Entropy and Heat Kernel on Generalized Ricci Flow

Xilun Li SMS, Peking University, Beijing 100871, China [email protected]

Abstract.

We introduce analogous geometric quantities and prove some geometric and analytic bounds in [Bam20a] to generalized Ricci flow.

1. Introduction

Generalized Ricci flow is a triple $(M,g_{t},H_{t})$ , where $M$ is a smooth manifold, $g_{t}$ , $H_{t}$ is a one-parameter family of metrics and closed three-forms on $M$ , which satisfies the equation

(1.1)

\displaystyle\left\{\begin{aligned} \frac{\partial}{\partial t}g&=-2\mathrm{Ric}+\frac{1}{2}H^{2},\\ \frac{\partial}{\partial t}H&=-dd^{*}_{g}H,\end{aligned}\right.

where $H^{2}(X,Y):=\left<i_{X}H,i_{Y}H\right>$ .

This flow arises in many aspects, such as generalized geometry [GFS21], complex geometry and mathematical physics [Str22a]. In complex geometry, G. Tian and J. Streets introduce the pluriclosed flow in [ST10] to study the geometry of non-Kähler manifold as extensions of the Kähler Ricci flow. Let $(M^{2n},J,\omega)$ be a complex manifold, the pluriclosed flow is

(1.2)

\frac{\partial}{\partial t}\omega=-\rho_{B}^{1,1},

where $\rho_{B}$ is the Ricci form of Bismut connection. In [ST13], regarding $M$ as a real manifold, pluriclosed flow is just the generalized Ricci flow up to a smooth family of diffeomorphism induced by the Lee form $\theta=-Jd^{*}\omega$ , where the 3-form $H$ is the torsion of Bismut connection $H=d^{c}\omega=i(\bar{\partial}-\partial)\omega$ . Thus the closeness of $H$ is equivalent to the pluriclosed condition, i.e. $\partial\bar{\partial}\omega=0$ .

The main motivation of pluriclosed flow is to provide a geometric-analytic approach to classify the non-Kähler complex surfaces, especially the class VII surfaces, as what Ricci flow does in Geometrization Conjectures and Kähler-Ricci flow does in Kähler manifolds. The pluriclosed flow preserves the pluriclosed condition. According to Gauduchon’s results[Gau77], each complex surfaces admit a pluriclosed metric. For this reason, we should mainly concern about the flows in complex dimension 2, i.e. real dimension 4.

A symmetry reduction of the generalized Ricci flow will give another interesting flow, which we called Ricci-Yang-Mills flow, see [Str22a]. When the manifold is the total space of a principal bundle, the phenomenon of the flow will become more clear. The Ricci-Yang-Mills flow not only is a symmetric solution of the generalized Ricci flow, but also has arisen in mathematical physics literature and as a tool to understand the geometry of principal bundles.

The classical Ricci flow is just the special case where $H\equiv 0$ . In general, since $H^{2}(X,X)=|i_{X}H|^{2}\geqslant 0$ , the generalized Ricci flow is actually a kind of super Ricci flow, i.e.

(1.3)

\frac{\partial}{\partial t}g\geqslant-2\mathrm{Ric}.

R. H. Bamler developed a compactness and regularity theory of super Ricci flow in [Bam20a, Bam20b, Bam20c], which states that any sequence of pointed super Ricci flows of bounded dimensions subsequentially converges to metric flow in a certain sense, which he called the $\mathbb{F}$ -limit. Moreover, the noncollapsed $\mathbb{F}$ -limits of Ricci flows is smooth away a set of codimension at least 4 in parabolic sense. Since the super Ricci flow is a quite big class of flows, it seems unlikely to get structure theory for general super Ricci flow. However, for some particular cases, such as generalized Ricci flow, it’s hopeful to have similar structure results as Ricci flow. A key step in the study of pluriclosed flow is the existence conjecture, see [ST13, Conjecture 5.2]. The structure theory, if exists, will be useful to deal with the conjecture.

In this paper, we will prove some theorems on generalized Ricci flow, which are analogous to the results on Ricci flow in [Bam20a]. The paper will be organized as follows.

In Section 2, we introduce the generalized definition of the pointed Nash and Perelman entropy, which has monotonicity along generalized Ricci flow. Then we will show that the lower bound of generalized point Nash entropy implies the weighted noncollapsed condition.

In Section 3, we will show the upper bounds of the heat kernel and its gradient.

In Section 4, we show an $\varepsilon$ -regularity theorem, which roughly states that the generalized Nash entropy is close to that in Euclidean implies the geometry is close to Euclidean.

In Section 5, we show some estimates on $H$ , particularly in lower dimensional cases, i.e. $n\leqslant 4$ .

Acknowledgements. I am grateful to my advisor Professor Gang Tian for his helpful guidance. I thank Yanan Ye and Shengxuan Zhou for inspiring discussions.

2. Generalized Entropy

2.1. Generalized Nash Entropy and Perelman Entropy

J. Streets[Str22b] introduced several concepts in generalized Ricci flow, we list them here for convenience.

Definition 2.1 ([Str22b, Definition 2.2]).

Given $(M,g_{t},H_{t})$ a solution to generalized Ricci flow, a one-parameter family $\phi_{t}$ is called the dilaton flow if

(2.1)

\square\phi=\frac{1}{6}|H|^{2},

where $\square:=\partial_{t}-\Delta_{g(t)}$ is the heat operator coupled with the flow.

Definition 2.2 ([Str22b, Definition 2.1]).

Given a smooth manifold $M$ , a triple $(g,H,f)$ of a Riemannian metric, closed three-form and function, generalized scalar curvature is

(2.2)

R^{H,f}:=R-\frac{1}{12}|H|^{2}+2\Delta f-|\nabla f|^{2},

generalized Ricci curvature, or twisted Bakry-Emery curvature is

(2.3)

\mathrm{Ric}^{H,f}:=\mathrm{Ric}-\frac{1}{4}H^{2}+\nabla^{2}f-\frac{1}{2}(d^{*}_{g}H+i_{\nabla f}H).

Proposition 2.3 ([Str22b, Proposition 2.4]).

Let $(M,g,H)$ be the generalized Ricci flow, $\phi_{t}$ dilaton flow,

(2.4)

\square R^{H,\phi}=2|\mathrm{Ric}^{H,\phi}|^{2}.

Remark 2.4.

Note that the generalized scalar curvature is not equal to the trace of generalized Ricci curvature, so it may not be bounded from below by a constant only depends on time and dimension as Ricci flow.

Corollary 2.5.

Let $(M,g,H)$ be the generalized Ricci flow, $\phi_{t}$ dilaton flow,

(2.5)

\frac{1}{12}|H|^{2}+|\nabla\phi|^{2}\leqslant R+2\Delta\phi-R^{\phi}_{0},

where $R^{\phi}_{0}:=\min_{x}R^{H,\phi}(x,0)$ . Integrate the inequality, we have

(2.6)

\int_{M}\left(\frac{1}{12}|H|^{2}+|\nabla\phi|^{2}\right)\mathrm{d}g_{t}\leqslant\int_{M}\left(R-R^{\phi}_{0}\right)\mathrm{d}g_{t}.

Proof.

This follows from maximum principle applied to (2.4). ∎

Definition 2.6 ([Str22b, Definition 3.1]).

Let $(M,g_{t},H_{t},\phi_{t})$ be a solution to generalized Ricci flow. Define the conjuate heat operator

\square^{*}:=-\partial_{t}-\Delta_{g(t)}+R-\frac{1}{4}|H|^{2}.

Also define the weighted conjugate heat operator

\square_{\phi}^{*}u:=e^{\phi}\square^{*}\left(ue^{-\phi}\right)=-\partial_{t}u-\Delta_{g(t)}u+2\left<\nabla\phi,\nabla u\right>+R^{H,\phi}u.

These are conjugate operators in the following sense:

	$\displaystyle\int_{t_{1}}^{t_{2}}\int_{M}(\square u)v\mathrm{d}g_{t}\mathrm{d}t$	$\displaystyle=\left.\int_{M}uv\mathrm{d}g_{t}\right\|_{t_{1}}^{t_{2}}+\int_{t_{1}}^{t_{2}}\int_{M}u(\square^{*}v)\mathrm{d}g_{t}\mathrm{d}t,$
	$\displaystyle\int_{t_{1}}^{t_{2}}\int_{M}(\square u)ve^{-\phi}\mathrm{d}g_{t}\mathrm{d}t$	$\displaystyle=\left.\int_{M}uve^{-\phi}\mathrm{d}g_{t}\right\|_{t_{1}}^{t_{2}}+\int_{t_{1}}^{t_{2}}\int_{M}u(\square^{*}v)e^{-\phi}\mathrm{d}g_{t}\mathrm{d}t.$

Definition 2.7 ([Str22b, Definition 4.1]).

Assume $v=(4\pi\tau)^{-n/2}e^{-f}e^{-\phi}$ , $\mathrm{d}\nu=v\mathrm{d}g$ , $\int_{M}\mathrm{d}\nu=1$ , define the weighted Nash entropy by

\mathcal{N}^{\phi}[g,f,\tau]:=\int f\mathrm{d}\nu-\frac{n}{2}.

And define the weighted Perelman entropy by

\mathcal{W}^{\phi}[g,f,\tau]:=\int(\tau R^{f+\phi}+f-n)\mathrm{d}\nu=\int[\tau(R^{\phi}+|\nabla f|^{2})+f-n]\mathrm{d}\nu.

Proposition 2.8.

Suppose $\square^{*}v=0$ , $\frac{\mathrm{d}}{\mathrm{d}t}\tau=-1$ , $(M,g,H,\phi)$ is generalized Ricci flow, then

\frac{d}{d\tau}(\tau\mathcal{N}^{\phi})=\mathcal{W}^{\phi},\ \ \ \frac{d}{d\tau}\mathcal{W}^{\phi}=-2\tau\int|\mathrm{Ric}^{f+\phi}-\frac{1}{2\tau}g|^{2}\mathrm{d}\nu+\frac{1}{3}\int|H|^{2}\mathrm{d}\nu.

Proof.

Since $f=-\log v-\phi-\frac{n}{2}\log(4\pi\tau)$ , $\square^{*}v=0$ , then

\square f=2v^{-1}\Delta v-|\nabla(f+\phi)|^{2}-R+\frac{1}{12}|H|^{2}+\frac{n}{2\tau},

\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{N}=\int\square f\mathrm{d}\nu=-\int\left(|\nabla(f+\phi)|^{2}+R-\frac{1}{12}|H|^{2}-\frac{n}{2\tau}\right)\mathrm{d}\nu.

Note that integration by part, we have $\int|\nabla(f+\phi)|^{2}\mathrm{d}\nu=\int\Delta(f+\phi)\mathrm{d}\nu$ , then

\frac{d}{d\tau}(\tau\mathcal{N})=\mathcal{N}-\tau\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{N}=\int\left[\tau\left(R-\frac{1}{12}|H|^{2}+|\nabla(f+\phi)|^{2}\right)+f-n\right]\mathrm{d}\nu=\mathcal{W}.

