The Ricci iteration towards cscK metrics

A long standing problem in Kähler geometry is to find constant scalar curvature Kähler (cscK) metrics in a given Kähler class. Namely, for a compact Kähler manifold $(X,\omega)$ of dimension $n$, we want to search for a Kähler form $\omega^{*}\in\{\omega\}$ that satisfies

$$\mathrm{tr}_{\omega^{*}}\operatorname{Ric}(\omega^{*})=\bar{R},$$

where $\bar{R}:=2\pi n\frac{-K_{X}\cdot\{\omega\}^{n-1}}{\{\omega\}^{n}}$ is the average of the scalar curvature.

Regarding the existence of such metrics, the influential Tian’s properness conjecture (cf. [41, Remark 5.2],[38, Conjecture 7.12]) predicts that the existence of cscK metrics is equivalent to some suitable notion of properness of Mabuchi’s K-energy functional. Tian’s conjecture is central in Kähler geometry and has attracted much work over the past two decades including motivating much work on equivalence between algebro-geometric notions of stability and existence of canonical metrics, as well as on the interface of pluripotential theory and Monge–-Ampère equations. We refer to the surveys [37, 31, 39, 28, 35].

In [20], using the Finsler geometry of the space of Kähler metrics, the authors reduce Tian’s conjecture to a purely PDE regularity problem, which has been recently solved in [13]. Therefore we now have complete solution to Tian’s properness conjecture.

On the other hand, provided the properness of the K-energy, how to produce a cscK metric is also a challenging problem in its own right. In [13] the authors show that certain continuity path provides an approach towards cscK metrics. In this work we show that one can also produce a cscK metric using some dynamical system.

To present our results, we begin by recalling an elementary result in Kähler geometry.

Therefore, $\omega^{*}$ is cscK if and only if $\operatorname{Ric}(\omega^{*})$ is a harmonic form with respect to $\omega^{*}$. This viewpoint is by no means new, which was explored in early works of Calabi, Futaki, Bando and Mabuchi; see, e.g., [2]. When combined with the framework of geometric flows, this motivates one to consider the following variant of the Kähler Ricc flow:

(1.1)

$$\partial_{t}\omega_{t}=-\operatorname{Ric}(\omega_{t})+\mathrm{HRic}(\omega_{t}),\ \omega_{0}=\omega.$$

Here, given any Kähler form $\alpha$, $\mathrm{HRic}(\alpha)$ denotes the harmonic part of $\operatorname{Ric}(\alpha)$ with respect to $\alpha$. If the flow (1.1) smoothly converges to a limit $\omega_{\infty}$, then one has

$$\operatorname{Ric}(\omega_{\infty})=\mathrm{HRic}(\omega_{\infty}),$$

namely, $\omega_{\infty}$ is a cscK metric.

The flow (1.1) first appeared in [21] (see also [36] for a related flow) and was then briefly studied in [34, 33]. Later in [15], this flow was systematically investigated and the authors call it the pseudo-Calabi flow. Note that the flow (1.1) can be viewed as a variant of the Calabi flow [10], and it can be reduced to a coupled system of parabolic equations. Indeed, it is easy to show that (1.1) is equivalent to the following coupled equations:

$$\begin{cases}\omega_{t}^{n}=e^{F_{t}}\omega^{n},\\ \dot{F_{t}}=\Delta_{t}F_{t}+\bar{R}-\mathrm{tr}_{\omega_{t}}\operatorname{Ric}(\omega).\end{cases}$$

So we obtain a parabolic version of the coupled equations for cscK metrics that are studied in [12, 13]. This being said, it is still a highly non-trivial problem to study this flow, with its long time existence and limiting behavior largely open.

In this paper we adopt a somewhat different viewpoint. We consider the discretization of the pseudo-Calabi flow (1.1) that was first proposed by Rubinstein [34, 33]. More precisely, given $\tau>0$, we investigate the following Ricci iteration that appeared in [34, Definition 2.1] and [33, (41)]:

(1.2)

$$\frac{\omega_{i+1}-\omega_{i}}{\tau}=-\operatorname{Ric}(\omega_{i+1})+\mathrm{HRic}(\omega_{i+1}),\ i\in{\mathbb{N}},\ \omega_{0}=\omega.$$

Part of the interest in this Ricci iteration is that, clearly, cscK metrics are fixed points. Therefore (1.2) aims to provide a natural theoretical and numerical approach to uniformization in the challenging case of cscK metrics. In [34, Conjecture 2.1], Rubinstein proposed the following.

