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# Double Schubert polynomials do have saturated Newton polytopes

Federico Castillo  ,  Yairon Cid-Ruiz  ,  Fatemeh Mohammadi  and  Jonathan
Montaño  Dedicated to Bernd Sturmfels, on the occasion of his 60th birthday,
for his far-reaching contributions and impact on the careers of many young
mathematicians.

###### Abstract.

We prove that double Schubert polynomials have the Saturated Newton Polytope
property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are
motivated by the theory of multidegrees. We introduce a notion of
standardization of ideals that enables us to study non-standard multigradings.
This allows us to show that the support of the multidegree polynomial of each
Cohen-Macaulay prime ideal in a non-standard multigrading, and in particular,
that of each Schubert determinantal ideal is a discrete polymatroid.

###### Key words and phrases:

Double Schubert polynomials; Schubert ideals; Multidegrees; Polymatroids;
Newton polytopes

###### 2010 Mathematics Subject Classification:

13H15, 14M15, 14C17, 52B40

##  1\. Introduction

Schubert polynomials are classical and important objects in algebraic
combinatorics. They were introduced by Lascoux and Schützenberger
[SCUBERT_POLY_L_S] to study the cohomology classes of Schubert varieties.
Since then, Schubert polynomials have played a fundamental role in algebraic
combinatorics (see, e.g., [fink2018schubert, monical2019newton,
BERGERON_BILLEY, BILLEY_STANLEY, KNUTSON_MILLER_SUBWORD,
KNUTSON_MILLER_SCHUBERT] and the references therein).

We first recall the definition of Schubert polynomials. Let
${\mathfrak{S}}_{p}$ be the symmetric group on the set $[p]=\\{1,\ldots,p\\}$.
For every $i\in[p-1]=\\{1,\ldots,p-1\\}$ we have the transposition
$\mathfrak{s}_{i}=(i,i+1)\in{\mathfrak{S}}_{p}$. Recall that the set
$\mathfrak{T}=\\{\mathfrak{s}_{i}\mid 1\leq i<p\\}$ generates
${\mathfrak{S}}_{p}$. The _length_ $\ell(\pi)$ of a permutation $\pi$ is the
least amount of elements in $\mathfrak{T}$ needed to obtain $\pi$ from the
identity permutation. The permutation $\pi_{0}=(p,p-1,\ldots,2,1)$ (in one-
line notation) is the longest permutation that has length $\frac{p(p-1)}{2}$.

######  Definition .

The double Schubert polynomial
${\mathfrak{S}}_{\pi}({\mathbf{t}},{\mathbf{s}})\in\mathbb{Z}[t_{1},\ldots,t_{p},s_{1},\dots,s_{p}]$
of a permutation $\pi\in{\mathfrak{S}}_{p}$ is defined recursively in the
following way. First we define ${\mathfrak{S}}_{\pi_{0}}=\prod_{i+j\leq
p}(t_{i}-s_{j})$, and for any permutation $\pi$ and transposition
$\mathfrak{s}_{i}$ with $\ell(\mathfrak{s}_{i}\pi)<\ell(\pi)$ we set

${\mathfrak{S}}_{\mathfrak{s}_{i}\pi}\,=\,\dfrac{{\mathfrak{S}}_{\pi}-\mathfrak{s}_{i}{\mathfrak{S}}_{\pi}}{t_{i}-t_{i+1}},$

where ${\mathfrak{S}}_{p}$ acts only on $\mathbb{Z}[t_{1},\ldots,t_{p}]$ by
permutation of variables. The _(ordinary) Schubert polynomial_
${\mathfrak{S}}_{\pi}({\mathbf{t}},\mathbf{0})\in\mathbb{Z}[t_{1},\ldots,t_{p}]$
is obtained from ${\mathfrak{S}}_{\pi}$ by setting $s_{j}=0$ for all $j$.

The monomial expansion of ordinary Schubert polynomials has been
combinatorially analyzed using different objects such as compatible sequences
[BILLEY_JOCKUSH_STANLEY], reduced pipe dreams [fomin1994grothendieck,
BERGERON_BILLEY], and Kohnert diagrams [KOHNERT]. The description using pipe
dreams also works for the double Schubert polynomials
[miller2005combinatorial, Corollary 16.30].

Following [monical2019newton], we say that a polynomial
$f=\sum_{{\mathbf{n}}}c_{{\mathbf{n}}}{\mathbf{x}}^{\mathbf{n}}\in\mathbb{Z}[x_{1},\ldots,x_{n}]$
has the _Saturated Newton Polytope property_ (SNP property for short) if the
support ${\text{supp}}(f)=\\{{\mathbf{n}}\in\mathbb{N}^{n}\mid
c_{\mathbf{n}}\neq 0\\}$ of $f$ is equal to
$\text{Newton}(f)\cap\mathbb{N}^{n}$, where
$\text{Newton}(f)=\text{ConvexHull}\\{{\mathbf{n}}\in\mathbb{N}^{n}\mid
c_{\mathbf{n}}\neq 0\\}$ denotes the _Newton polytope_ of $f$; in other words,
if the support of $f$ consists of the integer points of a polytope.

The main goal of this paper is to confirm the following challenging conjecture
by Monical, Tokcan and Yong that appeared in [monical2019newton, Conjecture
5.2].

######  Conjecture  ([monical2019newton]).

Double Schubert polynomials have the Saturated Newton Polytope property.

We confirm the conjecture by proving a stronger result that the support of
each double Schubert polynomial is a discrete polymatroid. A discrete
polymatroid $\mathcal{P}$ on $[n]=\\{1,\ldots,n\\}$ is a collection of points
in $\mathbb{N}^{n}$ of the following form

$\mathcal{P=}\left\\{(x_{1},\ldots,x_{n})\in\mathbb{N}^{n}\;\mid\;\sum_{j\in{\mathfrak{J}}}x_{j}\leq
r({\mathfrak{J}}),\;\forall{\mathfrak{J}}\subsetneq[n],\;\sum_{i\in[n]}x_{i}=r([n])\right\\}$

with $r$ being a rank function on $[n]$. A _rank function on_ $[n]$ is a
function $r:2^{[n]}\rightarrow\mathbb{N}$ satisfying the following three
properties: (i) $r(\emptyset)=0$, (ii) $r({\mathfrak{J}}_{1})\leq
r({\mathfrak{J}}_{2})$ if
${\mathfrak{J}}_{1}\subseteq{\mathfrak{J}}_{2}\subseteq[n]$, and (iii)
$r({\mathfrak{J}}_{1}\cap{\mathfrak{J}}_{2})+r({\mathfrak{J}}_{1}\cup{\mathfrak{J}}_{2})\leq
r({\mathfrak{J}}_{1})+r({\mathfrak{J}}_{2})$ if
${\mathfrak{J}}_{1},{\mathfrak{J}}_{2}\subseteq[n]$.

The following is the main theorem of this article.

######  Theorem A.

Let $\pi\in{\mathfrak{S}}_{p}$ be a permutation and
${\mathfrak{S}}_{\pi}({\mathbf{t}},{\mathbf{s}})\in\mathbb{Z}[t_{1},\ldots,t_{p},s_{1},\dots,s_{p}]$
be the corresponding double Schubert polynomial. Then, the support
${\text{supp}}({\mathfrak{S}}_{\pi})\subset\mathbb{N}^{2p}$ of
${\mathfrak{S}}_{\pi}$ is a discrete polymatroid on $[2p]=\\{1,\ldots,2p\\}$.
In particular, the statement of Section missing 1 holds.

Our approach to prove Theorem A can be summarized in the following quote by
Miller and Sturmfels [miller2005combinatorial, Introduction to Chapter 15]:
_“We consider the finest possible multigrading, which demands the refined
toolkit of a new generation of combinatorialists.”_ More precisely, we utilize
the result that double Schubert polynomials equal the multidegree polynomial
of Schubert determinantal ideals with the aforementioned _“finest possible
multigrading”_ (see [miller2005combinatorial, Theorem 15.40]), and then we
develop a method of _standardization of ideals_. This process of
standardization allows us to study multidegrees in certain non-standard
multigradings by reducing the problem to a standard multigraded setting. Our
main tool is Theorem 2.1 from [castillo2020multidegrees] which shows that the
support of the multidegree polynomial of any multihomogeneous prime ideal
(with usual standard multigrading) is a discrete polymatroid. Here we extend
this theorem to the family of non-standard multigradings that we study.

Much interest has been paid to the important conjectures proposed in
[monical2019newton] and a number of them have already been confirmed (see
[fink2018schubert]). Therefore, Theorem A settles a hitherto remaining
conjecture from [monical2019newton] and gives further evidence to the ubiquity
of the SNP property in many “combinatorially defined polynomials”.

The structure of the paper is as follows. We review the notion of multidegrees
in Section missing 2 and recall the connection between double Schubert
polynomials and Schubert determinantal ideals in Section missing 3. Section
missing 4 contains our main results, in particular the proof of Theorem A.

Acknowledgments. F.C. thanks Ghent University for their hospitality. Y.C.R.
was partially supported by an FWO Postdoctoral Fellowship (1220122N). F.M. was
partially supported by BOF/STA/201909/038, and FWO grants (G023721N,
G0F5921N). J.M. was partially supported by NSF Grant DMS #2001645.

##  2\. A short recap on multidegrees

In this short section, we briefly recall the notion of multidegrees and some
of its basic properties; for more details the reader is referred to
[miller2005combinatorial, cidruiz2021mixed].

Let $\mathbb{k}$ be a field and $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a
$\mathbb{Z}^{p}$-graded polynomial ring (for now, we do not assume the grading
to be positive). Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded
module and $F_{\bullet}$ be a $\mathbb{Z}^{p}$-graded free $R$-resolution
$F_{\bullet}:\;\cdots\rightarrow F_{i}\rightarrow
F_{i-1}\rightarrow\cdots\rightarrow F_{1}\rightarrow F_{0}$ of $M$. Let
$t_{1},\ldots,t_{p}$ be variables over $\mathbb{Z}$ and consider the
polynomial ring $\mathbb{Z}[{\mathbf{t}}]=\mathbb{Z}[t_{1},\ldots,t_{p}]$,
where the variable $t_{i}$ corresponds with the $i$-th elementary vector
${\mathbf{e}}_{i}\in\mathbb{Z}^{p}$. If we write
$F_{i}=\bigoplus_{j}R(-\mathbf{b}_{i,j})$ with
$\mathbf{b}_{i,j}=(\mathbf{b}_{i,j,1},\ldots,\mathbf{b}_{i,j,p})\in\mathbb{Z}^{p}$,
then we define the Laurent polynomial
$\left[F_{i}\right]_{\mathbf{t}}\,:=\,\sum_{j}{\mathbf{t}}^{\mathbf{b}_{i,j}}=\sum_{j}t_{1}^{\mathbf{b}_{i,j,1}}\cdots
t_{p}^{\mathbf{b}_{i,j,p}}.$ Then, the _K-polynomial_ of $M$ is defined by

$\mathcal{K}(M;{\mathbf{t}})\,:=\,\sum_{i}{(-1)}^{i}\left[F_{i}\right]_{\mathbf{t}}.$

It turns out that, even if the grading of $R$ is non-positive and we do not
have a well-defined notion of Hilbert series, the above definition of
K-polynomial is an invariant of the module $M$ and it does not depend on the
chosen free $R$-resolution $F_{\bullet}$ (see [miller2005combinatorial,
Theorem 8.34]).

######  Definition .

The _multidegree polynomial_ of a finitely generated $\mathbb{Z}^{p}$-graded
$R$-module $M$ is the homogeneous polynomial
$\mathcal{C}(M;{\mathbf{t}})\in\mathbb{Z}[{\mathbf{t}}]$ given as the sum of
all terms in

$\mathcal{K}(M;\mathbf{1}-{\mathbf{t}})=\mathcal{K}(M;1-t_{1},\ldots,1-t_{p})$

having total degree ${\text{codim}}(M)=n-\dim(M)$.

One case of particular interest is when $R$ is a standard multigraded
polynomial ring. We say that $R$ is _standard $\mathbb{Z}^{p}$-graded_ if the
total degree of each variable $x_{i}$ is equal to one (i.e., for each $1\leq
i\leq n$, we have $\deg(x_{i})={\mathbf{e}}_{k_{i}}\in\mathbb{Z}^{p}$ with
$1\leq k_{i}\leq p$). The study of standard multigraded algebras is of utmost
importance as they correspond with closed subschemes of a product of
projective spaces (see, e.g., [castillo2020multidegrees] and the references
therein). Since the coefficients of the multidegree polynomial are non-
negative in the standard multigraded case, it becomes natural to address the
positivity of these coefficients. For each subset
$\mathfrak{J}=\\{j_{1},\ldots,j_{k}\\}\subseteq[p]=\\{1,\ldots,p\\}$ denote by
$R_{({\mathfrak{J}})}$ the $\mathbb{Z}^{k}$-graded $\mathbb{k}$-algebra given
by

$R_{({\mathfrak{J}})}:=\bigoplus_{\begin{subarray}{c}i_{1}\geq
0,\ldots,i_{p}\geq 0\\\ i_{j}=0\text{ if
}j\not\in{\mathfrak{J}}\end{subarray}}{\left[R\right]}_{(i_{1},\ldots,i_{p})},$

and for any $R$-homogeneous ideal $I\subset R$ we define
$I_{({\mathfrak{J}})}$ as the contraction $I_{({\mathfrak{J}})}:=I\cap
R_{({\mathfrak{J}})}$. The following theorem completely characterizes the
positivity of multidegrees and is our main tool to prove Theorem A.