Let $u:=e^{\phi}v$ , then $\square^{*}_{\phi}u=e^{\phi}\square^{*}(ue^{-\phi})=0$ . Denote by $W^{H,F}:=\tau R^{H,F}+F-n$ , then by the local computation in [Str22b, Proposition 3.12],

	$\displaystyle\frac{d}{d\tau}\mathcal{W}$	$\displaystyle=\frac{d}{d\tau}\int(W-\phi)ue^{-\phi}\mathrm{d}g=\int\left(\square^{}_{\phi}(Wu)+\square\phi u-\phi\square^{}_{\phi}u\right)e^{-\phi}\mathrm{d}g$
		$\displaystyle=-2\tau\int\|\mathrm{Ric}^{f+\phi}-\frac{1}{2\tau}g\|^{2}\mathrm{d}\nu+\frac{1}{3}\int\|H\|^{2}\mathrm{d}\nu.$

∎

Proposition 2.9 ([Gue02, Theorem 2.1]).

There exists the unique heat kernel $K(x,t;y,s)\in C^{\infty}(\{x,y\in M,t>s\};\mathbb{R}_{+})$ satisfies

\displaystyle\left\{\begin{aligned} &\square_{x,t}K(x,t;y,s)=0,\\ &\lim_{t\searrow s}K(x,t;y,s)=\delta_{y}(x).\end{aligned}\right.

Moreover, the conjugate heat kernel is exactly the same function:

\displaystyle\left\{\begin{aligned} &\square^{*}_{y,s}K(x,t;y,s)=0,\\ &\lim_{s\nearrow t}K(x,t;y,s)=\delta_{x}(y).\end{aligned}\right.

As in [Bam20a, Definition 5.1], we can define the pointed entropy, i.e. taking the conjugate heat kernel as the function $v$ :

Definition 2.10.

Given $(x,t)$ , $\mathrm{d}\nu_{x,t;s}(y):=K(x,t;y,s)\mathrm{d}g_{s}(y)=(4\pi\tau)^{-\frac{n}{2}}e^{-(f+\phi)}\mathrm{d}g_{s}$ , the pointed Nash entropy is defined by

\mathcal{N}^{\phi}_{x,t}(\tau):=\mathcal{N}^{\phi}[g_{t-\tau},f_{t-\tau},\tau],\ \ \mathcal{N}^{\phi*}_{s}(x,t):=\mathcal{N}^{\phi}_{x,t}(t-s),

the pointed Perelman entropy is defined by

\mathcal{W}^{\phi}_{x,t}(\tau):=\mathcal{W}^{\phi}[g_{t-\tau},f_{t-\tau},\tau].

The pointed entropy is not monotone due to the existence of the torsion $H$ , we need to modify the definition of entropy to get the monotonicity.

Definition 2.11.

Define

\Psi_{x,t}(\tau):=\int_{t-\tau}^{t}\int_{M}|H|^{2}\mathrm{d}\nu_{x,t;s}\mathrm{d}s,\ \ P_{x,t}(\tau):=\int_{0}^{\tau}s^{-1}\Psi_{x,t}(s)\mathrm{d}s.

Then we define the generalized pointed Nash entropy by

\mathcal{N}^{H}_{x,t}(\tau):=\mathcal{N}^{\phi}_{x,t}(\tau)-\frac{1}{3}P_{x,t}(\tau).

And define the generalized pointed Perelman entropy by

\mathcal{W}^{H}_{x,t}(\tau):=\mathcal{W}^{\phi}_{x,t}(\tau)-\frac{1}{3}\left(\Psi_{x,t}(\tau)+P_{x,t}(\tau)\right).

Remark 2.12.

Note that $\Psi_{x,t}(0)=0$ , $\Psi^{\prime}_{x,t}(0)=|H|^{2}(x,t)$ , so the function $P_{x,t}$ is well-defined.

Proposition 2.13.

For the generalized pointed entropy, we have

\frac{\mathrm{d}}{\mathrm{d}\tau}(\tau\mathcal{N}^{H}_{x,t}(\tau))=\mathcal{W}^{H}_{x,t}(\tau),\ \ \frac{d}{d\tau}\mathcal{W}^{H}_{x,t}(\tau)=-2\tau\int_{M}|\mathrm{Ric}^{f+\phi}-\frac{1}{2\tau}g|^{2}\mathrm{d}\nu-\frac{1}{3\tau}\Psi_{x,t}(\tau)\leqslant 0.

Proof.

\frac{\mathrm{d}}{\mathrm{d}\tau}(\tau\mathcal{N}^{H}_{x,t}(\tau))=\frac{\mathrm{d}}{\mathrm{d}\tau}(\tau\mathcal{N}^{\phi}_{x,t}(\tau))-\frac{1}{3}\frac{\mathrm{d}}{\mathrm{d}\tau}(\tau P_{x,t}(\tau))=\mathcal{W}^{\phi}_{x,t}(\tau)-\frac{1}{3}\left(P_{x,t}(\tau)+\Psi_{x,t}(\tau)\right).

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\tau}\mathcal{W}^{H}_{x,t}(\tau)$	$\displaystyle=\frac{\mathrm{d}}{\mathrm{d}\tau}\mathcal{W}^{\phi}_{x,t}(\tau)-\frac{1}{3}\frac{\mathrm{d}}{\mathrm{d}\tau}\left(P_{x,t}(\tau)+\Psi_{x,t}(\tau)\right)$
		$\displaystyle=-2\tau\int_{M}\|\mathrm{Ric}^{f+\phi}-\frac{1}{2\tau}g\|^{2}\mathrm{d}\nu_{x,t;t-\tau}+\frac{1}{3}\int_{M}\|H\|^{2}\mathrm{d}\nu-\frac{1}{3}\left(\int_{M}\|H\|^{2}\mathrm{d}\nu+\frac{1}{\tau}\Psi_{x,t}(\tau)\right)$
		$\displaystyle=-2\tau\int_{M}\|\mathrm{Ric}^{f+\phi}-\frac{1}{2\tau}g\|^{2}\mathrm{d}\nu_{x,t;t-\tau}-\frac{1}{3\tau}\Psi_{x,t}(\tau)\leqslant 0.$

∎

Remark 2.14.

Perelman entropy in Ricci flow becomes constant implies $g$ is a gradient shrinking soliton. Our generalized pointed Perelman entropy becomes constant implies the flow is actually Ricci flow and $g$ is also gradient shrinking soliton. It’s natural since by [GFS21, Proposition 4.28], the generalized shrinking soliton is just the shrinking soliton with $H\equiv 0$ .

Corollary 2.15.

We have

\mathcal{N}^{H}_{x,t}(\tau)=\frac{1}{\tau}\int^{\tau}_{0}\mathcal{W}^{H}_{x,t}(s)\mathrm{d}s\geqslant\mathcal{W}^{H}_{x,t}(\tau),\ \ \frac{\mathrm{d}}{\mathrm{d}\tau}\mathcal{N}^{H}_{x,t}(\tau)\leqslant 0,

\int_{M}\tau(R^{\phi}+|\nabla f|^{2})\mathrm{d}\nu_{t-\tau}\leqslant\frac{n}{2}+\frac{1}{3}\Psi_{x,t}(\tau).

Proof.

The first follows from $\tau\frac{\mathrm{d}}{\mathrm{d}\tau}\mathcal{N}^{H}=\mathcal{W}^{H}-\mathcal{N}^{H}\leqslant 0$ . The second conclusion follows from $\mathcal{W}^{H}=\tau\int(R^{\phi}+|\nabla f|^{2})\mathrm{d}\nu+\mathcal{N}^{H}-\frac{n}{2}-\frac{1}{3}\Psi_{x,t}(\tau)$ . ∎

We need an oscillation estimation of Nash entropy analogous to [Bam20a, Corollary 5.11], which can be divided by two parts to estimate. The first term, weighted entropy, can be dealt as [Bam20a, Theorem 5.9].

Proposition 2.16.

Suppose $R^{\phi}(\cdot,s)\geqslant R^{\phi}_{\min}$ , then

|\nabla\mathcal{N}^{\phi*}_{s}|\leqslant\left(\frac{n}{2(t-s)}-R^{\phi}_{\min}+\frac{1}{3(t-s)}\Psi_{x,t}(t-s)\right)^{\frac{1}{2}},\ \ \ -\frac{n}{2(t-s)}\leqslant\square\mathcal{N}^{\phi*}_{s}\leqslant 0.

Proof.

The proof follows the argument in [Bam20a, Proof of Theorem 5.9] with a modification. We assume $s=0$ after application of a time-shift.

\mathcal{N}^{\phi*}_{0}(x,t)=-\int K(x,t;y,0)(\log K(x,t;y,0)+\phi(y,0))\mathrm{d}g_{0}(y)-\frac{n}{2}\log(4\pi t)-\frac{n}{2}.

For any vector $v\in T_{x}M$ with $|v|_{t}=1$ ,

	$\displaystyle\partial_{v}\mathcal{N}^{\phi*}_{0}(x,t)$	$\displaystyle=-\int\partial_{v}K(x,t;y,0)(\log K(x,t;y,0)+1+\phi(y,0))\mathrm{d}g_{0}(y)$
		$\displaystyle=\int\partial_{v}K(x,t;y,0)\left(f-\frac{n}{2}-\mathcal{N}^{\phi*}_{0}(x,t)\right)\mathrm{d}g_{0}(y)$
		$\displaystyle=\int\frac{\partial_{v}K(x,t;y,0)}{K(x,t;y,0)}\left(f-\frac{n}{2}-\mathcal{N}^{\phi*}_{0}(x,t)\right)\mathrm{d}\nu_{x,t;0}(y).$

\int\left(f-\frac{n}{2}-\mathcal{N}^{\phi*}_{0}(x,t)\right)^{2}\mathrm{d}\nu(y)\leqslant 2t\int|\nabla f|^{2}\mathrm{d}\nu\leqslant n-2tR^{\phi}_{\min}+\frac{2}{3}\Psi_{x,t}(t).

|\partial_{v}\mathcal{N}^{\phi*}_{0}|^{2}(x,t)\leqslant\frac{1}{2t}\left(n-2tR^{\phi}_{\min}+\frac{2}{3}\Psi_{x,t}(t)\right).

For the last two inequality, we use the $L^{2}$ -Poincaré inequality in [HN14, Theorem 1.10] and the integral bounds on the gradient of the heat kernel in [Bam20a, Proposition 4.2]. Both of them hold for super Ricci flow.

The proof of the second result is same as that in [Bam20a].