In addition, the Ricci iteration could be a source of new insights for the study of the flow (1.1), which is known to be a rather difficult problem in the field of geometric flows. For instance, just as in the case of Calabi flow, the long time existence of the flow (1.1) is still unknown (see [15, Conjecture 8.3]). However, after discretization, we can prove the following long time existence result for the sequence (1.2).

This result gives a strong evidence for the long time existence of the flow (1.1). Indeed, sending $\tau\to 0$, the iteration sequence $\{\omega_{i}\}$ is expected to converge to the flow (1.1) (this is interestingly still a conjecture even for the Ricci iteration associated to the Kähler Ricci flow; compare the classical Rothe’s method for parabolic equations). The fact that the sequence $\{\omega_{i}\}$ exists for all $i$ should imply that the flow (1.1) exists for all $t$. This is of course a heuristic viewpoint, which hopefully can be made more rigorous in future study.

It is proved in [15, Theorem 3.1] that Mabuchi’s K-energy decreases along the flow (1.1). We show that this is also the case for the Ricci iteration (1.2), which was previously only known in the case where $c_{1}(X)=\lambda\{\omega\}$ (see [33, Proposition 4.2]).

Since a cscK metric, if exists, minimizes the K-energy. The above result suggests that the iteration sequence (1.2) has the tendency to be attracted by a cscK metric in a suitable sense. We show that this is indeed the case, thus confirming Conjecture 1.3.

Our results extend those in the previous works [34, 33, 26, 5, 19, 25], where $c_{1}(X)$ is assumed to be proportional to $\{\omega\}$. Moreover, Theorem 1.6 also gives strong evidence that the flow (1.1) shall converge to a cscK metric, if there is one in $\{\omega\}$ (cf. [33, Conjecture 7.4] and [15, Question 8.5]). In view of Tian’s properness conjecture, Theorem 1.6 also shows that the properness of K-energy (modulo group actions, in the sense of Definition 3.13) implies that one can find a cscK metrics using the dynamical system (1.2).

Compared to the recent work of Darvas–Rubinstein [19] in the Fano case, the main difficulty we are faced with is the lack of Ding functional in our general setting. As we shall see, this technicality can be circumvented with the help of the estimates in [13, §3], which are needed for the smooth convergence in Theorem 1.6.

For the direction of Ricci iteration in the real case, we refer the reader to [32, 25, 9]. See also [22] for a Ricci iteration in the local setting.

Organization. After recalling some standard notions and facts in §2 and §3, we prove Theorem 1.4 and Theorem 1.5 in §4. Relying on [13, §3], we will derive some a priori estimates for the Ricci iteration in §5, which allows us to prove Theorem 1.6 in §6. Finally in §7 we point out that our work can be extended to the setting of twisted cscK metrics.

Acknowledgements. It is the author’s great pleasure to dedicate this paper to Prof. Tian, on the occasion of his 65th birthday, for his constant guidance over the years. The author is also grateful to T. Darvas, W. Jian, Y. Rubinstein and Y. Shi for helpful discussions and comments. Part of this work was done during the pleasant and inspiring visit at the Tianyuan Mathematics Research center in Oct. 2023.

The author is supported by NSFC grants 12101052, 12271040, and 12271038.

We recall several standard functionals that will be used throughout this paper.

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and set

$$\mathcal{H}_{\omega}:=\{\varphi\in C^{\infty}(X,\mathbb{R})|\omega_{\varphi}:=\omega+\sqrt{-1}\partial\bar{\partial}\varphi>0\}.$$

Put $V:=\int_{X}\omega^{n}.$ And let

$$\operatorname{Ric}(\omega_{\varphi}):=-\sqrt{-1}\partial\bar{\partial}\log\det\omega_{\varphi}\in 2\pi c_{1}(X),\ R(\omega_{\varphi}):=\mathrm{tr}_{\omega_{\varphi}}\operatorname{Ric}(\omega_{\varphi}),$$

$$\bar{R}:=\frac{1}{V}\int_{X}R(\omega)\omega^{n}=\frac{n}{V}\int_{X}\operatorname{Ric}(\omega)\wedge\omega^{n-1}=2\pi n\frac{c_{1}(X)\cdot\{\omega\}^{n-1}}{\{\omega\}^{n}}.$$

For any $u,v\in\mathcal{H}_{\omega}$, define

$$I(u,v)=I(\omega_{u},\omega_{v}):=\frac{1}{V}\int_{X}(v-u)(\omega^{n}_{u}-\omega^{n}_{v}).$$

$$E(u,v):=\frac{1}{(n+1)V}\int_{X}(v-u)\sum_{i=0}^{n}\omega^{i}_{u}\wedge\omega^{n-i}_{v}.$$

$$J(u,v)=J(\omega_{u},\omega_{v}):=\frac{1}{V}\int_{X}(v-u)\omega^{n}_{u}-E(u,v).$$

$$Ent(u,v)=Ent(\omega_{u},\omega_{v}):=\frac{1}{V}\int_{X}\log\frac{\omega^{n}_{v}}{\omega^{n}_{u}}\omega^{n}_{v}.$$

Note that by Jensen’s inequality, it always holds that $Ent(u,v)\geq 0$.