######  Theorem 2.1 ([castillo2020multidegrees]).

Let $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a standard $\mathbb{Z}^{p}$-graded
polynomial ring. Let $I\subset R$ be an $R$-homogeneous prime ideal. Write the
multidegree polynomial of $\mathcal{C}(R/I;{\mathbf{t}})$ as

$\mathcal{C}(R/I;{\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|={\text{codim}}(I)\end{subarray}}c_{{\mathbf{n}}}{\mathbf{t}}^{\mathbf{n}}\;\in\;\mathbb{N}[t_{1},\ldots,t_{p}].$

Then, for all ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{n}}|={\text{codim}}(I)=n-\dim(R/I)$, we have that
$c_{\mathbf{n}}>0$ if and only if for each
${\mathfrak{J}}=\\{j_{1},\ldots,j_{k}\\}\subseteq[p]$ the inequality
$n_{j_{1}}+\cdots+n_{j_{k}}\,\geq\,{\text{codim}}\left(I_{({\mathfrak{J}})}\right)$
holds. Furthermore, the support of $\mathcal{C}(R/I;{\mathbf{t}})$ is a
discrete polymatroid.

###### Proof.

We embed $X=\text{MultiProj}(R)$ as a closed subscheme of a multiprojective
space
${\mathbb{P}}:={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
From [castillo2020multidegrees, Remark 2.9] we have that
${\mathbf{n}}\in{\text{supp}}(\mathcal{C}(R/I;{\mathbf{t}}))$ if and only if
$\deg_{\mathbb{P}}^{{\mathbf{m}}-{\mathbf{n}}}(X)>0$ where
${\mathbf{m}}-{\mathbf{n}}=(m_{1}-n_{1},\ldots,m_{p}-n_{p})$. Then,
[castillo2020multidegrees, Theorem A] implies that
${\mathbf{n}}\in{\text{supp}}(\mathcal{C}(R/I;{\mathbf{t}}))$ if and only if
$|{\mathbf{n}}|={\text{codim}}(I)$ and
$\sum_{j\in{\mathfrak{J}}}n_{j}\geq{\text{codim}}\left(I_{({\mathfrak{J}})}\right)$
for each ${\mathfrak{J}}\subseteq[p]$. Equivalently, we obtain that
${\mathbf{n}}\in{\text{supp}}(\mathcal{C}(R/I;{\mathbf{t}}))$ if and only if
$|{\mathbf{n}}|={\text{codim}}(I)$ and

$\sum_{j\in{\mathfrak{J}}}n_{j}\,\leq\,{\text{codim}}(I)-{\text{codim}}\left(I_{([p]\setminus{\mathfrak{J}})}\right)\,=\,\sum_{j\in{\mathfrak{J}}}m_{j}+r([p]\setminus{\mathfrak{J}})-r([p])$

for each ${\mathfrak{J}}\subseteq[p]$, where $r:2^{[p]}\rightarrow\mathbb{N}$
is the rank function
$r({\mathfrak{J}}):=\dim\big{(}\text{MultiProj}\big{(}R_{({\mathfrak{J}})}/I_{({\mathfrak{J}})}\big{)}\big{)}$
(see [castillo2020multidegrees, Proposition 5.1]). Finally, we can check that
$s:2^{[p]}\rightarrow\mathbb{N}$ with
$s({\mathfrak{J}}):=\sum_{j\in{\mathfrak{J}}}m_{j}+r([p]\setminus{\mathfrak{J}})-r([p])$
is a rank function (see, e.g., [schrijver2003combinatorial, §44.6f]), and so
it follows that ${\text{supp}}(\mathcal{C}(R/I;{\mathbf{t}}))$ is a
polymatroid. ∎

##  3\. Schubert determinantal ideals

Here we recall the connection between double Schubert polynomials and Schubert
determinantal ideals (for more details, the reader is referred to
[miller2005combinatorial, Chapters 15, 16]).

First, we define matrix Schubert varieties and Schubert determinantal ideals
by following [miller2005combinatorial, Chapter 15]. Let $\mathbb{k}$ be an
algebraically closed field and $M_{p}(\mathbb{k})$ be the $\mathbb{k}$-vector
space of $p\times p$ matrices with entries in $\mathbb{k}$. As an affine
variety we define its coordinate ring as
$\widetilde{R}=\mathbb{k}[x_{i,j}\mid(i,j)\in[p]\times[p]]$. Furthermore, we
consider a $(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-grading on $\widetilde{R}$
by setting
$\deg(x_{i,j})={\mathbf{e}}_{i}\oplus-{\mathbf{e}}_{j}\in\mathbb{Z}^{p}\oplus\mathbb{Z}^{p}$,
where ${\mathbf{e}}_{i}\in\mathbb{Z}^{p}$ denotes the $i$-th elementary
vector.

######  Definition .

Let $\pi$ be a permutation matrix. The _matrix Schubert variety_
$\overline{X_{\pi}}\subset M_{p}(\mathbb{k})$ is the subvariety given by
$\overline{X_{\pi}}=\\{Z\in M_{p}(\mathbb{k})\mid\text{rank}(Z_{m\times
n})\leq\text{rank}(\pi_{m\times n})\;\text{ for all }\;m,n\\},$ where
$Z_{m\times n}$ is the restriction to the first $m$ rows and $n$ columns. The
_Schubert determinantal ideal_ $I_{\pi}\subset\widetilde{R}$ is the
$\widetilde{R}$-homogeneous ideal generated by all minors in
$\mathbf{X}_{m\times n}$ of size $1+\text{rank}(\pi_{m\times n})$ for all $m$
and $n$, where $\mathbf{X}=(x_{i,j})$ is the $p\times p$ matrix with the
variables of $\widetilde{R}$.

The following theorem collects several results of fundamental importance to
our approach. In particular, it shows that double Schubert polynomials equal
the multidegree polynomial of matrix Schubert varieties. To define
multidegrees over $\widetilde{R}$ with its
$(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-grading, we consider the polynomial
ring
$\mathbb{Z}[{\mathbf{t}},{\mathbf{s}}]=\mathbb{Z}[t_{1},\ldots,t_{p},s_{1},\ldots,s_{p}]$,
where $t_{i}$ corresponds with
${\mathbf{e}}_{i}\oplus\mathbf{0}\in\mathbb{Z}^{p}\oplus\mathbb{Z}^{p}$ and
$s_{i}$ corresponds with
$\mathbf{0}\oplus{\mathbf{e}}_{i}\in\mathbb{Z}^{p}\oplus\mathbb{Z}^{p}$.

######  Theorem 3.1.

Let $\pi\in{\mathfrak{S}}_{p}$ be a permutation and denote also by $\pi$ the
corresponding permutation matrix. Then, the following statements hold:

  1. (i)

$I_{\pi}$ is a prime ideal, and so it coincides with the ideal
$I(\overline{X_{\pi}})$ of polynomials vanishing on the matrix Schubert
variety $\overline{X_{\pi}}$. [miller2005combinatorial, Corollary 16.29]

  2. (ii)

$\widetilde{R}/I_{\pi}$ is a Cohen-Macaulay ring. [miller2005combinatorial,
Corollary 16.44]

  3. (iii)

${\mathfrak{S}}_{\pi}({\mathbf{t}},\mathbf{s})=\mathcal{C}(\widetilde{R}/I_{\pi};{\mathbf{t}},{\mathbf{s}})$.
[miller2005combinatorial, Theorem 15.40]

The next technical lemma will allow us to substitute the grading of
$\widetilde{R}$ which has negative components for the degrees of the
variables. Let $R=\mathbb{k}[x_{i,j}\mid(i,j)\in[p]\times[p]]$ with induced
$(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-grading by setting
$\deg(x_{i,j})={\mathbf{e}}_{i}\oplus{\mathbf{e}}_{j}\in\mathbb{Z}^{p}\oplus\mathbb{Z}^{p}$.
As for $\widetilde{R}$, define multidegrees over $R$ in the polynomial ring
$\mathbb{Z}[{\mathbf{t}},{\mathbf{s}}]$.

######  Lemma .

Let $I\subset\widetilde{R}$ be an $\widetilde{R}$-homogeneous ideal, and
denote also by $I$ the corresponding $R$-homogeneous ideal in $R$. Set
$\widetilde{P}({\mathbf{t}},{\mathbf{s}})=\mathcal{C}(\widetilde{R}/I;{\mathbf{t}},{\mathbf{s}})\in\mathbb{Z}[{\mathbf{t}},{\mathbf{s}}]$
and
$P({\mathbf{t}},{\mathbf{s}})=\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})\in\mathbb{Z}[{\mathbf{t}},{\mathbf{s}}]$.
Then we have

$\widetilde{P}({\mathbf{t}},{\mathbf{s}})=\widetilde{P}(t_{1},\ldots,t_{p},s_{1},\ldots,s_{p})=P(t_{1},\ldots,t_{p},-s_{1},\ldots,-s_{p})=P({\mathbf{t}},-{\mathbf{s}}).$

###### Proof.

Notice that, if $\widetilde{F}_{\bullet}$ is a
$(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-graded free $\widetilde{R}$-resolution
of $\widetilde{R}/I$ with
$\widetilde{F}_{i}=\bigoplus_{j}\widetilde{R}(-\mathbf{a}_{i,j},-\mathbf{b}_{i,j})$,
then there is a corresponding $(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-graded
free $R$-resolution $F_{\bullet}$ of $R/I$ with
$F_{i}=\bigoplus_{j}R(-\mathbf{a}_{i,j},\mathbf{b}_{i,j})$. By definition,
this yields the equality of $K$-polynomials

$\mathcal{K}(\widetilde{R}/I;{\mathbf{t}},{\mathbf{s}})=\mathcal{K}(\widetilde{R}/I;t_{1},\ldots,t_{p},s_{1},\ldots,s_{p})=\mathcal{K}(R/I;t_{1},\ldots,t_{p},s_{1}^{-1},\ldots,s_{p}^{-1})=\mathcal{K}(R/I;{\mathbf{t}},{\mathbf{s}}^{\mathbf{-1}}).$

From [miller2005combinatorial, Claim 8.54], we have
$\mathcal{K}(R/I;\mathbf{1-t},\mathbf{1-s})=\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})+Q({\mathbf{t}},{\mathbf{s}})$,
where $Q({\mathbf{t}},{\mathbf{s}})$ is a polynomial with terms of degree at
least ${\text{codim}}(I)+1$. Equivalently, we get
$\mathcal{K}(R/I;{\mathbf{t}},{\mathbf{s}})=\mathcal{C}(R/I;\mathbf{1-t},\mathbf{1-s})+Q(\mathbf{1-t},\mathbf{1-s})$.
It then follows that

$\mathcal{K}(\widetilde{R}/I;\mathbf{1-t},\mathbf{1-s})=\mathcal{C}(R/I;t_{1},\ldots,t_{p},1-\tfrac{1}{1-s_{1}},\ldots,1-\tfrac{1}{1-s_{p}})+Q(t_{1},\ldots,t_{p},1-\tfrac{1}{1-s_{1}},\ldots,1-\tfrac{1}{1-s_{p}}).$

By expanding the right hand side of the above equality, the result of the
lemma is obtained. ∎

##  4\. Standardization of ideals

In this section, we develop a process of standardization of ideals in a
certain non-standard multigrading. This process will allow us to show that the
support of the multidegree polynomial of any Cohen-Macaulay prime ideal is a
discrete polymatroid in the non-standard multigradings that we consider. The
following setup is used throughout this section.

######  Setup .

Let $p\geq 1$ be a positive integer and $\mathbb{k}$ be a field. Let $R$ and
$S$ be the polynomial rings $R=\mathbb{k}[{\mathbf{x}}]$ and
$S=\mathbb{k}[\mathbf{w},\mathbf{z}]$ over the set of variables
${\mathbf{x}}=\\{x_{i,j}\\}_{1\leq i,j\leq p}$,
$\mathbf{w}=\\{w_{i,j}\\}_{1\leq i,j\leq p}$ and
$\mathbf{z}=\\{z_{i,j}\\}_{1\leq i,j\leq p}$. We consider $R$ and $S$ as
$(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-graded rings by setting that

$\deg(x_{i,j})={\mathbf{e}}_{i}\oplus{\mathbf{e}}_{j},\quad\deg(w_{i,j})={\mathbf{e}}_{i}\oplus\mathbf{0}\quad\text{and}\quad\deg(z_{i,j})=\mathbf{0}\oplus{\mathbf{e}}_{j},$

where ${\mathbf{e}}_{i}\in\mathbb{Z}^{p}$ denotes the $i$-th elementary vector
and $\mathbf{0}\in\mathbb{Z}^{p}$ denotes the zero vector. We define the
$\mathbb{k}$-algebra homomorphism

$\phi:R=\mathbb{k}[{\mathbf{x}}]\longrightarrow
S=\mathbb{k}[\mathbf{w},\mathbf{z}],\quad\phi(x_{i,j})=w_{i,j}z_{i,j}.$

For an $R$-homogeneous ideal $I\subset R$, we say that the extension
$\phi(I)S$ is the _standardization_ of $I$, as $\phi(I)S$ is an
$S$-homogeneous ideal in the standard multigraded polynomial ring $S$. Let
${\mathbf{t}}=\\{t_{1},\ldots,t_{p}\\}$ and
${\mathbf{s}}=\\{s_{1},\ldots,s_{p}\\}$ be variables indexing the
$(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-grading, where $t_{i}$ corresponds with
${\mathbf{e}}_{i}\oplus\mathbf{0}\in\mathbb{Z}^{p}\oplus\mathbb{Z}^{p}$ and
$s_{i}$ corresponds with
$\mathbf{0}\oplus{\mathbf{e}}_{i}\in\mathbb{Z}^{p}\oplus\mathbb{Z}^{p}$. Given
a finitely generated graded $R$-module $M$ and a finitely generated graded
$S$-module $N$, by a slight abuse of notation, we consider both multidegrees
$\mathcal{C}(M;{\mathbf{t}},{\mathbf{s}})$ and
$\mathcal{C}(N;{\mathbf{t}},{\mathbf{s}})$ as elements of the same polynomial
ring
$\mathbb{Z}[{\mathbf{t}},{\mathbf{s}}]=\mathbb{Z}[t_{1},\ldots,t_{p},s_{1},\ldots,s_{p}]$.