	$\displaystyle\square\mathcal{N}^{\phi*}_{0}(x,t)$	$\displaystyle=-\int_{M}\square_{x,t}\left(K(x,t;y,0)(\log K(x,t;y,0)+\phi(y,0))\right)\mathrm{d}g_{0}(y)-\frac{n}{2t}$
		$\displaystyle=-\int_{M}\left(\square_{x,t}K(\log K+\phi(y,0)+1)-\frac{\|\nabla_{x}K(x,t;y,0)\|^{2}}{K(x,t;y,0)}\right)\mathrm{d}g_{0}(y)-\frac{n}{2t}$
		$\displaystyle=\int_{M}\frac{\|\nabla_{x}K(x,t;y,0)\|^{2}}{K(x,t;y,0)}\mathrm{d}g_{0}(y)-\frac{n}{2t}.$

Then the second results also follow from the integral bounds of the heat kernel, see [Bam20a, Proposition 4.2]. ∎

Definition 2.17.

Given two probability measure $\mu_{1},\mu_{2}$ on $(M,g)$ , define the $W_{1}$ -Wasserstein distance by

d_{W_{1}}^{g}(\mu_{1},\mu_{2}):=\sup\left\{\int_{M}f\mathrm{d}(\mu_{1}-\mu_{2}):f\in C^{\infty}(M),|\nabla f|\leqslant 1\right\}.

Lemma 2.18.

Let $(M,g(t),H(t))$ be a generalized Ricci flow, $x_{1},x_{2}\in M$ , then

\displaystyle t\mapsto d_{W_{1}}^{g(t)}(\nu_{x_{1},t_{1}}(t),\nu_{x_{2},t_{2}}(t))

is non-decreasing for $t\leqslant t_{1},t_{2}$ .

Proof.

Note that the generalized Ricci flow is a super Ricci flow, then it follows from [Bam20a, Lemma 2.7]. ∎

Corollary 2.19.

Suppose $R^{\phi}(\cdot,s)\geqslant R^{\phi}_{\min}$ , $|H|^{2}\leqslant A$ on $t\in[s,t^{*}]$ , $s<t^{*}\leqslant t_{1},t_{2}$ , then for $x_{1},x_{2}\in M$

\mathcal{N}^{\phi*}_{s}(x_{1},t_{1})-\mathcal{N}^{\phi*}_{s}(x_{2},t_{2})\leqslant\left(\frac{n}{2(t^{*}-s)}-R^{\phi}_{\min}+\frac{1}{3}A\right)^{\frac{1}{2}}d_{W_{1}}^{t^{*}}\left(\nu_{x_{1},t_{1}}(t^{*}),\nu_{x_{2},t_{2}}(t^{*})\right)+\frac{n}{2}\log\left(\frac{t_{2}-s}{t^{*}-s}\right).

Proof.

It follows from Proposition 2.16 with same argument as [Bam20a, Proof of Corollary 5.11]. ∎

We can get a gradient bound on $P^{*}_{s}$ by direct computation.

Proposition 2.20.

Suppose $|H|^{2}\leqslant A$ , $P^{*}_{s}(x,t):=P_{x,t}(t-s)$ , we have $|\nabla P^{*}_{s}(x,t)|\leqslant C_{n}A\sqrt{t-s}$ .

Proof.

By changing the order of integration, we have

	$\displaystyle P^{*}_{t-\tau}(x,t)$	$\displaystyle=\int_{0}^{\tau}s^{-1}\Psi_{x,t}(s)\mathrm{d}s=\int_{0}^{\tau}\int_{0}^{s}\int_{M}s^{-1}\|H\|^{2}\mathrm{d}\nu_{t-r}\mathrm{d}r\mathrm{d}s$
		$\displaystyle=\int_{0}^{\tau}\int_{r}^{\tau}\int_{M}s^{-1}\|H\|^{2}\mathrm{d}\nu_{t-r}\mathrm{d}s\mathrm{d}r=\int_{0}^{\tau}\log\left(\frac{\tau}{s}\right)\int_{M}\|H\|^{2}\mathrm{d}\nu_{t-s}\mathrm{d}s.$

	$\displaystyle\left\|\nabla P^{*}_{t-\tau}\right\|$	$\displaystyle\leqslant\int_{0}^{\tau}\log\left(\frac{\tau}{s}\right)\int_{M}\|H\|^{2}\frac{\left\|\nabla K(x,t;y,t-s)\right\|}{K(x,t;y,t-s)}\mathrm{d}\nu_{t-s}(y)\mathrm{d}s$
		$\displaystyle\leqslant C_{n}A\int_{0}^{\tau}s^{-\frac{1}{2}}\log\left(\frac{\tau}{s}\right)\mathrm{d}s=C_{n}A\sqrt{\tau}.$

where we use integral bounds on the gradient of heat kernel along super Ricci flow in [Bam20a, Proposition 4.2]. ∎

The oscillation bound of $P^{*}_{s}$ follows from a straightforward computation:

Proposition 2.21.

Suppose $0<t^{*}<t_{1}<t_{2}$ , $|H|^{2}+|\nabla H|^{2}\leqslant C$ , $d_{W_{1}}^{t^{*}}(\nu_{x_{1},t_{1}}(t^{*}),\nu_{x_{2},t_{2}}(t^{*}))\leqslant\delta$ , then

P^{*}_{0}(x_{1},t_{1})\leqslant P^{*}_{0}(x_{2},t_{2})+C\left(t_{1}-(1-\delta)\int_{0}^{t^{*}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s\right).

In particular, assume $t_{2}-t^{*}<\varepsilon<10^{-1}$ , $0<T^{-1}<t_{2}<T$ , then

P^{*}_{0}(x_{1},t_{1})\leqslant P^{*}_{0}(x_{2},t_{2})+C(\varepsilon T-\varepsilon\log\varepsilon+2\delta T).

Proof.

As the computation in Proof of Proposition 2.20, we have

\displaystyle P^{*}_{0}(x_{i},t_{i})=\int_{0}^{t_{i}}\log\left(\frac{t_{i}}{t_{i}-s}\right)\int_{M}|H|^{2}\mathrm{d}\nu_{x_{i},t_{i}}(s)\mathrm{d}s=\int_{0}^{t^{*}}+\int_{t^{*}}^{t_{i}},

where $i=1,2$ . For the second term, we have

0\leqslant\int_{t^{*}}^{t_{i}}\log\left(\frac{t_{i}}{t_{i}-s}\right)\int_{M}|H|^{2}\mathrm{d}\nu_{x_{i},t_{i}}(s)\mathrm{d}s\leqslant C\int_{t^{*}}^{t_{i}}\log\left(\frac{t_{i}}{t_{i}-s}\right)\mathrm{d}s,

where we use $|H|^{2}\leqslant C$ . Then we have

	$\displaystyle P^{}_{0}(x_{1},t_{1})-P^{}_{0}(x_{2},t_{2})\leqslant$	$\displaystyle\int_{0}^{t^{}}\log\left(\frac{t_{1}}{t_{1}-s}\right)\int_{M}\|H\|^{2}\mathrm{d}\nu_{x_{1},t_{1}}(s)\mathrm{d}s+C\int_{t^{}}^{t_{1}}\log\left(\frac{t_{1}}{t_{1}-s}\right)\mathrm{d}s$
		$\displaystyle-\int_{0}^{t^{*}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\int_{M}\|H\|^{2}\mathrm{d}\nu_{x_{2},t_{2}}(s)\mathrm{d}s.$

By lemma 2.18, we have $d_{W_{1}}^{t}(\nu_{x_{1},t_{1}}(t),\nu_{x_{2},t_{2}}(t))\leqslant\delta$ for any $t\leqslant t^{*}$ .

		$\displaystyle\int_{0}^{t^{}}\log\left(\frac{t_{1}}{t_{1}-s}\right)\int_{M}\|H\|^{2}\mathrm{d}\nu_{x_{1},t_{1}}(s)\mathrm{d}s-\int_{0}^{t^{}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\int_{M}\|H\|^{2}\mathrm{d}\nu_{x_{2},t_{2}}(s)\mathrm{d}s$
	$\displaystyle=$	$\displaystyle\ I_{1}+I_{2},$

where

	$\displaystyle I_{1}$	$\displaystyle:=\int_{0}^{t^{*}}\left[\log\left(\frac{t_{1}}{t_{1}-s}\right)-\log\left(\frac{t_{2}}{t_{2}-s}\right)\right]\int_{M}\|H\|^{2}\mathrm{d}\nu_{x_{1},t_{1}}(s)\mathrm{d}s$
		$\displaystyle\leqslant C\int_{0}^{t^{*}}\left[\log\left(\frac{t_{1}}{t_{1}-s}\right)-\log\left(\frac{t_{2}}{t_{2}-s}\right)\right]\mathrm{d}s,$
	$\displaystyle I_{2}$	$\displaystyle:=\int_{0}^{t^{*}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\int_{M}\|H\|^{2}\mathrm{d}\left(\nu_{x_{1},t_{1}}(s)-\nu_{x_{2},t_{2}}(s)\right)\mathrm{d}s$
		$\displaystyle\leqslant C\delta\int_{0}^{t^{*}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s.$

The inequality on $I_{1}$ is due to $|H|^{2}\leqslant C$ and the fact that $\log\left(\frac{t_{1}}{t_{1}-s}\right)-\log\left(\frac{t_{2}}{t_{2}-s}\right)\geqslant 0$ . The inequality on $I_{2}$ is due to the definition of $d_{W_{1}}$ and $\left|\nabla|H|^{2}\right|\leqslant 2|H||\nabla H|\leqslant C$ .

Combining above, we have

	$\displaystyle P^{}_{0}(x_{1},t_{1})-P^{}_{0}(x_{2},t_{2})\leqslant$	$\displaystyle C\left\{\int_{0}^{t^{}}\left[\log\left(\frac{t_{1}}{t_{1}-s}\right)-\log\left(\frac{t_{2}}{t_{2}-s}\right)\right]\mathrm{d}s+\delta\int_{0}^{t^{}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s\right.$
		$\displaystyle\left.+\int_{t^{*}}^{t_{1}}\log\left(\frac{t_{1}}{t_{1}-s}\right)\mathrm{d}s\right\}$
	$\displaystyle\leqslant$	$\displaystyle C\left\{\int_{0}^{t_{1}}\log\left(\frac{t_{1}}{t_{1}-s}\right)\mathrm{d}s-\int_{0}^{t^{}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s+\delta\int_{0}^{t^{}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s\right\}$
	$\displaystyle\leqslant$	$\displaystyle C\left(t_{1}-(1-\delta)\int_{0}^{t^{*}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s\right).$

If $t_{2}-t^{*}<\varepsilon<10^{-1}$ , $0<T^{-1}<t_{2}<T$ , then

\displaystyle\int_{0}^{t^{*}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s=t^{*}\log t_{2}-t_{2}\log t_{2}+t^{*}+(t_{2}-t^{*})\log(t_{2}-t^{*})\leqslant T+\log T\leqslant 2T,

	$\displaystyle t_{1}-\int_{0}^{t^{*}}\log\left(\frac{t_{2}}{t_{2}-s}\right)\mathrm{d}s=$	$\displaystyle(t_{1}-t^{})+(t_{2}-t^{})\log t_{2}-(t_{2}-t^{})\log(t_{2}-t^{})$
	$\displaystyle\leqslant$	$\displaystyle\varepsilon+\varepsilon\log T-\varepsilon\log\varepsilon$
	$\displaystyle\leqslant$	$\displaystyle\varepsilon T-\varepsilon\log\varepsilon.$

\displaystyle P^{*}_{0}(x_{1},t_{1})-P^{*}_{0}(x_{2},t_{2})\leqslant C(\varepsilon T-\varepsilon\log\varepsilon+2\delta T).

