-1}=&\,\\{x^{2}\partial_{x}\overline{\Gamma}_{\alpha\beta}^{x}\\}_{2m-1}\\\
=&\,\\{x\partial_{x}(x\overline{\Gamma}_{\alpha\beta}^{x})-x\overline{\Gamma}_{\alpha\beta}^{x}\\}_{2m-1}\\\
=&\,-\frac{1}{2}(2m-1)^{2}\\{\overline{g}_{\alpha\beta}\\}_{2m-1}+\frac{1}{2}(2m-1)\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\\\
&=-(2m-1)(m-1)\\{\overline{g}_{\alpha\beta}\\}_{2m-1}\ .\end{split}$ (A.7)

We do not calculate the components $x^{2}\overline{Rc}_{x\alpha}$ since these
do not come up in our arguments.

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