First, we show some basic properties of the process of standardization.

######  Proposition .

Assume Section missing 4. Let $I\subset R$ be an $R$-homogeneous ideal and
$J=\phi(I)S$ be its standardization. Then, the following statements hold:

  1. (i)

${\text{codim}}(I)={\text{codim}}(J)$.

  2. (ii)

$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})=\mathcal{C}(S/J;{\mathbf{t}},{\mathbf{s}})$.

  3. (iii)

If $R/I$ is a Cohen-Macaulay ring, then $S/J$ also is.

  4. (iv)

Let $>$ be a monomial order on $R$ and $>^{\prime}$ be a monomial order on $S$
which is compatible with $\phi$ (i.e., if $f,g\in R$ with $f>g$, then
$\phi(f)>^{\prime}\phi(g)$). Then
${\text{in}}_{>^{\prime}}(J)=\phi({\text{in}}_{>}(I))S$.

###### Proof.

Let $T$ be the polynomial ring
$T=\mathbb{k}[{\mathbf{x}},\mathbf{w},\mathbf{z}]\cong R\otimes_{\mathbb{k}}S$
with its natural $(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-grading induced from
the ones of $R$ and $S$. We now think of $R$ and $S$ as subrings of $T$.
Consider the quotient ring $T/IT$ and notice that
$\\{x_{i,j}-w_{i,j}z_{i,j}\\}_{1\leq i,j\leq p}$ is a regular sequence of
homogeneous elements over $T/IT$. We also have the following natural
isomorphism

$\frac{T}{IT+\left(\\{x_{i,j}-w_{i,j}z_{i,j}\\}_{1\leq i,j\leq
p}\right)}\;\cong\;S/J.$

As the natural inclusion $R\hookrightarrow T$ is a polynomial extension, we
have that $\dim(T/IT)=\dim(R/I)+\dim(S)=\dim(R)+\dim(S)-{\text{codim}}(I)$ and
that $T/IT$ is Cohen-Macaulay when $R/I$ is. So, by cutting out with the
regular sequence described above, we obtain that
$\dim(S/J)=\dim(T/IT)-\dim(R)=\dim(S)-{\text{codim}}(I)$ and that $S/J$ is
Cohen-Macaulay when $T/IT$ is. This completes the proofs of parts (i) and
(iii).

Let $F_{\bullet}:\cdots\xrightarrow{f_{2}}F_{1}\xrightarrow{f_{1}}F_{0}$ be a
graded free $R$-resolution of $R/I$. Since
$\\{x_{i,j}-w_{i,j}z_{i,j}\\}_{1\leq i,j\leq p}$ is a regular sequence on both
$T$ and $T/IT$, it follows that
${\text{Tor}}_{k}^{T}\left(T/IT,T/(\\{x_{i,j}-w_{i,j}z_{i,j}\\}_{1\leq i,j\leq
p})\right)=0$ for all $k>0$, and so
$G_{\bullet}=F_{\bullet}\otimes_{R}T/(\\{x_{i,j}-w_{i,j}z_{i,j}\\}_{1\leq
i,j\leq p})$ provides (up to isomorphism) a graded free $S$-resolution of
$S/J$. The identification of $G_{\bullet}$ as a resolution of $S$-modules is
the same as $\phi(F_{\bullet})$ (more precisely, $G_{\bullet}$ has the same
shiftings as $F_{\bullet}$ in the
$(\mathbb{Z}^{p}\oplus\mathbb{Z}^{p})$-grading and the $i$-th differential
matrix of $G_{\bullet}$ is given by the substitution $\phi(f_{i})$ of
$f_{i}$). Therefore, by definition, we obtain the equality
$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})=\mathcal{C}(S/J;{\mathbf{t}},{\mathbf{s}})$
that shows part (ii).

To show part (iv) we can use Buchberger’s algorithm (see, e.g., [EISEN_COMM,
Chapter 15]). Indeed, we can perform essentially the same steps of the
algorithm in a set of generators of $I$ and the corresponding set of
generators for $J$. ∎

The following theorem provides the main result of this section, it shows that
the support of the multidegree polynomial is a discrete polymatroid for Cohen-
Macaulay prime ideals in $R$. The proof is carried out by performing a
standardization process that allows us to invoke Theorem 2.1.

######  Theorem 4.1.

Assume Section missing 4. Let $I\subset R$ be an $R$-homogeneous Cohen-
Macaulay prime ideal. Then, the support of the multidegree polynomial
$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})$ is a discrete polymatroid.

###### Proof.

Let $\mathcal{L}=\\{(i,j)\mid x_{i,j}\in I\\}$ be the set of indices such that
the corresponding variable belongs to $I$. We consider the polynomial rings
$R^{\prime}=\mathbb{k}[x_{i,j}\mid(i,j)\not\in\mathcal{L}]\subset R$ and
$S^{\prime}=\mathbb{k}[w_{i,j},z_{i,j}\mid(i,j)\not\in\mathcal{L}]\subset S.$
Let $I^{\prime}\subset R^{\prime}$ be the (unique) ideal that satisfies the
condition $I=I^{\prime}R+\left(x_{i,j}\mid(i,j)\in\mathcal{L}\right)$. By
construction we have that $x_{i,j}\not\in I^{\prime}$ for all $x_{i,j}\in
R^{\prime}$. Since $R/I\cong R^{\prime}/I^{\prime}$, it follows that
$I^{\prime}$ is also a Cohen-Macaulay prime ideal.

Let $J^{\prime}=\phi(I^{\prime}R)S\cap S^{\prime}$. For any $w_{i,j}z_{i,j}\in
S^{\prime}$, Section missing 4(i) and the fact that the corresponding
$x_{i,j}$ does not belong to the prime $I^{\prime}$ imply that

${\text{codim}}(J^{\prime}S+w_{i,j}z_{i,j}S)={\text{codim}}(I^{\prime}R+x_{i,j}R)={\text{codim}}(I^{\prime}R)+1={\text{codim}}(J^{\prime}S)+1.$

By Section missing 4(iii), $S/J^{\prime}S$ is Cohen-Macaulay, and so it
necessarily follows that $w_{i,j}z_{i,j}$ is a non-zero-divisor over
$S/J^{\prime}S$ for all $w_{i,j}z_{i,j}\in S^{\prime}$. Consequently, we
obtain that $S^{\prime}/J^{\prime}$ is a domain if and only if
$\left(S^{\prime}/J^{\prime}\right)_{\prod w_{i,j}z_{i,j}}$ is a domain. Let
$B=\mathbb{k}[w_{i,j},z_{i,j},z_{i,j}^{-1}\mid(i,j)\not\in\mathcal{L}]$ and
consider the automorphism given by

$\psi:B\rightarrow B,\quad w_{i,j}\mapsto\frac{w_{i,j}}{z_{i,j}},\quad
z_{i,j}\mapsto{z_{i,j}}.$

The ideal $\psi(J^{\prime}B)$ coincides with the extension of $I^{\prime}$ in
$B$ under the ring homomorphism $R^{\prime}\rightarrow B,x_{i,j}\mapsto
w_{i,j}$, and so it follows that $\psi(J^{\prime}B)$ and, consequently,
$J^{\prime}B$ are prime ideals. We then conclude that $J^{\prime}$ is a prime
ideal.

Since the variables $x_{i,j}$ with indices in $\mathcal{L}$ form a regular
sequence over $R/I^{\prime}R$, we obtain the equation

(1)
$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})=\prod_{(i,j)\in\mathcal{L}}(t_{i}+s_{j})\cdot\mathcal{C}(R/I^{\prime}R;{\mathbf{t}},{\mathbf{s}})$

(see, e.g., [miller2005combinatorial, Exercise 8.12]). To conclude the proof,
it is now sufficient to show that the support of
$\mathcal{C}(R/I^{\prime}R;{\mathbf{t}},{\mathbf{s}})$ is a discrete
polymatroid; indeed, we would obtain that the support of
$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})$ is a Minkowski sum of a finite
number of discrete polymatroids which in turn is also a discrete polymatroid
by [schrijver2003combinatorial, Corollary 46.2c]. Finally, this condition
follows by applying Theorem 2.1 to the prime ideal $J^{\prime}S$ and
exploiting the equality
$\mathcal{C}(R/I^{\prime}R;{\mathbf{t}},{\mathbf{s}})=\mathcal{C}(S/J^{\prime}S;{\mathbf{t}},{\mathbf{s}})$
from Section missing 4(ii). ∎

We are now ready to prove the main result of this paper.

###### Proof of Theorem A.

As we already have all the necessary ingredients, the proof follows
straightforwardly by combining Theorem 3.1, Section missing 3 and Theorem 4.1.
∎

Furthermore, we determine the defining inequalities of the discrete
polymatroids in Theorem 4.1 and, accordingly, in Theorem A. Similarly to
Section missing 2, for any two subsets
${\mathfrak{J}}_{1},{\mathfrak{J}}_{2}\subseteq[p]$, we denote by
$R_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\subseteq R$ and
$S_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\subseteq S$ the
$(\mathbb{Z}^{|{\mathfrak{J}}_{1}|}\oplus\mathbb{Z}^{|{\mathfrak{J}}_{2}|})$-graded
$\mathbb{k}$-algebras obtained by restricting to the positions in
${\mathfrak{J}}_{1}$ for the first part $\mathbb{Z}^{p}\oplus\mathbf{0}$ of
the grading, and to the ones in ${\mathfrak{J}}_{2}$ for the second part of
the grading $\mathbf{0}\oplus\mathbb{Z}^{p}$.

######  Theorem 4.2.

Assume Section missing 4. Let $I\subset R$ be an $R$-homogeneous Cohen-
Macaulay prime ideal. Then, we have that the coefficient of
$\mathbf{t}^{\mathbf{r}}\mathbf{s}^{\mathbf{c}}=t_{1}^{r_{1}}\cdots
t_{p}^{r_{p}}s_{1}^{c_{1}}\cdots s_{p}^{c_{p}}$ is nonzero in
$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})$ if and only if

  1. (i)

$\sum_{j\in[p]}r_{j}+\sum_{j\in[p]}c_{j}={\text{codim}}(I)$

  2. (ii)

For every $\mathfrak{J}_{1},\mathfrak{J}_{2}\subseteq[p]$, we have that
$\sum_{j\in\mathfrak{J}_{1}}r_{j}+\sum_{j\in\mathfrak{J}_{2}}c_{j}\geq{\text{codim}}\big{(}I_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\big{)}$,
where $I_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}$ is the contracted ideal
$I_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}=I\cap
R_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}$.

###### Proof.

By Theorem 4.1, we know that the Newton polytope of
$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})$ is a base polymatroid polytope,
and so, under the condition
$\sum_{j\in[p]}r_{j}+\sum_{j\in[p]}c_{j}={\text{codim}}(I)$, all its defining
inequalities are of the form

(2) $\sum_{j\in\mathfrak{J}_{1}}r_{j}+\sum_{j\in\mathfrak{J}_{2}}c_{j}\geq
C({\mathfrak{J}}_{1},{\mathfrak{J}}_{2}),$

for some constant $C({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})$ that depends on
the subsets ${\mathfrak{J}}_{1},{\mathfrak{J}}_{2}\subseteq[p]$. We now
determine $C({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})$.

We keep the same notation of the proof of Theorem 4.1, in particular,
$I=I^{\prime}R+\left(x_{i,j}\mid(i,j)\in\mathcal{L}\right)$. Equation
(missing) 1 decomposes the Newton polytope of
$\mathcal{C}(R/I;{\mathbf{t}},{\mathbf{s}})$ as the Minkowski sum of the
Newton polytopes of $\prod_{(i,j)\in\mathcal{L}}(t_{i}+s_{j})$ and
$\mathcal{C}(S/J^{\prime}S;{\mathbf{t}},{\mathbf{s}})$, both of which are also
base polymatroid polytopes. So, we analyze the minimum of the sum in Equation
(missing) 2 with the two contributions.