∎

2.2. Weighted noncollapsed

Definition 2.22.

Define the $\phi$ -weighted volume by

|B(x,t,r)|_{\phi}:=\int_{B(x,t,r)}e^{-\phi(y,t)}\mathrm{d}g_{t}(y).

Remark 2.23.

Since $\square\phi=\frac{1}{6}|H|^{2}\geqslant 0$ , if we assume $\phi$ has nonnegative initial data, then $\phi(x,t)$ will always be nonnegative by maximum principle. Then we have $|B(x,t,r)|_{\phi}\leqslant|B(x,t,r)|_{g_{t}}$ . Thus weighted noncollapsed condition is stronger than the conventional condition.

Theorem 2.24.

Suppose $[t_{0}-r^{2},t_{0}]\subset I$ , $|R^{\phi}|+|H|^{2}\leqslant Cr^{-2}$ on $B(x_{0},t_{0},r)\times[t_{0}-r^{2},t_{0}]$ , $\phi\geqslant 0$ , then

|B(x_{0},t_{0},r)|_{\phi}\geqslant c\exp(\mathcal{N}^{H}_{x_{0},t_{0}}(r^{2}))r^{n}.

Proof.

The proof follows the argument in [Bam20a, Proof of Theorem 6.1] with a modification. After parabolic scaling, we can assume $r=1$ , $t_{0}=0$ .

Take $h(y):=\eta(d_{0}(x_{0},y))\in C^{\infty}_{c}(B(x_{0},0,1))$ , where $\eta$ is cut-off function, such that $0\leqslant\eta\leqslant 1$ , $\eta|_{[0,\frac{1}{2}]}\equiv 1$ , $\eta|_{[1,\infty)}\equiv 0$ , $0\geqslant\eta^{\prime}\geqslant-4$ . $a:=\int_{M}h^{2}e^{-\phi}\mathrm{d}g_{0}$ , then $|B(x_{0},0,\frac{1}{2})|_{\phi}\leqslant a\leqslant|B(x_{0},0,1)|_{\phi}$ .

Take $v\in C^{\infty}(M\times[-1,0])$ , such that $\square^{*}_{\phi}v=0$ and $v(\cdot,0)=a^{-1}h^{2}$ , then $\int ve^{-\phi}\mathrm{d}g=\int a^{-1}h^{2}e^{-\phi}\mathrm{d}g_{0}=1$ . $v_{t}=(4\pi\tau)^{-\frac{n}{2}}e^{-f_{t}}$ , where $\tau:=1-t$ . $f_{-1}=-\log v_{-1}-\frac{n}{2}\log(8\pi)$ , $f_{0}=-\log(a^{-1}h^{2})-\frac{n}{2}\log(4\pi)$ .

\mathcal{N}^{\phi}[g_{-1},f_{-1},2]=\int f_{-1}v_{-1}e^{-\phi}\mathrm{d}g_{-1}-\frac{n}{2}=-\int v_{-1}\log v_{-1}e^{-\phi}\mathrm{d}g_{-1}-C_{n}.

Note that $v_{-1}(z)e^{-\phi(z,-1)}=\int K(y,0;z,-1)a^{-1}h^{2}(y)e^{-\phi(y,0)}\mathrm{d}g_{0}(y)$ , then by Jensen inequality,

v_{-1}\log v_{-1}(z)\leqslant\int e^{\phi(z,-1)}K(y,0;z,-1)\left(\log K(y,0;z,-1)+\phi(z,-1)\right)a^{-1}h^{2}(y)e^{-\phi(y,0)}\mathrm{d}g_{0}(y).

Note that $\mathcal{N}^{\phi}_{y,0}(1)=-\int K(y,0;z,-1)\left(\log K(y,0;z,-1)+\phi(z,-1)\right)\mathrm{d}g_{-1}(z)+C_{n}$ , then

\mathcal{N}^{\phi}[g_{-1},f_{-1},2]\geqslant\int\mathcal{N}^{\phi}_{y,0}(1)a^{-1}h^{2}(y)e^{-\phi(y,0)}\mathrm{d}g_{0}(y)-C_{n}\geqslant\mathcal{N}^{\phi}_{x,0}(1)-C\geqslant\mathcal{N}^{H}_{x,0}(1)-C.

	$\displaystyle\int_{0}^{1}\mathcal{W}^{\phi}[g_{-s},f_{-s},1+s]\mathrm{d}s$	$\displaystyle=\int_{0}^{1}\frac{\mathrm{d}}{\mathrm{d}s}\left((1+s)\mathcal{N}^{\phi}[g_{-s},f_{-s},1+s]\right)\mathrm{d}s$
		$\displaystyle=2\mathcal{N}^{\phi}[g_{-1},f_{-1},2]-\mathcal{N}^{\phi}[g_{0},f_{0},1]$
		$\displaystyle\geqslant 2\mathcal{N}^{\phi}_{x,0}(1)-\mathcal{N}^{\phi}[g_{0},f_{0},1]-C,$

\int_{0}^{1}\mathcal{W}^{\phi}[g_{-s},f_{-s},1+s]\mathrm{d}s\leqslant\mathcal{W}^{\phi}[g_{0},f_{0},1]+C,

	$\displaystyle\mathcal{W}^{\phi}[g_{0},f_{0},1]+\mathcal{N}^{\phi}[g_{0},f_{0},1]$	$\displaystyle=\int\left(R^{\phi}+\|\nabla f_{0}\|^{2}+2f_{0}\right)\mathrm{d}\nu_{0}-C_{n}$
		$\displaystyle\leqslant a^{-1}\int(4\|\nabla h\|^{2}-2h^{2}\log h^{2})e^{-\phi}\mathrm{d}g_{0}+2\log a+C$
		$\displaystyle\leqslant a^{-1}(100+2e^{-1})\|B(x_{0},0,1)\|_{\phi}+2\log\|B(x_{0},0,1)\|_{\phi}+C$
		$\displaystyle\leqslant 200\frac{\|B(x_{0},0,1)\|_{\phi}}{\|B(x_{0},0,\frac{1}{2})\|_{\phi}}+2\log\|B(x_{0},0,1)\|_{\phi}+C.$

Then by parabolic rescale, we have

r^{-n}|B(x_{0},0,r)|_{\phi}\geqslant c\exp(\mathcal{N}^{H}_{x,0}(r^{2}))\exp\left(-100\frac{|B(x_{0},0,r)|_{\phi}}{|B(x_{0},0,\frac{r}{2})|_{\phi}}\right).

Then there exists $c_{0}=c_{0}(c,n)$ , we claim that $r^{-n}|B(x_{0},0,r)|_{\phi}\geqslant c_{0}\exp(\mathcal{N}^{H}_{x,0}(r^{2}))$ . If not, there exists $r_{0}>0$ , $r_{0}^{-n}|B(x_{0},0,r_{0})|_{\phi}<c_{0}\exp(\mathcal{N}^{H}_{x,0}(r_{0}^{2}))$ , then

\displaystyle\frac{|B(x_{0},0,r_{0})|_{\phi}}{|B(x_{0},0,r_{0}/2)|_{\phi}}>100^{-1}\log(\frac{c}{c_{0}})>2^{n}.

so $r_{0}/2$ is also a counterexample for the claim. Repeating this process, we get for any $k$ , $r_{k}=2^{-k}r_{0}$ violates the claim, however it’s impossible if we set $c_{0}<\omega_{n}$ since as $r\to 0$ , $r^{-n}|B(x_{0},0,r)|_{\phi}\to\omega_{n}e^{-\phi(x_{0},0)}$ , $\exp(\mathcal{N}^{H}_{x,0}(r_{0}^{2}))\to\exp(-\phi(x_{0},0)-\frac{1}{3}|H|^{2}(x_{0},0))$ . ∎

Remark 2.25.

Note that the key point in the proof is the monotonicity of the generalized entropy $\mathcal{N}^{H}_{x,t}$ . For the weighted entropy $\mathcal{N}^{\phi}_{x,t}$ , the last argument of the proof fails.

Bamler introduced the concept of $H_{n}$ -center instead of worldline to represent the same point in different time with error.

Definition 2.26 ([Bam20a, Definition 3.10]).

A point $(z,t)\in M\times I$ is called an $H_{n}$ -center of a point $(x_{0},t_{0})\in M\times I$ if $t\leqslant t_{0}$ and $\operatorname{Var}_{t}(\delta_{z},\nu_{x_{0},t_{0};t})\leqslant H_{n}(t_{0}-t)$ , where the variance $\operatorname{Var}(\mu_{1},\mu_{2}):=\int_{M\times M}d^{2}(x_{1},x_{2})\mathrm{d}\mu_{1}(x_{1})\mathrm{d}\mu_{2}(x_{2})$ .

In [Bam20a, Proposition 3.12], Bamler shows that along the super Ricci flow, $H_{n}$ -center always exists due to the $H_{n}$ -concentration of such flow. Considering the $H_{n}$ -center instead, we can have a similar result using the weaker entropy $\mathcal{N}^{\phi}_{x,t}$ .

Theorem 2.27.

Suppose $[t_{0}-r^{2},t_{0}]\subset I$ , $R^{\phi}(\cdot,t_{0}-r^{2})\geqslant R^{\phi}_{\min}$ , $|H|^{2}\leqslant Ar^{-2}$ , $(z,t_{0}-r^{2})$ is $H_{n}$ -center of $(x_{0},t_{0})$ , then

|B(z,t_{0}-r^{2},\sqrt{2H_{n}}r)|_{\phi}\geqslant c\exp(\mathcal{N}^{\phi}_{x_{0},t_{0}}(r^{2}))r^{n},

where $c=c\exp(-2(n-2R^{\phi}_{\min}r^{2}+\frac{2}{3}A)^{\frac{1}{2}})$ .

Proof.

The proof is analogous to the argument in [Bam20a, Proof of Theorem 6.2]. After parabolic scaling, we can assume $r=t_{0}=1$ . Denote by $\mathrm{d}\nu_{t}:=\mathrm{d}\nu_{x_{0},1;t}$ , $B:=B(z,0,\sqrt{2H_{n}})$ . By [Bam20a, Proposition 3.13], we have $\nu_{0}(B)\geqslant\frac{1}{2}$ .

\int_{M}\left|f-\frac{n}{2}-\mathcal{N}^{\phi}_{x_{0},1}(1)\right|\mathrm{d}\nu_{0}\leqslant\left(\int_{M}\left(f-\frac{n}{2}-\mathcal{N}^{\phi}_{x_{0},1}(1)\right)^{2}\mathrm{d}\nu_{0}\right)^{\frac{1}{2}}\leqslant\left(n-2R^{\phi}_{\min}+\frac{2}{3}\Psi_{x_{0},1}(1)\right)^{\frac{1}{2}}.

	$\displaystyle\frac{1}{\nu_{0}(B)}\int_{B}f\mathrm{d}\nu_{0}$	$\displaystyle\geqslant\frac{n}{2}+\mathcal{N}^{\phi}_{x_{0},1}(1)-\frac{1}{\nu_{0}(B)}\int_{M}\left\|f-\frac{n}{2}-\mathcal{N}^{\phi}_{x_{0},1}(1)\right\|\mathrm{d}\nu_{0}$
		$\displaystyle\geqslant\frac{n}{2}+\mathcal{N}^{\phi}_{x_{0},1}(1)-2\left(n-2R^{\phi}_{\min}+\frac{2}{3}\Psi_{x_{0},1}(1)\right)^{\frac{1}{2}}.$

Let $u:=\nu_{0}(B)^{-1}(4\pi)^{-\frac{n}{2}}e^{-f}$ , then $\int_{B}ue^{-\phi}\mathrm{d}g_{0}=1$ .

	$\displaystyle\int_{B}u\log ue^{-\phi}\mathrm{d}g_{0}$	$\displaystyle=-\frac{1}{\nu_{0}(B)}\int f\mathrm{d}\nu_{0}-\frac{n}{2}\log(4\pi)-\log(\nu_{0}(B))$
		$\displaystyle\leqslant-\mathcal{N}^{\phi}_{x_{0},1}(1)-\frac{n}{2}+2\left(n-2R^{\phi}_{\min}+\frac{2}{3}\Psi_{x_{0},1}(1)\right)^{\frac{1}{2}}+\log 2+\frac{n}{2}\log(4\pi).$

By Jensen’s inequality,

\frac{1}{|B|_{\phi}}\int_{B}u\log ue^{-\phi}\mathrm{d}g_{0}\geqslant\left(\frac{1}{|B|_{\phi}}\int_{B}ue^{-\phi}\mathrm{d}g_{0}\right)\log\left(\frac{1}{|B|_{\phi}}\int_{B}ue^{-\phi}\mathrm{d}g_{0}\right)=-\frac{1}{|B|_{\phi}}\log|B|_{\phi},

\log|B|_{\phi}\geqslant\mathcal{N}^{\phi}_{x_{0},1}(1)-2\left(n-2R^{\phi}_{\min}+\frac{2}{3}\Psi_{x_{0},1}(1)\right)^{\frac{1}{2}}-C_{n}.

∎

3. Heat kernel Estimate

Theorem 3.1.

Suppose $[s,t]\subset I$ , $R^{\phi}(\cdot,s)\geqslant R^{\phi}_{\min}$ , $|H|^{2}\leqslant A$ , then

K(x,t;y,s)\leqslant\frac{C}{(t-s)^{\frac{n}{2}}}\exp(-\mathcal{N}^{\phi*}_{s}(x,t)),

where $C=C(R^{\phi}_{\min}(t-s),A(t-s))$ .

Proof.

The proof follows from the argument in [Bam20a, Proof of Theorem 7.1]. After parabolic scaling, we can assume $s=0$ , $t=1$ . By [CCG⁺10, Theorem 26.25], we always have $K(x,t;y,0)\leqslant Zt^{-\frac{n}{2}}\exp(-\mathcal{N}^{\phi*}_{0}(x,t))$ for $t\in(0,1]$ , for some large $Z$ , however, $Z$ may depends on the flow $g(t)$ . We will show that there exists $\underline{Z}=\underline{Z}(R^{\phi}_{\min}(t-s),A(t-s))$ , such that if $Z\geqslant\underline{Z}$ , the bound will still hold after replacing $Z$ by $\frac{Z}{2}$ . Using this argument, we have $K(x,t;y,0)\leqslant\underline{Z}t^{-\frac{n}{2}}\exp(-\mathcal{N}^{\phi*}_{0}(x,t))$ . Note that the bound is parabolic scaling invariant, it suffices to show the bound holds for $t=1$ . Denote by $u(x,t):=K(x,t;y,0)$ , $\square u=0$ .

\frac{\mathrm{d}}{\mathrm{d}t}\int_{M}ue^{-\phi}\mathrm{d}g_{t}=-\int_{M}u(\square^{*}_{\phi}1)e^{-\phi}\mathrm{d}g_{t}=-\int_{M}R^{\phi}ue^{-\phi}\mathrm{d}g_{t}\leqslant-R^{\phi}_{\min}\int_{M}ue^{-\phi}\mathrm{d}g_{t}.

Then we have $\int_{M}ue^{-\phi}\mathrm{d}g_{t}\leqslant C\exp(-\phi(y,0))$ . Denote by $v:=(t-\frac{1}{2})|\nabla u|^{2}+u^{2}$ .

\square v\leqslant-2(t-\frac{1}{2})|\nabla^{2}u|^{2}-2|\nabla u|^{2}.

Then $\square v\leqslant 0$ for $t\in[\frac{1}{2},1]$ . We assume $t\geqslant\frac{1}{2}$ from now on, we have

	$\displaystyle v(x,t)$	$\displaystyle\leqslant\int_{M}K(x,t;y,\frac{1}{2})v(y,\frac{1}{2})\mathrm{d}g_{\frac{1}{2}}(y)=\int_{M}K(x,t;y,\frac{1}{2})u^{2}(y,\frac{1}{2})\mathrm{d}g_{\frac{1}{2}}(y)$
		$\displaystyle\leqslant 2^{n}Z^{2}\int_{M}K(x,t;y,\frac{1}{2})\exp(-2\mathcal{N}^{\phi*}_{0}(y,\frac{1}{2}))\mathrm{d}g_{\frac{1}{2}}(y).$

Take $(z,\frac{1}{2})$ be the $H_{n}$ -center of $(x,t)$ , we have $d_{W_{1}}^{\frac{1}{2}}(\delta_{z},\nu_{x,t}(\frac{1}{2}))\leqslant\sqrt{\frac{1}{2}H_{n}}$ . We have $-\mathcal{N}^{\phi*}_{0}(y,\frac{1}{2})\leqslant-\mathcal{N}^{\phi*}_{0}(z,\frac{1}{2})+Cd_{\frac{1}{2}}(y,z)\leqslant-\mathcal{N}^{\phi*}_{0}(x,t)+Cd_{\frac{1}{2}}(y,z)+C$ .

	$\displaystyle v(x,t)$	$\displaystyle\leqslant 2^{n}Z^{2}\int_{M}K(x,t;y,\frac{1}{2})\exp(-2\mathcal{N}^{\phi*}_{0}(y,\frac{1}{2}))\mathrm{d}g_{\frac{1}{2}}(y)$
		$\displaystyle\leqslant CZ^{2}\exp(-2\mathcal{N}^{\phi*}_{0}(x,t))\int_{M}K(x,t;y,\frac{1}{2})\exp(Cd_{\frac{1}{2}}(y,z))\mathrm{d}g_{\frac{1}{2}}(y).$

By [Bam20a, Proof of Theorem 7.1, Theorem 3.14], we have

\int_{M}K(x,t;y,\frac{1}{2})\exp(Cd_{\frac{1}{2}}(y,z))\mathrm{d}g_{\frac{1}{2}}(y)\leqslant C.

Note that $v(x,t)\geqslant(t-\frac{1}{2})|\nabla u|^{2}$ , we have for $t\in[\frac{3}{4},1]$ , $|\nabla u|(x,t)\leqslant CZ\exp(-\mathcal{N}^{\phi*}_{0}(x,t))$ .

Fix $(x_{0},1)$ , denote by $\mathrm{d}\nu:=K(x_{0},1;\cdot,\cdot)\mathrm{d}g$ , let $\rho\in(0,\frac{1}{2})$ to be determined, then $t_{1}:=1-\rho^{2}\in[\frac{3}{4},1]$ . Let $(z_{1},t_{1})$ be the $H_{n}$ -center of $(x_{0},1)$ .

-\mathcal{N}^{\phi*}_{0}(z_{1},t_{1})\leqslant-\mathcal{N}^{\phi*}_{0}(x_{0},1)+Cd_{W_{1}}^{t_{1}}(\delta_{z_{1}},\nu_{t_{1}})\leqslant-\mathcal{N}^{\phi*}_{0}(x_{0},1)+C\sqrt{H_{n}\rho^{2}}.

We have for $\rho\leqslant\bar{\rho}$ , $-\mathcal{N}^{\phi*}_{0}(z_{1},t_{1})\leqslant-\mathcal{N}^{\phi*}_{0}(x_{0},1)+\log 2$ . Then $-\mathcal{N}^{\phi*}_{0}(y,t_{1})\leqslant-\mathcal{N}^{\phi*}_{0}(x_{0},1)+\log 2+Cd_{t_{1}}(y,z_{1})$ . Denote by $B:=B(z_{1},t_{1},\sqrt{100H_{n}}\rho)$ .

u(x_{0},1)=\int_{M}u\mathrm{d}\nu_{t_{1}}=\int_{B}+\int_{M\setminus B}u\mathrm{d}\nu_{t_{1}}.

By Theorem 2.27, $|B|_{\phi}\geqslant c\exp(\mathcal{N}^{\phi*}_{0}(x_{0},1))\rho^{n}$ . For any $x^{\prime},x^{\prime\prime}\in B$ , we have $u(x^{\prime},t_{1})\leqslant u(x^{\prime\prime},t_{1})+CZ\exp(-\mathcal{N}^{\phi*}_{0}(x,t))\rho$ . Integrate by $|B|_{\phi}^{-1}e^{-\phi}\mathrm{d}g_{t_{1}}$ ,

	$\displaystyle u(x^{\prime},t_{1})$	$\displaystyle\leqslant\frac{1}{\|B\|_{\phi}}\int_{B}ue^{-\phi}\mathrm{d}g_{t_{1}}+CZ\exp(-\mathcal{N}^{\phi*}_{0}(x,t))\rho$
		$\displaystyle\leqslant C\exp(-\mathcal{N}^{\phi*}_{0}(x,t))(Z\rho+C\rho^{-n}).$

Then for the first term, we have $\int_{B}u\mathrm{d}\nu_{t_{1}}\leqslant C\exp(-\mathcal{N}^{\phi*}_{0}(x,t))(Z\rho+C\rho^{-n})$ . For the second term, $\nu_{t_{1}}(M\setminus B)\leqslant 100^{-1}$ , $\int_{M\setminus B}u\mathrm{d}\nu_{t_{1}}\leqslant Zt_{1}^{-\frac{n}{2}}\int_{M\setminus B}\exp(-\mathcal{N}^{\phi*}_{0}(y,t_{1}))\mathrm{d}\nu_{t_{1}}(y)$ . For $\rho\leqslant\bar{\rho}$ , since $-\mathcal{N}^{\phi*}_{0}(z_{1},t_{1})\leqslant-\mathcal{N}^{\phi*}_{0}(x_{0},1)+\log 2$ , we have

	$\displaystyle\int_{M\setminus B}u\mathrm{d}\nu_{t_{1}}$	$\displaystyle\leqslant 2Z\int_{M\setminus B}\exp(-\mathcal{N}^{\phi*}_{0}(y,t_{1}))\mathrm{d}\nu_{t_{1}}(y)$
		$\displaystyle\leqslant 4Z\exp(-\mathcal{N}^{\phi*}_{0}(x_{0},1))\int_{M\setminus B}\exp(Cd_{t_{1}}(y,z_{1}))\mathrm{d}\nu_{t_{1}}(y).$

By [Bam20a, (7.18)], we have

\int_{M\setminus B}\exp(Cd_{t_{1}}(y,z_{1}))\mathrm{d}\nu_{t_{1}}(y)\leqslant\nu(M\setminus B)+C\rho\leqslant\frac{1}{100}+C\rho.