  1. (a)

$\textrm{Newton}(\prod_{(i,j)\in\mathcal{L}}(t_{i}+s_{j}))$ is determined by
the equality $\sum_{j\in[p]}r_{j}+\sum_{j\in[p]}c_{j}=|\mathcal{L}|$ and the
inequalities
$\sum_{j\in\mathfrak{J}_{1}}r_{j}+\sum_{j\in\mathfrak{J}_{2}}c_{j}\geq\big{|}\\{(i,j)\in\mathcal{L}\mid
i\in{\mathfrak{J}}_{1}\text{ and }j\in{\mathfrak{J}}_{2}\\}\big{|}$.

  2. (b)

Due to Theorem 2.1,
$\textrm{Newton}(\mathcal{C}(S/J^{\prime}S;{\mathbf{t}},{\mathbf{s}}))$ is
determined by the equality
$\sum_{j\in[p]}r_{j}+\sum_{j\in[p]}c_{j}={\text{codim}}(J^{\prime}S)$ and the
inequalities
$\sum_{j\in\mathfrak{J}_{1}}r_{j}+\sum_{j\in\mathfrak{J}_{2}}c_{j}\geq{\text{codim}}\big{(}{(J^{\prime}S)}_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\big{)}$,
where ${(J^{\prime}S)}_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}$ is the
contracted ideal
${(J^{\prime}S)}_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}=J^{\prime}S\cap
S_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}$.

Notice that Section missing 4(i) yields the equality

${\text{codim}}\big{(}I_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\big{)}={\text{codim}}\big{(}J_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\big{)}={\text{codim}}\big{(}{(J^{\prime}S)}_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\big{)}+\big{|}\\{(i,j)\in\mathcal{L}\mid
i\in{\mathfrak{J}}_{1}\textrm{ and }j\in{\mathfrak{J}}_{2}\\}\big{|}.$

Therefore, since we can split the value of
$C({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})$ in terms of the sum of the defining
inequalities of the Newton polytopes of
$\prod_{(i,j)\in\mathcal{L}}(t_{i}+s_{j})$ and
$\mathcal{C}(S/J^{\prime}S;{\mathbf{t}},{\mathbf{s}})$, it follows that
$C({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})={\text{codim}}\big{(}I_{({\mathfrak{J}}_{1},{\mathfrak{J}}_{2})}\big{)}$.
This concludes the proof of the theorem. ∎

## References

Authors’ addresses:

Departamento de Matemáticas, Pontificia Universidad Católica de Chile,
Santiago, Chile<EMAIL_ADDRESS>
Department of Mathematics: Algebra and Geometry, Ghent University, Belgium
<EMAIL_ADDRESS>
Department of Mathematics: Algebra and Geometry, Ghent University, Belgium
<EMAIL_ADDRESS>
Department of Mathematics and Statistics, UiT - The Arctic University of
Norway, Tromsø, Norway

Department of Mathematical Sciences, New Mexico State University, NM, USA
<EMAIL_ADDRESS>


# Compact Star Twins with a Dark Matter Core††thanks: Presented at the XXVII
Cracow EPIPHANY Conference on Recent Advances in Astroparticle Physics (2022).

David Alvarez-Castillo
Michał Marczenko  Institute of Nuclear Physics, Polish Academy of Sciences,
31-342 Cracow, Poland  Incubator of Scientific Excellence - Centre for
Simulations of Superdense Fluids, University of Wrocław, plac Maksa Borna 9,
PL-50204 Wrocław, Poland

###### Abstract

We present a model of compact stars with a dark matter core. The hadronic
equation of state is based on the parity doublet model and does not present a
phase transition to quark matter. Instead, a strong first-order phase
transition to dark matter described by a constant speed of sound model leads
to the scenario of compact star mass twins. Compact star structural properties
which obey state-of-the-art measurements and constraints are presented.

##  1 Introduction

The long-standing problem of the clarification of dark matter (DM) has
persisted since its discovery. Astronomical observations, such as anomalies in
the rotation curves of galaxies or gravitational lensing, as well as
cosmological features, support the idea that DM is present in the universe
however hitherto unexplainable. Several attempts to describe DM include exotic
particles such as supersymmetric ones or axions, as well as modifications to
Einstein’s gravity or extra forces beyond the Standard Model. Dark matter is
thought to interact only gravitationally and perhaps changing its strength
over different scales, as for instance suggested in Chameleon models [1].
Consequently, DM might have some sizeable effect on compact stars in whose
interiors and surroundings gravity is very strong. Within this work, we focus
on neutron stars, motivated by the aforementioned facts. The idea that DM can
be found in neutron star (NS) interiors, exteriors, or in both cases [2, 3]
has been explored already in [4], where it is meant to be a separated fluid
weakly interacting gravitationally with normal matter. Furthermore, examples
of microscopic approaches to DM in compact stars that might serve as evidence
include axions and their emission resulting in cooling of compacts [5], or
sexaquarks [6] which might lead to stellar collapse [7] into a hybrid star
with a quark matter (QM) core.

We consider a particular scenario, where we assume that DM lies in the core of
compact stars without inquiring into its nature. We allow the condensed DM to
interact with normal matter through nuclear interactions. The resulting dark
matter equation of state undergoes a strong first-order phase transition to
hadronic matter in the interior of compact stars, producing the so-called
compact star mass twins configurations, stars of about the same gravitational
mass but different radius. This phenomenon has been studied in many works [8,
9, 10, 11, 12, 13, 14] where compact stars feature a deconfined quark matter
core. The astrophysical aspects of the twins have been studied in [15].
Moreover, in order to discriminate from the standard mass twins scenario, we
choose a hadronic equation of state that does not feature quark deconfinement
but includes chiral symmetry restoration. Thus, the resulting hybrid compact
stars bear a DM core.

##  2 The Equation of State

The most basic elements of compact star twins require a stiff enough hadronic
equation of star capable of describing stable high mass stars of more than the
observed 2M⊙ pulsars [16] and a strong first-order phase transition to high
densities. Following the softening at the phase transition, the EoS at the
high-density phase should be as well stiff enough to produce stable stars. For
compact star twins to appear, the Seidov condition
$\Delta\varepsilon>\Delta\varepsilon_{\rm crit}$ [17] should be fulfilled. The
expression

$\frac{\Delta\varepsilon_{\rm crit}}{\varepsilon_{\rm
trans}}=\frac{1}{2}+\frac{3}{2}\frac{p_{\rm trans}}{\varepsilon_{\rm trans}}\
.$ (1)

relates critical energy density difference at the transition
$\Delta\varepsilon_{\rm crit}$ with the energy density and pressure at the
onset of DM, $p_{\rm trans}$, and $\varepsilon_{\rm trans}$. As for the lower
neutron star densities, the NS crust is well described by the approach
developed in [18].

###  2.1 Hadronic Equation of State

Central densities of neutron stars lie up to a few times normal nuclear
density, so it is to be expected that baryons change their properties due to
the restoration of chiral symmetry. The recent lattice QCD (LQCD) results [19]
exhibit a clear manifestation of the parity doubling structure for the low-
lying baryons around the chiral crossover at finite temperature. Such
properties of the baryonic chiral partners can be described in the parity
doublet model [20]. It has been applied to hot and dense hadronic matter,
neutron stars, as well as the vacuum phenomenology of QCD (see, e.g., [21, 22,
23, 24, 25]). In the parity doublet model, the masses of the chiral partners,
$N_{\pm}$, are given by

$m_{\pm}=\frac{1}{2}\left(\sqrt{\left(g_{1}+g_{2}\right)^{2}\sigma^{2}+4m_{0}^{2}}\mp\left(g_{1}-g_{2}\right)\sigma\right)\textrm{,}$
(2)

where $g_{i}$’s are baryon-to-meson coupling constants and $m_{0}$ is the
chirally invariant mass parameter. From Eq. (2), it is clear that the chiral
symmetry breaking generates only the splitting between the two masses. When
the symmetry is restored, the masses become degenerate,
$m_{\pm}(\sigma=0)=m_{0}$. The positive-parity state, corresponds to the
nucleon $N(938)$. Its negative parity partner is identified with $N(1535)$. In
this work, we adopt the parametrization for purely nucleonic EoSs in the mean-
field approximation [25].

###  2.2 Dark Matter Equation of State

For simplicity, in order to parametrize the microscopically unknown dark
matter equation of state we consider the constant speed of sound (CSS)
approach, as introduced in [26].

$\varepsilon(p)=\left\\{\\!\begin{array}[]{ll}\varepsilon_{\rm NM}(p)&p<p_{\rm
trans}\\\ \varepsilon_{\rm NM}(p_{\rm trans})+\Delta\varepsilon+c_{\rm
DM}^{-2}(p-p_{\rm trans})&p>p_{\rm trans}\end{array}\right.\ ,$ (3)

where $\varepsilon_{\rm NM}$ is the energy density of nuclear matter, and
$c_{\rm DM}$ is the speed of sound of dark matter.

##  3 Results

The resulting sequences of compact stars with dark matter cores are displayed
in the mass-radius diagram of Fig. 1. They are obtained by integrating the
Tolman-Oppenheimer-Volkoff equations [27, 28]:

$\displaystyle\frac{\textrm{d}P(r)}{\textrm{d}r}$ $\displaystyle=$
$\displaystyle-\frac{\left(\varepsilon(r)+P(r)\right)\left(m(r)+4\pi
r^{3}P(r)\right)}{r\left(r-2m(r)\right)},$ (4)
$\displaystyle\frac{\textrm{d}m(r)}{\textrm{d}r}$ $\displaystyle=$
$\displaystyle 4\pi r^{2}\varepsilon(r).$ (5)

for each compact star defined by its central density $\varepsilon_{c}$ up to
the maximum mass where $\partial M/\partial\varepsilon_{c}>0$. For the
solution of these equations the EoS relation $P(\varepsilon)$ must be used as
an input. For our models, It can be seen that the three lower $m_{0}$ EoS
parameters can lead to stable configurations at low compact star masses
avoiding the excluded region by [29]. The remaining EoS with $m_{0}=700$MeV is
soft enough to avoid this constraint and support the 2M⊙ compact star, however
if dark matter sets in its core it would collapse due to the softness of its
nuclear matter possibly leading to a detectable radiation emission event. In
addition, we have computed other stellar properties like the moment of inertia
and tidal deformation, which have been already observed through
electromagnetic or gravitational radiation of compact stars and their mergers,
respectively. The moment of inertia is computed following [30]:

$I\simeq\frac{J}{1+2GJ/R^{3}c^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}J=\frac{8\pi}{3}\int_{0}^{R}r^{4}\left(\rho(r)+\frac{P(r)}{c^{2}}\right)\Lambda(r)dr.$
(6)

with $\Lambda(r)=[1-2Gm(r)/rc^{2}]^{-1}$. Tidal deformability is computed
following [31]. The expression $\Lambda=2\kappa_{2}R^{5}/3M^{5}$ relates the
Love number $\kappa_{2}$ and the mass $M$ and radius $R$ of the star. This
quantity has been estimated from the neutron star merger event GW170817 [32,
33]. Figure 2 shows the corresponding diagrams for these quantities and
astrophysical measurements, whereas table 1 shows the parameters found for our
EoS models.

Figure 1: Mass-radius diagram for compact stars which includes measurement
regions as well as exclusion regions. Each curve corresponds to a sequence of
compact stars for the same parameters within our model which is characterized
by the hadronic $m_{0}$ parameter. Description of the several regions in this
diagram as well as their usage within Bayesian studies can be found in [34].

$\begin{array}[]{cc}\includegraphics[width=216.81pt]{I-M_DM_Twins.pdf}&\includegraphics[width=216.81pt]{L-M_DM_Twins.pdf}\\\
\end{array}$

Figure 2: Stellar moment of inertia $I$ (left panel) and dimensionless tidal
deformability $\Lambda$ (right panel) related to the compact star mass $M$
with corresponding measurements in electromagnetic and gravitational signals,
respectively. A recent Bayesian estimation of the moment of inertia reports
$I_{*}=1.15^{+0.38}_{-0.24}\times 10^{45}$g cm2 [35].

Hadronic EoS | $M_{onset}$ | $\Delta\varepsilon$ | $c_{\rm DM}$ | $p_{\rm trans}$ | $\varepsilon_{\rm trans}$ | $n_{\rm trans}$
---|---|---|---|---|---|---
$m_{0}$ [MeV] | [M⊙] | [$\varepsilon_{\rm trans}$] | [c] | [MeV/fm3] | [MeV/fm3] | [fm-3]
550 | 1.325 | 0.75 | 0.80 | 30.95 | 281.32 | 0.286
600 | 1.325 | 0.75 | 0.90 | 30.95 | 281.32 | 0.286
650 | 1.325 | 0.70 | 0.98 | 30.95 | 281.32 | 0.286
700 | 2.000 | 0.75 | 1.00 | 417.63 | 1312.42 | 1.049

Table 1: EoS parameters found in this work.

##  4 Outlook

The search for DM in NS has become feasible due to the birth of multi-
messenger astronomy and progress on the determination of the cold, dense
matter EoS. The macroscopical properties of compact stars may be affected from
quantities like the cosmological constant [36] or the presence of DM, the
latter considered here. Observations of the effects of a strong phase
transition on supernova explosions, compact stars, and their mergers [37] or
of NS cooling as well as detection of associated cosmic rays [38], eccentric
binary orbits [39], emissions from evolutionary stages [40], or the growth of
black holes as DM in the interior of rotating NS [41], may allow for testing
our NS twins hypothesis. In our work, we have introduced a realistic equation
of state for the nuclear matter which features chiral symmetry restoration
without phase transition to deconfined QM in order to produce NS twins with a
DM core described by the CSS EoS. Under these specific assumptions, best upon
corroboration of the hadronic EoS implemented, detection of the mass twins
would allow for the study of the bulk properties of DM. Inclusion of more
refined DM models in NS is work in progress.

## Acknowledgements

The authors thank the COST Action CA16214 “PHAROS” for its support of their
networking activities. M.M. acknowledges support from the Polish National
Science Centre (NCN) under No. 2017/27/N/ST2/01973 Preludium Grant, and the
program Excellence Initiative–Research University of the University of Wroclaw
of the Ministry of Education and Science.

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# A direct approach to sharp Li-Yau estimates on closed manifolds with
negative Ricci lower bound

XingYu Song, Ling Wu and Meng Zhu  School of Mathematical Sciences, Key
Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP,
East China Normal University, Shanghai 200241, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Recently, Qi S.Zhang [30] has derived a sharp Li-Yau estimate for positive
solutions of the heat equation on closed Riemannian manifolds with the Ricci
curvature bounded below by a negative constant. The proof is based on an
integral iteration argument which utilizes Hamilton’s gradient estimate, heat
kernel Gaussian bounds and parabolic Harnack inequality.

In this paper, we show that the sharp Li-Yau estimate can actually be obtained
directly following the classical maximum principle argument, which simplifies
the proof in [30]. In addition, we apply the same idea to the heat and
conjugate heat equations under the Ricci flow and prove some Li-Yau type
estimates with optimal coefficients.

##  1\. Introduction

The renowned Li-Yau estimate [20] states that on an $n$-dimensional complete
Riemannian manifold $(M^{n},g)$ with the Ricci curvature
$\operatorname{Ric}\geq-K$, $K\geq 0$, any positive solution $u$ of the heat
equation

$\left(\Delta-\frac{\partial}{\partial t}\right)u=0$ (1.1)

satisfies

$\frac{|\nabla
u|^{2}}{u^{2}}-\alpha\frac{\partial_{t}u}{u}\leq\frac{n\alpha^{2}}{2t}+\frac{n\alpha^{2}K}{2(\alpha-1)},\
\forall\alpha>1,\ t>0.$ (1.2)

Although (1.2) was discovered earlier by Aronson-Bénilan on Euclidean spaces
[1], Li and Yau were the first to prove it on general differentiable manifolds
and demonstrated numerous crucial applications in the research of differential
geometry and geometric analysis. The Li-Yau estimate is closely related to
parabolic Harnack inequality, Sobolev and Poincaré inequalities, volume
doubling property, heat kernel estimates, estimates of the Green’s functions,
eigenvalue estimates, etc. Even more powerfully, when the Ricci curvature is
nonnegative, i.e., $K=0$, one can let $\alpha$ in (1.2) tend to $1$ and obtain
the sharp Li-Yau estimate

$\frac{|\nabla u|^{2}}{u^{2}}-\frac{\partial_{t}u}{u}\leq\frac{n}{2t},$ (1.3)

which in term gives an optimal heat kernel estimate, as well as the laplacian
and Bishop-Gromov volume comparison theorems.

Due to its significancy, many efforts have been made to improve or generalize
Li-Yau estimates. Assuming $\operatorname{Ric}$ bounded below by a negative
constant, the coefficient $\alpha$ in (1.2) was improved in e.g. [27, 14, 19,
24, 3]. The Li-Yau estimate was also generalized to various linear and
semilinear parabolic equations, as well as matrix versions (see e.g. [14, 7,
18, 8]). Li-Yau type estimates under time-dependent metrics, for example the
Ricci flow and Kähler-Ricci flow, were studied in the past two decades, and
are very useful tools in characterizing the singularities of the Ricci flow
(see e.g. [15, 16, 6, 2, 17, 23, 10, 9, 22]). Furthermore, the Ricci lower
bound assumption for Li-Yau estimate has recently been relaxed to integral
Ricci conditions and Bakry-Émery Ricci curvature condition, or even the
$RCD^{*}(K,N)$ spaces (see e.g. [32, 33, 4, 12, 25, 29, 21]).

Estimate (1.3) is sharp in the sense that the coefficient of
$\frac{\partial_{t}u}{u}$ is equal to $1$, and the equality is achieved by the
heat kernel $(4\pi t)^{-n/2}exp\\{-\frac{|x-y|^{2}}{4t}\\}$ on
$\mathbb{R}^{n}$. However, estimate (1.2) is not sharp for $K>0$. For example,
Hamilton [14] showed that

$\frac{|\nabla u|^{2}}{u^{2}}-e^{2Kt}\frac{\partial_{t}u}{u}\leq
e^{4Kt}\frac{n}{2t},$ (1.4)

where the constant $\alpha>1$ is improved to an exponential function
approaching $1$ as $t$ tending to $0$. Indeed, an open question asks if one
can take $\alpha=1$ even for $K>0$.

Very recently, Qi S. Zhang [30] has invented a method which answers the above
question affirmatively for closed manifolds. More precisely, he showed that

######  Theorem 1.1.

([30]) Let $(M^{n},g)$ be an $n$-dimensional closed Riemannian manifold with
$\operatorname{Ric}\geq-K$ for some constant $K\geq 0$, and $u$ a positive
solution of the heat equation (1.1) on $M\times[0,+\infty)$. Then there exist
constants $c_{1}$ and $c_{2}$ depending only on $n$ such that

$t\left(\frac{|\nabla
u|^{2}}{u^{2}}-\frac{\partial_{t}u}{u}\right)\leq\frac{n}{2}+\sqrt{2nK(1+Kt)(1+t)}diam_{M}+\sqrt{K(1+Kt)(c_{1}+c_{2}K)t}$
(1.5)

for all $(x,t)\in M\times[0,+\infty)$, where $diam_{M}$ is the diameter of
$M$.

The proof in [30] is mainly based on an integral iteration argument with
dedicated estimates, in which Hamilton’s gradient estimate for the heat
equation [14] , Gaussian bounds of the heat kernel and differential Harnack
inequality are used.

In this paper, we observe that Theorem 1.1 can actually be obtained by
directly following the classical maximum principle arguments, and the proof
can be simplified. Our proof is inspired by the usage of Hamilton’s gradient
estimate and the heat kernel bounds in [30].

In Theorem 1.1, the Laplace operator is associated with a fixed metric. We can
also consider the heat equation and conjugate heat equation under the Ricci
flow, in which the Laplace operator evolves as the metric changes in time. For
the heat equation, we show that

######  Theorem 1.2.

Let $M^{n}$ be an n-dimensional closed Riemannian manifold, and $g(x,t)$,
$(x,t)\in M\times[0,T]$, a solution to the Ricci flow
$\frac{\partial}{\partial t}g(x,t)=-2\operatorname{Ric}(x,t)$ on $M$. Assume
that for some constants $K_{1},K_{2}\geq 0$,
$-K_{1}g(x,t)\leq\operatorname{Ric}(x,t)\leq K_{2}g(x,t)$, $\forall(x,t)\in
M\times[0,T]$. Let $u$ be a smooth positive solution of the heat equation
$\left(\Delta_{t}-\frac{\partial}{\partial t}\right)u=0$ on $M\times[0,T]$,
where $\Delta_{t}$ is the Laplace operator of the metric $g(x,t)$. Then there
exist constants $c_{5}$ and $c_{6}$ depending only on $n$ such that

$t\left(\frac{|\nabla
u|^{2}}{u^{2}}-\frac{\partial_{t}u}{u}\right)\leq\frac{n}{2}+nK_{3}t+\sqrt{c_{5}K_{1}[e^{c_{6}K_{3}t}(t+d^{2}_{0}(M))]}$

for all $(x,t)\in M\times[0,T]$, where $d_{t}(M)$ is the diameter of
$(M^{n},g(x,t))$ and $K_{3}=\max\\{K_{1},K_{2}\\}$.

######  Remark 1.3.

When $K_{1}=0$, Theorem 1.2 reduces to Theorem 2.9 in [2].

In Theorem 1.2, the Ricci curvature is assumed to be uniformly bounded up to
time $T$ before the singular time. The main reason is that the heat kernel
Gaussian bounds may not hold uniformly till the singular time. This assumption
can be relaxed to a so-called ”Type-I” Ricci curvature bound, if we assume in
addition that Perelman’s $\nu$ entropy is bounded from below.

Recall that Perelman’s $\mathcal{W}$ functional is defined as

$\mathcal{W}(g,f,\tau)=(4\pi\tau)^{-\frac{n}{2}}\int_{M}(\tau(R+|\nabla
f|^{2})+f-n)e^{-f}dg,$

where $f\in C^{\infty}(M)$ and $\tau>0$. Then the $\mu$ and $\nu$ entropy
functionals are

$\mu[g,\tau]:=\inf\left\\{\mathcal{W}(g,f,\tau);\ f\in C^{\infty}(M)\ and\
\int_{M}(4\pi\tau)^{-\frac{n}{2}}e^{-f}dg=1\right\\},$

and

$\nu[g,\tau]:=\inf\limits_{s\in[0,\tau]}\mu[g,s].$

######  Theorem 1.4.

Let $M^{n}$ be an n-dimensional closed Riemannian manifold and
$g(x,t),(x,t)\in M\times[0,2)$, a solution to the Ricci flow
$\frac{\partial}{\partial t}g(x,t)=-2\operatorname{Ric}(x,t)$ on $M$. Assume
that for some constant $K\geq 0$,
$-\frac{K}{2-t}g(x,t)\leq\operatorname{Ric}(x,t)\leq\frac{K}{2-t}g(x,t)$ and
$\nu[g_{0},4]\geq-nK$, $\forall(x,t)\in M\times[0,2)$. Let $u$ be a smooth
positive solution of the heat equation
$\left(\Delta_{t}-\frac{\partial}{\partial t}\right)u(x,t)=0$ on
$M\times[0,2)$. Then there exists a constant $c_{9}$ depending on $n,K$ such
that

$t\left(\frac{|\nabla
u|^{2}}{u^{2}}-\frac{\partial_{t}u}{u}\right)\leq\frac{n}{2}+\frac{nKt}{2-t}+\sqrt{\frac{Kc_{9}}{2-t}\left(t+d_{0}^{2}(M)t+d_{0}^{2}(M)\right)}$

for all $(x,t)\in M\times[0,2)$, where $d_{0}(M)$ is the diameter of
$(M,g(x,0))$.

Finally, we consider the Li-Yau estimate for positive solutions of the
conjugate heat equation under the Ricci flow.

######  Theorem 1.5.

Let $M^{n}$ be an n-dimensional closed Riemannian manifold and
$g(x,t),(x,t)\in M\times[0,T)$, a solution to the Ricci flow
$\frac{\partial}{\partial t}g(x,t)=-2\operatorname{Ric}(x,t)$ on $M$. Assume
that for some constants $K_{1},K_{2}\geq 0$,
$-K_{1}g(x,t)\leq\operatorname{Ric}(x,t)\leq K_{2}g(x,t)$, $|\nabla
R|(x,t)\leq K_{2}^{\frac{3}{2}}$ and $R(x,t)\geq 0$, $\forall(x,t)\in
M\times[0,T]$. Let $u$ be a positive solution of the conjugate heat equation
$\left(\Delta_{t}+\frac{\partial}{\partial t}-R\right)u=0$ on $M\times[0,T]$.
Then there exist constants $c_{10}$ and $c_{11}$ depending only on $n$ such
that

$\frac{|\nabla
u|^{2}}{u^{2}}+\frac{u_{t}}{u}\leq\frac{n}{T-t}+c_{10}e^{c_{11}K_{1}T}\left(1+\frac{d_{0}^{2}(M)}{T}\right)^{2}\left(\frac{1}{t}+K_{1}+K_{2}\right)$

for all $(x,t)\in M\times(0,\frac{T}{2}]$, where $d_{0}(M)$ is the diameter of
$(M,g(x,0))$.

The rest of the paper is organized as follows. In section 2, we give an
alternative proof of Theorem 1.1. Section 3 is devoted to the proof of Theorem
1.2. Theorem 1.4 is dealt with separately in section 4, since the curvature is
blowing up at the singular time and a different heat kernel estimate is
required. Finally, we show Theorem 1.5 in section 5.

##  2\. The heat equation with fixed metric

In this section, we prove the sharp Li-Yau estimate (1.5) .