Then we have $\int_{M\setminus B}u\mathrm{d}\nu_{t_{1}}\leqslant\exp(-\mathcal{N}^{\phi*}_{0}(x,t))(\frac{1}{10}Z+CZ\rho)$ .

u(x_{0},1)\leqslant\exp(-\mathcal{N}^{\phi*}_{0}(x,t))(CZ\rho+C\rho^{-n}+\frac{1}{10}Z).

First fix $\rho=\rho_{0}$ such that $C\rho_{0}\leqslant\frac{1}{10}$ , then for $Z\geqslant\underline{Z}=5C\rho_{0}^{-n}$ , we have

u(x_{0},1)\leqslant\frac{1}{2}Z\exp(-\mathcal{N}^{\phi*}_{0}(x,t)).

. ∎

Theorem 3.2.

Suppose $[s,t]\subset I$ , $R^{\phi}(\cdot,s)\geqslant R^{\phi}_{\min}$ , $|H|^{2}\leqslant A$ , then there exist constants $C,C_{0}$ depending on $R^{\phi}_{\min}(t-s)$ and $A(t-s)$ , such that

\frac{|\nabla_{x}K|}{K}(x,t;y,s)\leqslant\frac{C}{(t-s)^{\frac{1}{2}}}\sqrt{\log\left(\frac{C_{0}\exp(-\mathcal{N}^{\phi*}_{s}(x,t))}{(t-s)^{\frac{n}{2}}K(x,t;y,s)}\right)}.

Proof.

By [Bam20a, Proof of Theorem 7.5], Theorem 3.1 and Corollary 2.19 imply this theorem, since other argument holds for super Ricci flow. ∎

4. $\varepsilon$ -Regularity

We need some notions to state the theorem.

Definition 4.1.

Backward parabolic curvature radius is defined by

r_{\mathrm{Rm}}(x,t):=\sup\{r>0:|\mathrm{Rm}|\leqslant r^{-2}\text{on\ }B(x,t,r)\times[t-r^{2},t]\}.

Bamler introduce a new notion of parabolic neighborhood instead of conventional neighborhood in [Bam20a], which is an important concept in [Bam20a, Bam20b, Bam20c].

Definition 4.2 ([Bam20a]).

Suppose that $(x_{0},t_{0})\in M\times I$ , $r>0$ and $t_{0}-r^{2}\in I$ . The backward $P^{*}$ -parabolic $r$ -balls is defined by

P^{*}_{-}(x_{0},t_{0},r):=\{(x,t)\in M\times[t_{0}-r^{2},t_{0}]:d_{W_{1}}^{t_{0}-r^{2}}(\nu_{x_{0},t_{0};t_{0}-r^{2}},\nu_{x,t;t_{0}-r^{2}})<r\}.

Theorem 4.3.

Suppose $[t-r^{2},t]\subset I$ , $r^{2}|H|^{2}+r^{4}|\nabla H|^{2}\leqslant C$ , $R^{\phi}\geqslant-Cr^{-2}$ , there exists $\varepsilon(n,C)>0$ , if $\mathcal{N}^{H}_{x,t}(r^{2})\geqslant-\varepsilon$ , then $r_{\mathrm{Rm}}(x,t)\geqslant\varepsilon r$ .

Proof.

The proof follows from the argument in [Bam20a, Proof of Theorem 10.2]. After a parabolic scaling, we can assume $t=r=1$ . Assume there exists a sequence of counterexamples $(M_{i},(g_{i,t})_{t\in[0,1]},(H_{i,t}),(\phi_{i,t}),(x_{i},1))$ , $r_{i}:=r_{\mathrm{Rm}}(x_{i},1)<\varepsilon_{i}\to 0$ , $\mathcal{N}^{H}_{x_{i},1}(1)\geqslant-\varepsilon_{i}$ .

Choose a sequence $A_{i}\to\infty$ with $10A_{i}r_{i}\to 0$ . By a point-picking argument in [Bam20a, Claim 10.5], there exists $(x_{i}^{\prime},t_{i}^{\prime})\in P^{*}_{-}(x_{i},1,10A_{i}r_{i})$ satisfies $r_{i}^{\prime}:=r_{\mathrm{Rm}}(x_{i}^{\prime},t_{i}^{\prime})\leqslant r_{i}\to 0$ and $r_{\mathrm{Rm}}\geqslant\frac{1}{10}r_{i}^{\prime}$ on $P^{*}_{-}(x_{i}^{\prime},t_{i}^{\prime},A_{i}r_{i}^{\prime})$ .

By $R^{\phi}\geqslant-C$ , Corollary 2.19 and Proposition 2.21, we have

	$\displaystyle\mathcal{N}^{\phi*}_{0}(x_{i}^{\prime},t_{i}^{\prime})$	$\displaystyle\geqslant\mathcal{N}^{\phi*}_{0}(x_{i},1)-CA_{i}r_{i}-\frac{n}{2}\log\left(\frac{t_{i}^{\prime}}{1-(10A_{i}r_{i})^{2}}\right),$
	$\displaystyle P^{*}_{0}(x_{i}^{\prime},t_{i}^{\prime})$	$\displaystyle\leqslant P^{*}_{0}(x_{1},1)-C(A_{i}r_{i})^{2}\log(10A_{i}r_{i}).$

Then $\mathcal{N}^{H}_{x_{i},1}(1)\to 0$ , $A_{i}r_{i}\to 0$ implies $\mathcal{N}^{H}_{x_{i}^{\prime},t_{i}^{\prime}}(t_{i}^{\prime})\to 0$ . Note that $\int\phi_{i}(y,t_{i}^{\prime})\mathrm{d}\nu(y)=-\mathcal{N}^{\phi}_{x_{i}^{\prime},t_{i}^{\prime}}(0)\to 0$ , $\frac{\mathrm{d}}{\mathrm{d}\tau}\int\phi_{i}\mathrm{d}\nu=-\frac{1}{6}\int|H_{i}|^{2}\mathrm{d}\nu\leqslant 0$ . Then for $t\in[0,t_{i}^{\prime}]$ , $\int\phi_{i}(t)\mathrm{d}\nu$ uniformly converges to $0$ .

By [Bam20a, Corollary 9.6], we can take a sequence $A_{i}^{\prime}\leqslant A_{i}$ , $A_{i}^{\prime}\to\infty$ such that $P_{-}(x_{i}^{\prime},t_{i}^{\prime},A_{i}^{\prime}r_{i}^{\prime})\subset P^{*}_{-}(x_{i}^{\prime},t_{i}^{\prime},A_{i}r_{i}^{\prime})$ , then $r_{\mathrm{Rm}}\geqslant\frac{1}{10}r_{i}^{\prime}$ on $P_{-}(x_{i}^{\prime},t_{i}^{\prime},A_{i}^{\prime}r_{i}^{\prime})$ . Define $\tilde{g}_{i}(t):=r_{i}^{\prime-2}g_{i}(r_{i}^{\prime 2}t+t_{i}^{\prime}),$ $\tilde{H}_{i}(t):=r_{i}^{\prime-2}H_{i}(r_{i}^{\prime 2}t+t_{i}^{\prime})$ , we obtain a sequence of flows $(\tilde{M}_{i}$ , $(\tilde{g}_{i,t})_{t\in[-(A_{i}^{\prime})^{2},0]}$ , $(\tilde{H}_{i,t}),(\tilde{\phi}_{i,t})))$ satisfies that $r_{\mathrm{Rm}}\geqslant\frac{1}{10}$ on $P_{-}(x_{i}^{\prime},0,A_{i}^{\prime})$ , $r_{\mathrm{Rm}}(x_{i}^{\prime},0)=1$ , $|\tilde{H}_{i,t}|^{2}\leqslant Cr_{i}^{\prime 2}\to 0$ , and for any fixed $T>0$ , $\tilde{\mathcal{N}}^{H}_{x_{i}^{\prime},0}(T)=\mathcal{N}^{H}_{x_{i}^{\prime},t_{i}^{\prime}}(r_{i}^{\prime 2}T)\to 0$ , $\int\tilde{\phi}_{i}(T)\mathrm{d}\tilde{\nu}_{i}\to 0$ .

Then the injectivity radius at $(x_{i}^{\prime},0)$ is uniformly bounded from below by Theorem 2.24. Since $|\mathrm{Rm}|$ , $|H|$ , $|\nabla H|$ are all uniformly bounded, after passing to a subsequence, these flows converge to a smooth and complete pointed ancient flow $(M_{\infty},(g_{\infty,t})_{t\leqslant 0},x_{\infty},H_{\infty}=0)$ . Since $|\tilde{H}_{i}|^{2}\to 0$ , the limit flow is actually Ricci flow. By the upper bound of heat kernel and its gradient, $\tilde{K}_{i}(x_{i}^{\prime},0;\cdot,\cdot)$ converge locally uniformly to a positive solution $v_{\infty}\in C^{\infty}(M_{\infty}\times\mathbb{R}_{-})$ of conjugate heat equation. Since $\int\tilde{\phi}_{i}(T)\mathrm{d}\tilde{\nu}_{i}\to 0$ , $\square\tilde{\phi}_{i}=\frac{1}{6}|\tilde{H}_{i}|^{2}\to 0$ , $|\nabla\tilde{H}_{i}|^{2}\to 0$ , we have $\tilde{\phi}_{i}\to\phi_{\infty}\equiv 0$ .

Since for any fixed $T>0$ , $\tilde{\mathcal{N}}^{H}_{x_{i}^{\prime},0}(T)\to 0$ , then $\tilde{\mathcal{W}}^{H}_{x_{i}^{\prime},0}(T)\geqslant\int_{T}^{T+1}\tilde{\mathcal{W}}^{H}_{x_{i}^{\prime},0}(s)\mathrm{d}s=(T+1)\tilde{\mathcal{N}}^{H}_{x_{i}^{\prime},0}(T+1)-T\tilde{\mathcal{N}}^{H}_{x_{i}^{\prime},0}(T)\to 0$ , which implies

\int_{-T}^{0}\int_{M}\left(-2\tau\left|\mathrm{Ric}^{\tilde{f}_{i}+\tilde{\phi}_{i}}-\frac{1}{2\tau}\tilde{g}_{i}\right|^{2}-\frac{1}{3\tau}\tilde{\Psi}_{x_{i}^{\prime},0}\right)\mathrm{d}\nu\mathrm{d}t\to 0.