Proof of Theorem 1.1. Let $u=u(x,t)$ be a positive solution of the heat
equation on $M\times[0,+\infty)$. Denote

$Q=\frac{|\nabla u|^{2}}{u^{2}}-\frac{\partial_{t}u}{u}=|\nabla\ln
u|^{2}-\partial_{t}\ln u=-\Delta\ln u.$ (2.1)

From the Bochner formula and the Cauchy-Schwarz inequality, it follows that

$\displaystyle(\Delta-\partial_{t})Q+2\left<\nabla\ln u,\nabla Q\right>$
$\displaystyle=2\left[|\operatorname{Hess}\ln
u|^{2}+\operatorname{Ric}(\nabla\ln u,\nabla\ln u)\right]$ (2.2)
$\displaystyle\geq 2\left[\frac{Q^{2}}{n}-K|\nabla\ln u|^{2}\right].$

Hence,

$(\Delta-\partial_{t})(tQ)+2\left<\nabla\ln u,\nabla(tQ)\right>\geq
2t\left(\frac{1}{n}Q^{2}-K|\nabla\ln u|^{2}\right)-Q.$ (2.3)

For any $T>0$, the maximum of $tQ$ on $M\times[0,T]$ is achieved at some
$(x_{0},t_{0})$. We may assume that $tQ$ is positive at $(x_{0},t_{0})$, for
otherwise the theorem is trivial. Thus, $t_{0}>0$, and at $(x_{0},t_{0})$ we
have

$\nabla(tQ)=0,\ \frac{\partial}{\partial t}(tQ)\geq 0,\ and\ \Delta(tQ)\leq
0.$

Evaluating (2.3) at $(x_{0},t_{0})$, we obtain

$0\geq\frac{2t_{0}}{n}Q^{2}(x_{0},t_{0})-Q(x_{0},t_{0})-2Kt_{0}|\nabla\ln
u|^{2}(x_{0},t_{0}).$

Multiplying both sides above by $t_{0}$ gives

$0\geq\frac{2}{n}(t_{0}Q(x_{0},t_{0}))^{2}-t_{0}Q(x_{0},t_{0})-2Kt_{0}^{2}|\nabla\ln
u|^{2}(x_{0},t_{0}),$

which infers

$t_{0}Q(x_{0},t_{0})\leq\frac{n}{2}+\sqrt{nK}t_{0}|\nabla\ln u|(x_{0},t_{0}).$
(2.4)

Next we will bound $t_{0}|\nabla\ln u|(x_{0},t_{0})$. From [30], the heat
kernel $H(x,t,y)$, $x,y\in M$, $t>0$ satisfies the estimates

$t|\nabla_{x}\ln H(x,t,y)|^{2}\leq
2(1+Kt)\left(2C_{1}Kt+4C_{2}^{2}K+\frac{diam_{M}^{2}}{t}+2\ln C_{3}\right),\
\forall\ 0<t\leq 8,$ (2.5)

and

$t|\nabla_{x}\ln H(x,t,y)|^{2}\leq 2(1+Kt)(n\ln 2+2C_{4}K+diam_{M}^{2}),\
\forall\ t\geq 8,$ (2.6)

where the positive constants $C_{i}$, $1\leq i\leq 4$, depend only on $n$.
Actually, (2.5) comes from Hamilton estimate [14] and Gaussian bounds of the
heat kernel [20], and (2.6) relies further on the parabolic Harnack
inequality.

Combining (2.5) and (2.6), we deduce that

$t^{2}|\nabla_{x}\ln H(x,t,y)|^{2}\leq
2(1+KT)(C_{5}T+C_{6}KT+diam_{M}^{2}+diam_{M}^{2}T),\ \forall\ 0<t\leq T.$
(2.7)

Since $T$ is arbitrary, we conclude that

$t^{2}|\nabla_{x}\ln H(x,t,y)|^{2}\leq
2(1+Kt)(C_{5}t+C_{6}Kt+diam_{M}^{2}+diam_{M}^{2}t),\ \forall\ t>0\ and\ x,y\in
M.$ (2.8)

A short argument from [28] implies that the bound (2.8) also holds if one
replaces the heat kernel by any positive solution $u(x,t)$ of the heat
equation. For completeness, we present the proof below. By Duhamel’s
principle, we have $u(x,t)=\int_{M}H(x,t,y)u(y,0)dy$. Then

$\displaystyle|\nabla_{x}u(x,t)|^{2}$
$\displaystyle=\int_{M}\int_{M}\left<\nabla_{x}H(x,t,y),\nabla_{x}H(x,t,z)\right>u(y,0)u(z,0)dydz$
$\displaystyle\leq\frac{1}{2}\int_{M}\int_{M}\left(\frac{|\nabla_{x}H(x,t,y)|^{2}}{H(x,t,y)}H(x,t,z)+\frac{|\nabla_{x}H(x,t,z)|^{2}}{H(x,t,z)}H(x,t,y)\right)u(y,0)u(z,0)dydz.$

Using (2.8) in the integrand, one has

$t^{2}|\nabla\ln u(x,t)|^{2}\leq
2(1+Kt)(C_{5}t+C_{6}Kt+diam_{M}^{2}+diam_{M}^{2}t),\ \forall\ t\geq 0\ and\
x\in M.$ (2.9)

Plugging (2.9) into (2.4), we obtain

$TQ(x,T)\leq
t_{0}Q(x_{0},t_{0})\leq\frac{n}{2}+\sqrt{K(1+KT)(C_{7}T+C_{8}KT)}+\sqrt{2nK(1+KT)(1+T)}diam_{M}.$
(2.10)

By the arbitrariness of $T$ again, Theorem 1.1 follows from (2.10). ∎

######  Remark 2.1.

One may replace (2.4) by the Li-Yau type inequalities of Yau [27] or Bakry-
Qian [5] and obtain a similar sharp Li-Yau estimate. However, in the proof
above we only need this type of inequality at a maximal point, and (2.4) is
easier to derive.

######  Remark 2.2.

If $\operatorname{Ric}\geq 0$, then Theorem 1.1 becomes the sharp Li-Yau
estimate (1.3).

######  Remark 2.3.

Under the same condition in Theorem 1.1, for any $\alpha>1$, using Theorem 1.1
and (2.9), one gets that there exist constants $c_{i}$ $(i=1,2,3,4)$ depending
only on $n$ such that

$\displaystyle t\left(\alpha\frac{|\nabla
u|^{2}}{u^{2}}-\frac{u_{t}}{u}\right)$ $\displaystyle=t\left(\frac{|\nabla
u|^{2}}{u^{2}}-\frac{u_{t}}{u}\right)+(\alpha-1)t\frac{|\nabla u|^{2}}{u^{2}}$
$\displaystyle\leq\frac{n}{2}+\sqrt{2nK(1+Kt)(1+t)}diam_{M}+\sqrt{K(1+Kt)(c_{1}+c_{2}K)t}$
$\displaystyle\ \ \ \
+2(\alpha-1)\frac{1+Kt}{t}\left(c_{3}t+c_{4}Kt+diam_{M}^{2}+diam_{M}^{2}t\right)$

for all $(x,t)\in M\times(0,+\infty)$.

##  3\. The heat equation under the Ricci flow with uniform Ricci bound

In this section, we consider Li-Yau estimates in the Ricci flow case and prove
Theorem 1.2. First, we need the following Hamilton type estimate by Qi S.Zhang
[31] and X. Cao-R. Hamilton [9].

######  Lemma 3.1.

([31], [9]) Suppose that $M^{n}$ is an $n$-dimensional closed manifold, and
$g(x,t)$, $t\in[0,T]$ is a solution to the Ricci flow
$\frac{\partial}{\partial t}g(x,t)=-2\operatorname{Ric}(x,t)$ on $M$. Let $u$
be a positive solution of the heat equation
$\left(\Delta_{t}-\frac{\partial}{\partial t}\right)u(x,t)=0$ on
$M\times[0,T]$. Then it holds that

$\frac{|\nabla u|}{u}\leq\sqrt{\frac{1}{t}\ln\frac{\operatorname{A}}{u}}$

for all $(x,t)\in M\times(0,T]$ with $\operatorname{A}=\sup\limits_{x\in
M}u(x,0)$.

In [34], the third author provides Gaussian upper and lower bounds for the
heat kernel of the heat equation under the Ricci flow.

######  Lemma 3.2.

([34]) Let $(M^{n},g(x,t))_{t\in[0,T)}$ be a complete solution to the Ricci
flow $\frac{\partial}{\partial t}g(x,t)=-2\operatorname{Ric}(x,t)$, $(x,t)\in
M\times[0,T)$ for $T<\infty$. Suppose that $\operatorname{Ric}(x,t)\geq-
K_{1}g(x,t)$ on $M\times[0,T)$ for some nonnegative constant $K_{1}$ and
$\Lambda=\int_{0}^{T}\sup\limits_{x\in M}|\operatorname{Ric}|(x,t)dt<\infty$.
Let $H(x,t;y,l)$ be the heat kernel of the heat equation
$\left(\Delta_{t}-\frac{\partial}{\partial t}\right)u(x,t)=0$ with $0\leq
l<t<T$. Then we have the following upper and lower bounds:

$H(x,t;y,l)\leq\tilde{c}_{1}(n)e^{\tilde{c}_{2}(n)e^{{\tilde{c}_{3}(n)\Lambda+{\tilde{c}_{4}(n)K_{1}T}}}}\min\left\\{\frac{e^{-\frac{d_{t}^{2}(x,y)}{8e^{4K_{1}T}(t-l)}}}{\operatorname{Vol}_{l}(B_{l}(x,\sqrt{\frac{t-l}{8}}))},\
\frac{e^{-\frac{d_{t}^{2}(x,y)}{8e^{4K_{1}T}(t-l)}}}{\operatorname{Vol}_{t}(B_{t}(y,\sqrt{\frac{t-l}{8}}))}\right\\}$

and

$H(x,t;y,l)\geq\frac{\tilde{c}_{5}(n)e^{-\tilde{c}_{6}(n)e^{\tilde{c}_{7}(n)\Lambda+\tilde{c}_{8}(n)K_{1}T}}e^{-\frac{4d_{t}^{2}(x,y)}{t-l}}}{\operatorname{Vol}_{t}(B_{t}(y,\sqrt{\frac{t-l}{8}}))},$

where $B_{t}(x,\sqrt{\frac{t-l}{8}})$ is the geodesic ball of radius
$\sqrt{\frac{t-l}{8}}$ in $M$ at time $t$ and, $\operatorname{Vol}_{t}(.)$
denotes the volume of a region at time $t$.

Proof of Theorem 1.2. Denote by $f=\ln u$ and $F=t(|\nabla
f|^{2}-f_{t})=-t\Delta f$. By direct computation, we obtain

$\Delta F=t(2|\operatorname{Hess}f|^{2}+2\left<\nabla f,\nabla\Delta
f\right>+2\operatorname{Ric}(\nabla f,\nabla f)-(\Delta
f)_{t}+2\left<\operatorname{Ric},\operatorname{Hess}f\right>),$

where we have used the identity

$\Delta(f_{t})=(\Delta
f)_{t}-2\left<\operatorname{Ric},\operatorname{Hess}f\right>.$

On the other hand,

$F_{t}=(|\nabla f|^{2}-f_{t})+t(2\left<\nabla f,\nabla
f_{t}\right>+2\operatorname{Ric}(\nabla f,\nabla f)-f_{tt}).$

Since $(\Delta-\partial_{t})f=-|\nabla f|^{2}$, we have

$\displaystyle 2t$ $\displaystyle\left<\nabla f,\nabla\Delta f\right>-t(\Delta
f)_{t}-2t\left<\nabla f,\nabla f_{t}\right>+tf_{tt}$
$\displaystyle=2t\left<\nabla f,\nabla(\Delta f-f_{t})\right>+t(f_{t}-\Delta
f)_{t}$ $\displaystyle=-2t\left<\nabla f,\nabla|\nabla f|^{2}\right>+t(|\nabla
f|^{2})_{t}$ $\displaystyle=-2\left<\nabla
f,\nabla(F+tf_{t})\right>+2t\left<\nabla f,\nabla
f_{t}\right>+2t\operatorname{Ric}(\nabla f,\nabla f)$
$\displaystyle=-2\left<\nabla f,\nabla F\right>+2t\operatorname{Ric}(\nabla
f,\nabla f).$

Therefore,

$(\Delta-\partial_{t})F=-2\left<\nabla f,\nabla
F\right>+2t(|\operatorname{Hess}f|^{2}+\left<\operatorname{Ric},\operatorname{Hess}f\right>)+2t\operatorname{Ric}(\nabla
f,\nabla f)-(|\nabla f|^{2}-f_{t}).$ (3.1)

For any $0<a<1$, by the Cauchy-Schwarz inequality and the bound of the Ricci
curvature, one has

$\displaystyle|\operatorname{Hess}f|^{2}+\left<\operatorname{Ric},\operatorname{Hess}f\right>$
$\displaystyle=\sum_{i,j=1}^{n}(f^{2}_{ij}+R_{ij}f_{ij})$ (3.2)
$\displaystyle\geq\sum_{i,j=1}^{n}\left(f_{ij}^{2}-(1-a)f^{2}_{ij}-\frac{1}{4(1-a)}R_{ij}^{2}\right)$
$\displaystyle\geq\sum_{i,j=1}^{n}af_{ij}^{2}-\frac{n}{4(1-a)}K_{3}^{2}$
$\displaystyle\geq\frac{a}{n}(\Delta f)^{2}-\frac{n}{4(1-a)}K_{3}^{2}$
$\displaystyle=\frac{a}{n}(f_{t}-|\nabla
f|^{2})^{2}-\frac{n}{4(1-a)}K_{3}^{2}.$

Plugging (3.2) into (3.1), we see that

$(\Delta-\partial_{t})F+2\left<\nabla f,\nabla
F\right>\geq\frac{2a}{n}\frac{F^{2}}{t}-\frac{F}{t}-\frac{nt}{2(1-a)}K_{3}^{2}-2K_{1}t|\nabla
f|^{2}$ (3.3)

for any $0<a<1$.
For any $0<\tau\leq T$, let $(x_{0},t_{0})\in M\times[0,\tau]$ be a maximum
point of $F$. Without loss of generality, we may assume $F(x_{0},t_{0})>0$.
Then $t_{0}>0$, and by the maximum principle, we have

$\Delta F(x_{0},t_{0})\leq 0,\ \frac{\partial}{\partial t}F(x_{0},t_{0})\geq
0,\ and\ \nabla F(x_{0},t_{0})=0.$ (3.4)

It follows from (3.3) and (3.4) that at $(x_{0},t_{0})$,

$\frac{2a}{n}F^{2}(x_{0},t_{0})-F(x_{0},t_{0})-\frac{nt_{0}^{2}}{2(1-a)}K_{3}^{2}-2K_{1}t_{0}^{2}|\nabla
f|^{2}(x_{0},t_{0})\leq 0,$

which yields

$F(x_{0},t_{0})\leq\frac{n}{4a}\left(1+\sqrt{1+\frac{4at_{0}^{2}}{1-a}K_{3}^{2}}\right)+\sqrt{\frac{n}{a}K_{1}}t_{0}|\nabla
f|(x_{0},t_{0}).$

By choosing $a=\frac{1+K_{3}t_{0}}{1+2K_{3}t_{0}}\in[\frac{1}{2},1)$, one has

$F(x_{0},t_{0})\leq nK_{3}t_{0}+\frac{n}{2}+\sqrt{2nK_{1}}t_{0}|\nabla
f|(x_{0},t_{0}).$ (3.5)

Again, to estimate $t|\nabla f|$, we first find a bound for $t|\nabla\ln
H(x,t;y,0)|$ with $x,y\in M$ and $t\in(0,\tau]$, where $H(x,t;y,0)$ is the
heat kernel of the heat equation $\left(\Delta_{t}-\frac{\partial}{\partial
t}\right)H(x,t;y,0)=0$.

For a time $0<t_{1}\leq\frac{\tau}{2}$, we denote by $h(x,t)=H(x,t+t_{1};y,0)$
with $x,y\in M$ and $t\in[0,t_{1}]$.
Since $h(x,t)$ is a positive solution of the heat equation, by Lemma 3.1, we
derive

$t|\nabla\ln H(x,t+t_{1};y,0)|^{2}\leq\ln\frac{A}{H(x,t+t_{1};y,0)},\ (x,t)\in
M\times(0,t_{1}],$ (3.6)

where $A=\sup\limits_{x\in M}H(x,t_{1};y,0)$.

According to Lemma 3.2, since $\Lambda\leq 2\sqrt{n}K_{3}t_{1}$, the following
upper and lower bounds for $h(x,t)$ hold

$\frac{\tilde{c}_{5}(n)e^{-\tilde{c}_{6}(n)e^{\tilde{c}_{10}(n)K_{3}t_{1}}}e^{-\frac{4d^{2}_{t+t_{1}}(M)}{t_{1}}}}{\operatorname{Vol}_{t+t_{1}}(B_{t+t_{1}}(y,\sqrt{\frac{t_{1}}{4}}))}\leq
h(x,t)\leq\frac{\tilde{c}_{1}(n)e^{\tilde{c}_{2}(n)e^{\tilde{c}_{9}(n)K_{3}t_{1}}}}{\operatorname{Vol}_{t+t_{1}}(B_{t+t_{1}}(y,\sqrt{\frac{t+t_{1}}{8}}))},\
t\in[0,t_{1}].$

The upper bound gives

$A=\sup\limits_{x\in
M}H(x,t_{1};y,0)\leq\frac{\tilde{c}_{1}(n)e^{\tilde{c}_{2}(n)e^{\tilde{c}_{9}(n)K_{3}t_{1}}}}{\operatorname{Vol}_{t_{1}}(B_{t_{1}}(y,\sqrt{\frac{t_{1}}{8}}))},$

which, combining with the lower bound of $h(x,t)$, yields that

$\frac{A}{H(x,t+t_{1};y,0)}\leq\tilde{c}_{11}(n)e^{\tilde{c}_{12}(n)e^{\tilde{c}_{13}(n)K_{3}t_{1}}}e^{\frac{4d^{2}_{t+t_{1}}(M)}{t_{1}}}\frac{\operatorname{Vol}_{t+t_{1}}(B_{t+t_{1}}(y,\sqrt{\frac{t_{1}}{4}}))}{\operatorname{Vol}_{t_{1}}(B_{t_{1}}(y,\sqrt{\frac{t_{1}}{8}}))}.$
(3.7)

The uniform boundedness of the Ricci curvature implies that $g(x,t)$ is
uniformly equivalent to the initial metric $g(0)$ (see Corollary 6.11 in [11]
), that is ,

$e^{-2K_{2}t}g(0)\leq g(x,t)\leq e^{2K_{1}t}g(0).$ (3.8)

Then, according to the Bishop-Gromov volume comparison theorem, we have

$\displaystyle\frac{\operatorname{Vol}_{t+t_{1}}(B_{t+t_{1}}(y,\sqrt{\frac{t_{1}}{4}}))}{\operatorname{Vol}_{t_{1}}(B_{t_{1}}(y,\sqrt{\frac{t_{1}}{8}}))}$
$\displaystyle\leq\frac{e^{3nt_{1}K_{3}}\operatorname{Vol}_{t_{1}}(B_{t_{1}}(y,\sqrt{\frac{t_{1}}{4}}))}{\operatorname{Vol}_{t_{1}}(B_{t_{1}}(y,\sqrt{\frac{t_{1}}{8}}))}$
$\displaystyle\leq e^{3nt_{1}K_{3}}(\sqrt{2})^{n}e^{\sqrt{(n-1)K_{3}t_{1}}}$
$\displaystyle\leq e^{\tilde{c}_{14}(n)(K_{3}t_{1}+1)}.$

Plugging this into (3.7), we see that (3.6) becomes

$t|\nabla\ln
H(x,t+t_{1};y,0)|^{2}\leq\tilde{c}_{15}(n)\left(e^{\tilde{c}_{16}(n)K_{3}t_{1}}+\frac{d^{2}_{t+t_{1}}(M)}{t_{1}}\right),\
\forall t\in(0,t_{1}].$

Taking $t=t_{1}$ and using the arbitrariness of $t_{1}$, one concludes that

$t^{2}|\nabla\ln
H(x,t;y,0)|^{2}\leq\tilde{c}_{17}(n)\left(e^{\tilde{c}_{19}(n)K_{3}t}(t+d^{2}_{0}(M))\right),\forall\
0<t\leq\tau.$ (3.9)

Since the solution $u$ to the heat equation can also be written as

$u(x,t)=\int_{M}H(x,t;y,0)u(y,0)d\mu_{0}(y),$

where $d\mu_{0}$ is the volume element at $t=0$, the same bound as in (3.9)
actually holds for any positive solution of the heat equation, that is,

$t^{2}|\nabla\ln
u(x,t)|^{2}\leq\tilde{c}_{17}(n)\left(e^{\tilde{c}_{19}(n)K_{3}t}(t+d^{2}_{0}(M))\right),\forall\
0\leq t\leq\tau.$ (3.10)

Combining (3.10) with (3.5), we obtain

$F(x,\tau)\leq F(x_{0},t_{0})\leq
nK_{3}\tau+\frac{n}{2}+\sqrt{\tilde{c}_{20}(n)K_{1}[e^{\tilde{c}_{19}(n)K_{3}\tau}(\tau+d^{2}_{0}(M))]}.$
(3.11)

Since $\tau\in(0,T]$ can be chosen arbitrarily, it finishes the proof of
Theorem 1.2. ∎

######  Remark 3.3.

Under the assumptions and notations as in Theorem 1.2, similar to Remark 2.3,
we have for any $\alpha>1$, there exist constants $c_{i}$ $(i=5,6,7,8)$
depending only on $n$ such that

$\displaystyle t\left(\alpha\frac{|\nabla
u|^{2}}{u^{2}}-\frac{\partial_{t}u}{u}\right)$
$\displaystyle\leq\frac{n}{2}+nK_{3}t+\sqrt{c_{5}K_{1}[e^{c_{6}K_{3}t}(t+d^{2}_{0}(M))]}$
$\displaystyle\ \ \ \
+(\alpha-1)\frac{1}{t}c_{7}e^{c_{8}K_{3}t}(t+d_{0}^{2}(M))$

for all $(x,t)\in M\times(0,T]$.

##  4\. The heat equation under the Ricci flow with blowing-up Ricci curvature

To prove Theorem 1.4, we will need the following Gaussian upper and lower
bounds for the heat kernel near the singular time.

######  Lemma 4.1.

([13]) Let $(M^{n},g(x,t))_{t\in[-2,0)}$ be a closed solution of Ricci flow
satisfying $\nu[g_{-2},4]\geq-K$ and $|R(x,t)|\leq K|t|^{-1}$ or some constant
$K\geq 0$. Let $H(x,t;y,l)$ be the heat kernel of the equation
$\left(\Delta_{t}-\frac{\partial}{\partial t}\right)H(x,t;y,l)=0$ with $-2\leq
l<t<0$ and $x,y\in M$. Then, for any $x,y\in M$ and $-\frac{1}{2}\leq l<t<0$,
there exists a constant $C^{*}=C^{*}(K,n)$ such that

$\frac{1}{C^{*}(t-l)^{\frac{n}{2}}}e^{-\frac{C^{*}d^{2}_{l}(x,y)}{t-l}}\leq
H(x,t;y,l)\leq\frac{C^{*}}{(t-l)^{\frac{n}{2}}}e^{-\frac{d^{2}_{l}(x,y)}{C^{*}(t-l)}}.$

Proof of Theorem 1.4. Denote $f=\ln u$ and $F=t(|\nabla f|^{2}-f_{t})$.

For any $0<T<2$, let $(x_{0},t_{0})\in M\times(0,T]$ be a maximum point of $F$
and $F(x_{0},t_{0})>0$. Similar to (3.5), we get

$F(x_{0},t_{0})\leq
nt_{0}\frac{K}{2-t_{0}}+\frac{n}{2}+\sqrt{\frac{2nK}{2-t_{0}}}t_{0}|\nabla
f|(x_{0},t_{0})\ \ for\ \ 0<t_{0}\leq T.$ (4.1)

Since the Ricci curvature bound implies $|R(x,t)|\leq\frac{nK}{2-t}$,
combining the condition $\nu[g_{0},4]\geq-nK$, by applying Lemma 4.1, there
exists a constant $\bar{C_{1}}(n,K)$ such that for $\frac{3}{2}<t<2$ and any
$x,y\in M$,

$\frac{1}{\bar{C}_{1}(n,K)\left(t-\frac{3}{2}\right)^{\frac{n}{2}}}e^{-\frac{\bar{C_{1}}(n,K)d^{2}_{\frac{3}{2}}(x,y)}{t-\frac{3}{2}}}\leq
H(x,t;y,\frac{3}{2})\leq\frac{\bar{C}_{1}(n,K)}{\left(t-\frac{3}{2}\right)^{\frac{n}{2}}}e^{-\frac{d^{2}_{\frac{3}{2}}(x,y)}{\bar{C}_{1}(n,K)\left(t-\frac{3}{2}\right)}}.$
(4.2)

For a time $\frac{3}{2}<t_{1}<\frac{8}{5}$, we denote
$h(x,t)=H(x,t+t_{1};y,\frac{3}{2})$, $t\in\left[0,\frac{1}{4}t_{1}\right]$,
then
$t+t_{1}\in\left[t_{1},\frac{5}{4}t_{1}\right]\subset\left(\frac{3}{2},2\right)$.
Hence, (4.2) implies

$\bar{C}_{2}(n,K)e^{-\frac{\bar{C}_{1}(n,K)d^{2}_{\frac{3}{2}}(M)}{t_{1}-\frac{3}{2}}}\leq
H(x,t+t_{1};y,\frac{3}{2})\leq\frac{\bar{C}_{1}(n,K)}{\left(t+t_{1}-\frac{3}{2}\right)^{\frac{n}{2}}}\
\ for\ \ t\in\left[0,\frac{1}{4}t_{1}\right].$

The upper bound gives

$A=\sup\limits_{x\in
M}H(x,t_{1};y,\frac{3}{2})\leq\frac{\bar{C}_{1}(n,K)}{(t_{1}-\frac{3}{2})^{\frac{n}{2}}}.$

By lemma 3.1,

$t|\nabla\ln
H(x,t+t_{1};y,\frac{3}{2})|^{2}\leq\bar{C}_{3}(n,K)+\ln(t_{1}-\frac{3}{2})^{-\frac{n}{2}}+\frac{\bar{C}_{1}(n,K)d^{2}_{\frac{3}{2}}(M)}{t_{1}-\frac{3}{2}}.$

Since

$-\frac{K}{2-t}g(x,t)\leq\operatorname{Ric}(x,t)\leq\frac{K}{2-t}g(x,t),$

then

$\left(\frac{2}{2-t}\right)^{-2K}g(0)\leq
g(x,t)\leq\left(\frac{2}{2-t}\right)^{2K}g(0),$

therefore

$t|\nabla\ln
H(x,t+t_{1};y,\frac{3}{2})|^{2}\leq\bar{C}_{3}(n,K)+\ln(t_{1}-\frac{3}{2})^{-\frac{n}{2}}+\frac{\bar{C}_{4}(n,K)d^{2}_{0}(M)}{t_{1}-\frac{3}{2}}.$