Then $\mathrm{Ric}^{f_{\infty}+\phi_{\infty}}-\frac{1}{2\tau}g_{\infty}=0$ . $\phi_{\infty}=0$ implies $\mathrm{Ric}+\nabla^{2}f_{\infty}-\frac{1}{2\tau}g_{\infty}=0$ , thus $g_{\infty}$ is a gradient shrinking soliton. Since $|\mathrm{Rm}|$ is uniformly bounded, $g_{\infty}$ must be flat, which contradicts with $r_{\mathrm{Rm}}(x_{\infty},0)=1<\infty$ . ∎

5. Some Estimates on $H$

In the previous sections, a bound on $|H|$ is always needed. Such bounds may not always hold without assumptions, so we will derive some estimates under centain curvature conditions.

5.1. Integral bounds on $H$

By the monotonicity of the weighted scalar curvature, (2.6) gives the first integral bounds depends on the scalar curvature $R$ and the volume $\mathrm{Vol}(g_{t})$ :

\int_{M}|H|^{2}\mathrm{d}g_{t}\leqslant 12\int_{M}\left(R-R^{\phi}_{0}\right)\mathrm{d}g_{t}.

Proposition 5.1.

We have

\mathrm{Vol}(g_{t})\leqslant e^{-3R^{\phi}_{0}t}\left(\mathrm{Vol}(g_{0})+\int_{0}^{t}\int_{M}2e^{3R^{\phi}_{0}s}R\mathrm{d}g_{s}\mathrm{d}s\right).

In particular, if $\int_{M}R\mathrm{d}g_{t}\leqslant C_{0}$ , then we have

•

If $R^{\phi}_{0}>0$ , then $\mathrm{Vol}(g_{t})\leqslant C$ , $\int_{M}|H|^{2}\mathrm{d}g_{t}\leqslant C$ .
•

If $R^{\phi}_{0}=0$ , then $\mathrm{Vol}(g_{t})\leqslant C(t+1)$ , $\int_{M}|H|^{2}\mathrm{d}g_{t}\leqslant C$ .
•

If $R^{\phi}_{0}<0$ , then $\mathrm{Vol}(g_{t})\leqslant Ce^{-3R^{\phi}_{0}t}$ , $\int_{M}|H|^{2}\mathrm{d}g_{t}\leqslant Ce^{-3R^{\phi}_{0}t}$ .

In particular, if $R\leqslant C_{0}$ , then we have

\mathrm{Vol}(g_{t})\leqslant Ce^{\left(2C_{0}-3R^{\phi}_{0}\right)t},\ \ \int_{M}|H|^{2}\mathrm{d}g_{t}\leqslant Ce^{Ct}.

Proof.

\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{Vol}(g_{t})=\int_{M}\left(\frac{1}{4}|H|^{2}-R\right)\mathrm{d}g_{t}\leqslant 2\int_{M}R\mathrm{d}g_{t}-3R^{\phi}_{0}\mathrm{Vol}(g_{t}).

Then the results follow from the Gronwall’s inequality. ∎

Proposition 5.2.

\frac{\mathrm{d}}{\mathrm{d}t}\int_{M}|H|^{2}\mathrm{d}g_{t}=\int_{M}\left(-2|d^{*}H|^{2}-\frac{3}{2}|H^{2}|^{2}+\frac{1}{4}|H|^{4}+6\left<\mathrm{Ric},H^{2}\right>-R|H|^{2}\right)\mathrm{d}g_{t}.

Proof.

Direct calculation. ∎

5.2. Pointwise bounds on $H$

In many cases, the integral bound is not enough. In previous sections, we need to estimate the term

\Psi_{x,t}(\tau)=\int_{t-\tau}^{t}\int_{M}|H|^{2}\mathrm{d}\nu_{x,t;s}\mathrm{d}s.

Note that $\Psi_{x,t}^{\prime}(0)=\int_{M}|H|^{2}\mathrm{d}\nu_{x,t;t}=|H|^{2}(x,t)$ , then $\Psi_{x,t}(\tau)=|H|^{2}(x,t)\tau+O(\tau^{2})$ . Thus to estimate the term $\Psi_{x,t}$ , the pointwise bound on $H$ is necessary.

Proposition 5.3.

(5.1)

\square|H|^{2}=-2|\nabla H|^{2}-\frac{3}{2}|H^{2}|^{2}-6R_{H}\leqslant-2|\nabla H|^{2}-\frac{3}{2n}|H|^{4}+C_{n}|\mathrm{Rm}||H|^{2},

where $R_{H}:=R_{ijkl}H^{ijr}H^{kls}g_{rs}$ .

Proof.

\frac{\partial}{\partial t}|H|^{2}=3\left<2\mathrm{Ric}-\frac{1}{2}H^{2},H^{2}\right>+2\left<\Delta_{d}H,H\right>,

where $\Delta_{d}=-(dd^{*}+d^{*}d)$ is Hodge Laplacian. Denote the connection Laplacian by $\Delta$ , then by the Weitzenböck formula, we have

2\left<\Delta_{d}H,H\right>=2\left<\Delta H-\mathrm{Ric}(H),H\right>=\Delta|H|^{2}-2|\nabla H|^{2}-2\left<\mathrm{Ric}(H),H\right>,

where $\mathrm{Ric}(H)$ is the Weitzenböck curvature operator on the tensor defined by

\mathrm{Ric}(H)(X_{1},\cdots,X_{k}):=\sum_{i,j}(R(e_{j},X_{i})H)(X_{1},\cdots,e_{j},\cdots,X_{k}).

In the normal coordinate,

	$\displaystyle(\mathrm{Ric}(H))_{ijk}=-\left(R_{sisr}H_{rjk}+R_{sijr}H_{srk}+R_{sikr}H_{sjr}+R_{sjir}H_{rsk}+R_{sjsr}H_{irk}+R_{sjkr}H_{isr}\right.$
	$\displaystyle\left.+R_{skir}H_{rjs}+R_{skjr}H_{irs}+R_{sksr}H_{ijr}\right),$

where

-R_{sisr}H_{rjk}=R_{ir}H_{rjk},\ -R_{sjsr}H_{irk}=R_{jr}H_{irk},\ -R_{sksr}H_{ijr}=R_{kr}H_{ijr}.

By Bianchi identity,

-R_{sijr}H_{srk}-R_{sjir}H_{rsk}=R_{ijsr}H_{srk}+R_{jsir}H_{srk}-R_{sjir}H_{rsk}=R_{ijsr}H_{srk}.

Thus

(\mathrm{Ric}(H))_{ijk}=R_{ir}H_{rjk}+R_{jr}H_{irk}+R_{kr}H_{ijr}+R_{ijsr}H_{srk}+R_{iksr}H_{sjr}+R_{jksr}H_{isr},

\left<\mathrm{Ric}(H),H\right>=3\left<\mathrm{Ric},H^{2}\right>+3R_{ijsr}H_{srk}H_{ijk}=3\left<\mathrm{Ric},H^{2}\right>+3R_{H},

	$\displaystyle\frac{\partial}{\partial t}\|H\|^{2}$	$\displaystyle=6\left<\mathrm{Ric},H^{2}\right>-\frac{3}{2}\|H^{2}\|^{2}+\Delta\|H\|^{2}-2\|\nabla H\|^{2}-6\left<\mathrm{Ric},H^{2}\right>-6R_{H}$
		$\displaystyle=\Delta\|H\|^{2}-2\|\nabla H\|^{2}-\frac{3}{2}\|H^{2}\|^{2}-6R_{H}.$

The inequality of (5.1) follows from $|A|^{2}\geqslant\frac{1}{n}(\operatorname{tr}A)^{2}$ and $|R_{H}|=|\mathrm{Rm}*H*H|\leqslant C_{n}|\mathrm{Rm}||H|^{2}.$ ∎

Corollary 5.4.

Suppose $|\mathrm{Rm}|\leqslant K$ , then $|H|^{2}\leqslant\max\left(C_{n}K,\max_{x}|H|^{2}(x,0)\right)$ .

Proof.

Apply maximum principle to (5.1). ∎

In general cases, bounds on Riemann curvature tensor seems to be necessary. However, the more important cases of generalized Ricci flow is in the lower dimension. For the dimensional reasons, the equation will be a bit simplified. Since the 3-form in 2-manifolds is trivial, the generalized Ricci flow is exactly the same as the Ricci flow. In dimension $n=3$ , the tensor $H$ will be deduced to a scalar function:

Proposition 5.5 ([GFS21, Proposition 4.38]).

Suppose $n=3$ , $(g_{t},H_{t})$ solves the generalized Ricci flow, the function $h_{t}\in C^{\infty}(M)$ satisfies $H_{t}=h_{t}\mathrm{d}g_{t}$ , then

\left\{\begin{aligned} \frac{\partial}{\partial t}g&=-2\mathrm{Ric}+h^{2}g,\\ \frac{\partial}{\partial t}h&=\Delta h+Rh-\frac{3}{2}h^{3}.\end{aligned}\right.

Corollary 5.6.

Suppose $n=3$ , $|R|\leqslant C_{0}$ , then $|H|\leqslant C$ .

Proof.

Assume $h$ attains its maximum at $(x_{0},t_{0})$ , $t_{0}>0$ , $h(x_{0},t_{0})=\max_{t\in[0,T]}h(x,t)$ .

0\leqslant\square h(x_{0},t_{0})\leqslant Rh(x_{0},t_{0})-\frac{3}{2}h^{3}(x_{0},t_{0}).

Then $h(x_{0},t_{0})\leqslant\left(\frac{2}{3}C_{0}\right)^{\frac{1}{2}}$ . The lower bound is similar. The result follows from $|H|^{2}=6h^{2}$ . ∎

In dimension $n=4$ , the bounds on scalar curvature $R$ seems only to give a time-dependent bound on $||H||_{L^{2}}$ . However, by a careful computation, we will show the Ricci curvature bound implies the pointwise bound on $H$ .

Proposition 5.7.

Suppose $n=4$ , for a suitable orthonormal basis $\{e_{i}\}$ of $T_{p}M$ , we have

(H^{2})_{ij}(p)=\frac{1}{3}|H|^{2}\left(\begin{matrix}1&&&\\ &1&&\\ &&1&\\ &&&0\end{matrix}\right).

Thus there exists a vector field $X_{H}$ as the eigenvector of $0$ . In fact, we can take $X_{H}=\left(\star H\right)^{\sharp}\in\Gamma^{\infty}(TM)$ .

Proof.

It’s obvious if $H(p)=0$ . So we assume $H(p)\neq 0$ . Take $\left(\star H\right)^{\sharp}=\lambda e_{4}$ and extend $e_{4}$ to an orthonormal basis $\{e_{i}\}$ . Then

\displaystyle H=-\star(\lambda e^{4})=\lambda e^{1}\wedge e^{2}\wedge e^{3}.

So $\lambda^{2}=\frac{1}{6}|H|^{2}$ . $(H^{2})_{4k}=(H^{2})_{k4}=\left<i_{e_{4}}H,i_{e_{k}}H\right>=0$ for $k\in\{1,2,3,4\}$ . For $i,j\in\{1,2,3\}$

\displaystyle(H^{2})_{ij}=\sum_{k,l\in\{1,2,3\}}H_{ikl}H_{jkl}=\delta_{ij}\sum_{k,l\in\{1,2,3\}}H^{2}_{ikl}=2\lambda^{2}\delta_{ij}=\frac{1}{3}|H|^{2}\delta_{ij}.

∎

Proposition 5.8.

Suppose $n=4$ , we have

R_{H}=\frac{1}{3}R|H|^{2}-2\left<\mathrm{Ric},H^{2}\right>=-\frac{1}{3}R|H|^{2}+\frac{2}{3}\mathrm{Ric}(X_{H})|H|^{2}.

Combine with (5.1), we have

\square|H|^{2}=-2|\nabla H|^{2}-\frac{1}{2}|H|^{4}+2R|H|^{2}-4\mathrm{Ric}(X_{H})|H|^{2}.

Thus $\mathrm{Ric}\geqslant-K$ , $R\leqslant K$ implies $|H|^{2}\leqslant\max(12K,\max_{x}|H|^{2}(x,0))$ .

Proof.

Take the normal coordinate and let $\mathrm{Ric}=(R_{ij})$ to be diagonal.

R_{H}=\sum_{i,j,k,l,r}R_{ijkl}H_{ijr}H_{klr}=\sum_{\{i,j\}=\{k,l\}}+\sum_{\{i,j\}\neq\{k,l\}},

	$\displaystyle\sum_{\{i,j\}=\{k,l\}}=-4\left[K_{12}(H^{2}_{123}+H^{2}_{124})+K_{13}(H^{2}_{123}+H^{2}_{134})+K_{14}(H^{2}_{124}+H^{2}_{134})\right.$
	$\displaystyle\left.+K_{23}(H^{2}_{123}+H^{2}_{234})+K_{24}(H^{2}_{124}+H^{2}_{234})+K_{34}(H^{2}_{134}+H^{2}_{234})\right],$

where $K_{ij}=R_{ijji}$ is the sectional curvature. Note that

(K_{12}+K_{13}+K_{23})H^{2}_{123}=\left(\frac{1}{2}R-R_{44}\right)H^{2}_{123}.

Thus we have

\sum_{\{i,j\}=\{k,l\}}=-2R\left(H^{2}_{123}+H^{2}_{124}+H^{2}_{134}+H^{2}_{234}\right)+4\left(R_{11}H^{2}_{234}+R_{22}H^{2}_{134}+R_{33}H^{2}_{124}+R_{44}H^{2}_{123}\right).

Using

(H^{2})_{11}=2(H^{2}_{123}+H^{2}_{124}+H^{2}_{134})=2\left(\frac{1}{6}|H|^{2}-H^{2}_{234}\right),

we have

(5.2)		$\displaystyle\sum_{\{i,j\}=\{k,l\}}$	$\displaystyle=-\frac{1}{3}R\|H\|^{2}+\frac{2}{3}\|H\|^{2}\sum_{i}R_{ii}-2\sum_{i}R_{ii}(H^{2})_{ii}$
(5.2)			$\displaystyle=\frac{1}{3}R\|H\|^{2}-2\left<\mathrm{Ric},H^{2}\right>.$

(5.3)	$\displaystyle\sum_{\{i,j\}\neq\{k,l\}}=8$	$\displaystyle\left[(R_{1213}+R_{2434})H_{124}H_{134}+(R_{1214}+R_{2334})H_{123}H_{143}+(R_{1223}+R_{1434})H_{124}H_{234}\right.$
		$\displaystyle\left.(R_{1224}+R_{1334})H_{123}H_{243}+(R_{1314}+R_{2324})H_{132}H_{142}+(R_{1323}+R_{1424})H_{134}H_{234}\right]$
	$\displaystyle=8$	$\displaystyle\left(-R_{23}H_{124}H_{134}-R_{24}H_{123}H_{143}+R_{13}H_{124}H_{234}+R_{14}H_{123}H_{243}\right.$
		$\displaystyle\ \ \left.-R_{34}H_{132}H_{142}-R_{12}H_{134}H_{234}\right)$
	$\displaystyle=0$	.

Combine (5.2) and (5.3), we have

R_{H}=\frac{1}{3}R|H|^{2}-2\left<\mathrm{Ric},H^{2}\right>.

For the suitable orthonormal basis in Proposition 5.7, $X_{H}=e_{4}$ .

\left<\mathrm{Ric},H^{2}\right>=\sum_{i}R_{ii}(H^{2})_{ii}=\frac{1}{3}|H|^{2}\left(R_{11}+R_{22}+R_{33}\right)=\frac{1}{3}R|H|^{2}-\frac{1}{3}R_{44}|H|^{2},

R_{H}=-\frac{1}{3}R|H|^{2}+\frac{2}{3}\mathrm{Ric}(X_{H})|H|^{2}.

∎

Recently, an interesting bound of torsion is found in the case of the pluriclosed flow on complex surface with special initial data, see [Ye23].

Theorem 5.9 ([Ye23, Theorem 1.3]).

For a compact complex surface $(M^{4},J,g(t))$ that admits a pluriclosed flow with Hermitian-symplectic initial data on $[0,\tau)$ , if the Chern scalar curvature satisfies

\displaystyle R^{C}(x,t)\leqslant K,\ x\in M,t\in[0,\tau),

then we have

\displaystyle|H(x,t)|^{2}_{g(x,t)}\leqslant\max\left\{2K,\max_{x}|H(x,0)|^{2}_{g(x,0)}\right\},\ x\in M,t\in[0,\tau).

References

[Bam20a] R. H. Bamler, Entropy and heat kernel bounds on a Ricci flow background, 2020. arXiv:2008.07093.
[Bam20b] by same author, Compactness theory of the space of Super Ricci flows, 2020. arXiv:2008.09298.
[Bam20c] by same author, Structure theory of non-collapsed limits of Ricci flows, 2020. arXiv:2009.03243.
[CCG⁺10] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci flow: techniques and applications. Part III. Geometric-analytic aspects, Mathematical Surveys and Monographs, vol. 163, American Mathematical Society, Providence, RI, 2010. https://doi.org/10.1090/surv/163.
[Gau77] P. Gauduchon, Le théorème de l’excentricité nulle, C. R. Acad. Sci. Paris Sér. A-B 285 (1977) A387–A390.
[GFS21] M. Garcia-Fernandez and J. Streets, Generalized Ricci flow, University Lecture Series, vol. 76, American Mathematical Society, Providence, RI, [2021] ©2021. https://doi.org/10.1090/ulect/076.
[Gue02] C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (2002) 425–436.
[HN14] H.-J. Hein and A. Naber, New logarithmic Sobolev inequalities and an $\epsilon$ -regularity theorem for the Ricci flow, Comm. Pure Appl. Math. 67 (2014) 1543–1561.
[ST10] J. Streets and G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. IMRN (2010) 3101–3133.
[ST13] by same author, Regularity results for pluriclosed flow, Geom. Topol. 17 (2013) 2389–2429.
[Str22a] J. Streets, Ricci-Yang-Mills flow on surfaces and pluriclosed flow on elliptic fibrations, Adv. Math. 394 (2022) Paper No. 108127, 31.
[Str22b] J. Streets, Scalar curvature, entropy, and generalized Ricci flow, 2022. arXiv:2207.13197.
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	$\displaystyle\mathcal{W}^{\phi}[g_{0},f_{0},1]+\mathcal{N}^{\phi}[g_{0},f_{0},1]$	$\displaystyle=\int\left(R^{\phi}+\|\nabla f_{0}\|^{2}+2f_{0}\right)\mathrm{d}\nu_{0}-C_{n}$
		$\displaystyle\leqslant a^{-1}\int(4\|\nabla h\|^{2}-2h^{2}\log h^{2})e^{-\phi}\mathrm{d}g_{0}+2\log a+C$
		$\displaystyle\leqslant a^{-1}(100+2e^{-1})\|B(x_{0},0,1)\|_{\phi}+2\log\|B(x_{0},0,1)\|_{\phi}+C$
		$\displaystyle\leqslant 200\frac{\|B(x_{0},0,1)\|_{\phi}}{\|B(x_{0},0,\frac{1}{2})\|_{\phi}}+2\log\|B(x_{0},0,1)\|_{\phi}+C.$

Entropy and Heat Kernel on Generalized Ricci Flow

Abstract.

1. Introduction

2. Generalized Entropy

2.1. Generalized Nash Entropy and Perelman Entropy

Definition 2.1 ([Str22b, Definition 2.2]).

Definition 2.2 ([Str22b, Definition 2.1]).

Proposition 2.3 ([Str22b, Proposition 2.4]).

Remark 2.4.

Corollary 2.5.

Proof.

Definition 2.6 ([Str22b, Definition 3.1]).

Definition 2.7 ([Str22b, Definition 4.1]).

Proposition 2.8.

Proof.

Proposition 2.9 ([Gue02, Theorem 2.1]).

Definition 2.10.

Definition 2.11.

Remark 2.12.

Proposition 2.13.

Proof.

Remark 2.14.

Corollary 2.15.

Proof.

Proposition 2.16.

Proof.

Definition 2.17.

Lemma 2.18.

Proof.

Corollary 2.19.

Proof.

Proposition 2.20.

Proof.

Proposition 2.21.

Proof.

2.2. Weighted noncollapsed

Definition 2.22.

Remark 2.23.

Theorem 2.24.

Proof.

Remark 2.25.

Definition 2.26 ([Bam20a, Definition 3.10]).

Theorem 2.27.

Proof.

3. Heat kernel Estimate

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

4. ε\varepsilon-Regularity

Definition 4.1.

Definition 4.2 ([Bam20a]).

Theorem 4.3.

Proof.

5. Some Estimates on HH

5.1. Integral bounds on HH

Proposition 5.1.

Proof.

Proposition 5.2.

Proof.

5.2. Pointwise bounds on HH

Proposition 5.3.

Proof.

Corollary 5.4.

Proof.

Proposition 5.5 ([GFS21, Proposition 4.38]).

Corollary 5.6.

Proof.

Proposition 5.7.

Proof.

Proposition 5.8.

Proof.

Theorem 5.9 ([Ye23, Theorem 1.3]).

References

4. $\varepsilon$ -Regularity

5. Some Estimates on $H$

5.1. Integral bounds on $H$

5.2. Pointwise bounds on $H$