We take $t=\frac{1}{4}t_{1}$ and use
$t_{1}\in\left(\frac{3}{2},\frac{8}{5}\right)$ to conclude that

$\frac{1}{5}t|\nabla\ln
H(x,t;y,\frac{3}{2})|^{2}\leq\bar{C}_{3}(n,K)+\ln\left(\frac{4}{5}t-\frac{3}{2}\right)^{-\frac{n}{2}}+\frac{\bar{C}_{4}(n,K)d^{2}_{0}(M)}{\frac{4}{5}t-\frac{3}{2}},\
\ \forall t\in\left(\frac{15}{8},2\right),$

in particular,

$t|\nabla\ln H(x,t;y,\frac{3}{2})|^{2}\leq\bar{C}_{5}(n,K)(1+d^{2}_{0}(M)),\ \
\forall t\in\left(\frac{17}{9},2\right).$

Since
$u(x,t)=\int_{M}H(x,t;y,\frac{3}{2})u(y,\frac{3}{2})d\mu_{\frac{3}{2}}(y)$,
$t\geq\frac{3}{2}$, the above upper bound actually implies

$t|\nabla\ln u(x,t)|^{2}\leq\bar{C}_{5}(n,K)(1+d^{2}_{0}(M)),\ \ \forall
t\in\left(\frac{17}{9},2\right).$ (4.3)

For $t\in\left[0,\frac{17}{9}\right]$, since the Ricci curvature is uniformly
bounded, from (3.10), we derive

$t^{2}|\nabla\ln
u(x,t)|^{2}\leq\bar{C}_{6}(n)\left(e^{\bar{C}_{7}(n)Kt}(t+d_{0}^{2}(M))\right),\
\ \forall t\in\left[0,\frac{17}{9}\right].$ (4.4)

Combining (4.3) and (4.4), we conclude that

$t^{2}|\nabla\ln
u(x,t)|^{2}\leq\bar{C}_{8}(n,K)(t+d^{2}_{0}(M)t+d^{2}_{0}(M)).\ \ \forall
t\in[0,2).$

Plugging above into (4.1) gives

$F(x_{0},t_{0})\leq\frac{n}{2}+\frac{nKt_{0}}{2-t_{0}}+\sqrt{\frac{K\bar{C}_{9}(n,K)}{2-t_{0}}(t_{0}+d_{0}^{2}(M)t_{0}+d_{0}^{2}(M))},$

which implies

$F(x,T)\leq
F(x_{0},t_{0})\leq\frac{n}{2}+\frac{nKT}{2-T}+\sqrt{\frac{K\bar{C}_{9}(n,K)}{2-T}(T+d_{0}^{2}(M)T+d_{0}^{2}(M))}.$

The arbitrariness of $T$ gives the theorem.
∎

##  5\. The Conjugate heat equation under the Ricci flow

In this section, we deal with the conjugate heat equation under the Ricci flow
and prove Theorem 1.5.

First, we recall the following Hamilton type estimate for the positive
solution to the conjugate heat equation under the Ricci flow.

######  Lemma 5.1.

([11]) Let $(M^{n},g(t))_{t\in[0,T]}$ be a solution to the Ricci flow
$\frac{\partial}{\partial t}g(x,t)=-2\operatorname{Ric}(x,t)$. For a fixed
point $o\in M$, assume that for some constants $K_{1},\ K_{2}\geq 0$,
$\operatorname{Ric}(x,t)\geq-K_{1}g(x,t)$ and $|\nabla R|(x,t)\leq
K_{2}^{\frac{3}{2}}$ in $B_{\tau}(o,2r)\times[0,T]$, $\tau=T-t$. Let $u$ be a
solution to the conjugate heat equation
$\left(\Delta_{t}+\frac{\partial}{\partial t}-R\right)u=0$ in
$B_{\tau}(o,2r)\times[0,T]$ satisfying $0<u\leq A$. Then there exist
nonnegative constants $\tilde{C_{1}}(n)$ and $\tilde{C_{2}}(n)$ such that

$\displaystyle\sup\limits_{x\in B_{\tau}(o,r)}\frac{|\nabla
u|^{2}}{u^{2}}(x,\tau)$
$\displaystyle\leq\left(1+\ln\frac{A}{u}\right)^{2}\left(\frac{1}{\tau}+\tilde{C_{2}}(n)K_{1}+2K_{2}+\frac{1}{r^{2}}(\tilde{C_{1}}(n)\sqrt{K_{1}}r\coth(\sqrt{K_{1}}r)+\tilde{C_{2}}(n))\right).$

######  Remark 5.2.

If the conditions $\operatorname{Ric}(x,t)\geq-K_{1}g(x,t)$, $|\nabla
R|(x,t)\leq K_{2}^{\frac{3}{2}}$ and $0<u\leq A$ are satisfied on
$M\times[0,T]$, then by letting $R\rightarrow\infty$, we have that

$\frac{|\nabla
u|^{2}}{u^{2}}(x,\tau)\leq\left(1+\ln\frac{A}{u}\right)^{2}\left(\frac{1}{\tau}+\tilde{C_{3}}(n)(K_{1}+K_{2})\right)$

holds on $M\times(0,T]$.

In [17], Kuang-Zhang established a Li-Yau type estimate for positive solutions
of the conjugate heat equation under the Ricci flow.

######  Lemma 5.3.

([17]) Let $M^{n}$ be an n-dimensional closed Riemannian manifold and
$g(x,t),t\in[0,T)$ a solution to the Ricci flow $\frac{\partial}{\partial
t}g(x,t)=-2\operatorname{Ric}(x,t)$ on $M$. Assume that $R(x,t)\geq 0$,
$(x,t)\in M\times[0,T)$. Let $u$ be a positive solution to the conjugate heat
equation $\left(\Delta_{t}+\frac{\partial}{\partial t}-R\right)u=0$ on
$M\times[0,T)$. Let $u=(4\pi\tau)^{-\frac{n}{2}}e^{-f}$ with $\tau=T-t$. Then

$2\Delta f-|\nabla f|^{2}+R\leq\frac{2n}{\tau},$

or equivalently,

$\frac{|\nabla u|^{2}}{u^{2}}+2\frac{u_{t}}{u}-R\leq\frac{2n}{T-t}$

holds on $M\times[0,T)$.

Proof of Theorem 1.5. Since the heat kernel $G(y,l;x,t)$ of the conjugate heat
equation satisfies $G(y,l;x,t)=H(x,t;y,l)$, where $H(x,t;y,l)$ is the heat
kernel of the heat equation and $x,y\in M$ and $0\leq l<t\leq T$, by Lemma
3.2, we have

$\frac{\tilde{C_{7}}(n)e^{-\tilde{C_{8}}(n)e^{\tilde{C_{9}}(n)K_{1}T}}e^{-\frac{4d_{T}^{2}(M)}{T-l}}}{\operatorname{Vol}_{T}(B_{T}(y,\sqrt{\frac{T-l}{8}}))}\leq
G(y,l;x,T)\leq\frac{\tilde{C_{4}}(n)e^{\tilde{C_{5}}(n)e^{\tilde{C_{6}}(n)K_{1}T}}}{\operatorname{Vol}_{l}(B_{l}(x,\sqrt{\frac{T-l}{8}}))}.$
(5.1)

For a time $0<s\leq\frac{T}{2}$, we denote $h(y,l)=G(y,l;x,T)$, $x,y\in M$ and
$l\in[0,s]$.
Since $h(y,l)$ is a positive solution of the conjugate heat equation
$\left(\Delta_{l,y}+\frac{\partial}{\partial l}-R\right)h(y,l)=0$, by Remark
5.2, we derive

$|\nabla_{y}\ln
G(y,l;x,T)|^{2}\leq\left(1+\ln\frac{A}{G(y,l;x,T)}\right)^{2}\left(\frac{1}{l}+\tilde{C}_{10}(n)(K_{1}+K_{2})\right),\
(y,l)\in M\times(0,s],$ (5.2)

where $A=\sup\limits_{(y,l)\in M\times[0,s]}G(y,l;x,T)$.

The upper bound in (5.1) gives

$A\leq\frac{\tilde{C_{4}}(n)e^{\tilde{C}_{11}(n)e^{\tilde{C}_{12}(n)K_{1}T}}}{\operatorname{Vol}_{0}(B_{0}(x,\sqrt{\frac{T}{16}}))}.$

Hence, combining with the lower bound in (5.1), we have

$\displaystyle\frac{A}{G(y,l;x,T)}$
$\displaystyle\leq\tilde{C}_{13}(n)e^{\tilde{C}_{14}(n)e^{\tilde{C}_{15}(n)K_{1}T}}e^{\frac{8d^{2}_{T}(M)}{T}}\frac{\operatorname{Vol}_{0}(B_{0}(y,\sqrt{\frac{T}{8}}))}{\operatorname{Vol}_{0}(B_{0}(x,\sqrt{\frac{T}{16}}))}$
(5.3)
$\displaystyle\leq\tilde{C}_{13}(n)e^{\tilde{C}_{14}(n)e^{\tilde{C}_{15}(n)K_{1}T}}e^{\frac{8d^{2}_{T}(M)}{T}}\frac{\operatorname{Vol}_{0}(B_{0}(x,d_{0}(M)+\sqrt{\frac{T}{8}}))}{\operatorname{Vol}_{0}(B_{0}(x,\sqrt{\frac{T}{16}}))}.$

According to the Bishop-Gromov volume comparison theorem, we compute

$\frac{\operatorname{Vol}_{0}(B_{0}(x,d_{0}(M)+\sqrt{\frac{T}{8}}))}{\operatorname{Vol}_{0}(B_{0}(x,\sqrt{\frac{T}{16}}))}\leq\left(\frac{d_{0}(M)+\sqrt{\frac{T}{8}}}{\sqrt{\frac{T}{16}}}\right)^{n}e^{\tilde{C}_{16}(n)\sqrt{K_{1}}(d_{0}(M)+\sqrt{T})}.$
(5.4)

Combining (5.3) and (5.4), one has

$\frac{A}{G(y,l;x,T)}\leq\tilde{C}_{13}(n)e^{\tilde{C}_{14}(n)e^{\tilde{C}_{15}(n)K_{1}T}}e^{\left[\frac{8d^{2}_{T}(M)}{T}+\tilde{C}_{17}(n)(\frac{d_{0}(M)}{\sqrt{T}}+1+d_{0}(M)\sqrt{K_{1}}+\sqrt{K_{1}T})\right]}.$

Plugging this into (5.2) results in

$|\nabla_{y}\ln
G(y,l;x,T)|^{2}\leq\tilde{C}_{20}(n)e^{\tilde{C}_{21}(n)K_{1}T}\left(1+\frac{d^{2}_{0}(M)}{T}\right)^{2}\left(\frac{1}{l}+K_{1}+K_{2}\right),\
(y,l)\in M\times(0,s].$

We take $l=s$ and use the arbitrariness of $0<s\leq\frac{T}{2}$ to conclude

$|\nabla_{y}\ln
G(y,l;x,T)|^{2}\leq\tilde{C}_{20}(n)e^{\tilde{C}_{21}(n)K_{1}T}\left(1+\frac{d^{2}_{0}(M)}{T}\right)^{2}\left(\frac{1}{l}+K_{1}+K_{2}\right),\
(y,l)\in M\times(0,\frac{T}{2}].$

Since a positive solution $u(x,t)$ to the conjugate heat equation can be
written as $u(x,t)=\int_{M}G(x,t;y,T)u(y,T)d\mu_{T}(y)$ with $t\in[0,T)$, the
above estimate holds for any $u(x,t)$, i.e.,

$|\nabla\ln
u(x,t)|^{2}\leq\tilde{C}_{20}(n)e^{\tilde{C}_{21}(n)K_{1}T}\left(1+\frac{d^{2}_{0}(M)}{T}\right)^{2}\left(\frac{1}{t}+K_{1}+K_{2}\right),\
(x,t)\in M\times(0,\frac{T}{2}].$

Combining Lemma 5.3, we achieve that

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}+\frac{u_{t}}{u}$
$\displaystyle=\frac{1}{2}\left(\frac{|\nabla
u|^{2}}{u^{2}}+2\frac{u_{t}}{u}-R\right)+\frac{1}{2}|\nabla\ln
u|^{2}+\frac{1}{2}R$
$\displaystyle\leq\frac{n}{T-t}+\tilde{C}_{22}(n)e^{\tilde{C}_{21}(n)K_{1}T}\left(1+\frac{d^{2}_{0}(M)}{T}\right)^{2}\left(\frac{1}{t}+K_{1}+K_{2}\right),$

where $(x,t)\in M\times(0,\frac{T}{2}].$ This completes the proof of Theorem
1.5. ∎

## Acknowledgements

We would like to thank Prof. Qi S. Zhang for inspiring discussions and
invaluable suggestions. We are also grateful for Prof. Huai-Dong Cao and Prof.
Bin Qian for very helpful comments. Research is partially supported by NSFC
Grant No. 11971168, Shanghai Science and Technology Innovation Program Basic
Research Project STCSM 20JC1412900, and Science and Technology Commission of
Shanghai Municipality (STCSM) No. 22DZ2229014.

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