For any closed $(1,1)$ form $\chi$, define

$$\mathcal{J}^{\chi}(u,v):=\frac{1}{V}\int_{X}(v-u)\chi\wedge\sum_{i=0}^{n-1}\omega_{u}^{i}\wedge\omega^{n-1-i}_{v}-\bar{\chi}E(u,v),$$

where

$$\bar{\chi}:=\frac{n}{V}\int_{X}\chi\wedge\omega^{n-1}=n\frac{\{\chi\}\cdot\{\omega\}^{n-1}}{\{\omega\}^{n}}.$$

The K-energy is defined by

$$K(u,v)=K(\omega_{u},\omega_{v}):=Ent(u,v)+\mathcal{J}^{-\operatorname{Ric}(\omega_{u})}(u,v).$$

The $\chi$-twisted K-energy is

$$K^{\chi}(u,v)=K^{\chi}(\omega_{u},\omega_{v}):=K(u,v)+\mathcal{J}^{\chi}(u,v).$$

If we choose $\chi:=\omega_{u}$, then integration by parts gives

(2.1)

$$\mathcal{J}^{\omega_{u}}(u,v)=(I-J)(u,v).$$

More generally, if $\chi:=\omega_{w}$ for some $w\in\mathcal{H}_{\omega}$, then

$$\mathcal{J}^{\omega_{w}}(u,v)=(I-J)(u,v)+\frac{1}{V}\int_{X}(w-u)(\omega^{n}_{v}-\omega^{n}_{u}),$$

so in particular we have that

(2.2)

$$\mathcal{J}^{\omega_{w}}(u,w)=-J(u,w).$$

One has the following variation formulas (for any $u,v\in\mathcal{H}_{\omega}$ and $f\in C^{\infty}(X,\mathbb{R})$):

(2.3)

$$\begin{cases}\frac{d}{dt}\big{|}_{t=0}E(u,v+tf)=\frac{1}{V}\int_{X}f\omega^{n}_{v},\\ \frac{d}{dt}\big{|}_{t=0}\mathcal{J}^{\chi}(u,v+tf)=\frac{1}{V}\int_{X}f(\mathrm{tr}_{\omega_{v}}\chi-\bar{\chi})\omega^{n}_{v},\\ \frac{d}{dt}\big{|}_{t=0}K^{\chi}(u,v+tf)=\frac{1}{V}\int_{X}f(\bar{R}-\bar{\chi}+\mathrm{tr}_{\omega_{v}}\chi-R(\omega_{v}))\omega^{n}_{v}.\\ \end{cases}$$

They imply the well known cocycle relations (for $u,v,w\in\mathcal{H}_{\omega}$):

$$E(u,v)+E(v,w)=E(u,w).$$

$$\mathcal{J}^{\chi}(u,v)+\mathcal{J}^{\chi}(v,w)=\mathcal{J}^{\chi}(u,w).$$

$$K^{\chi}(u,v)+K^{\chi}(v,w)=K^{\chi}(u,w).$$

One can then further deduce the following cocycle relations:

$$J(u,v)+J(v,w)=J(u,w)+\frac{1}{V}\int_{X}(v-w)(\omega^{n}_{u}-\omega^{n}_{v}).$$

$$(I-J)(u,v)+(I-J)(v,w)=(I-J)(u,w)+\frac{1}{V}\int_{X}(v-u)(\omega^{n}_{w}-\omega^{n}_{v}).$$

The following result proved in Tian’s work [40] will be used repeatedly.

Convention. Given an energy functional $F\in\{I,J,\mathcal{J}^{\chi},K,K^{\chi}\}$ and $u\in\mathcal{H}_{\omega}$, we also use the notation

$$F_{\omega}(u)=F_{\omega}(\omega_{u}):=F(0,u)\text{ and }E_{\omega}(u):=E(0,u)$$

in the circumstances where $\omega$ is viewed as a background metric.

We will work with the finite energy space $\mathcal{E}_{\omega}^{1}$ introduced in [23] and use the $d_{1}$-distance on it introduced by Darvas [18]. They provide useful tools for proving our main result concerning convergence of the Ricci iteration. We briefly recall the machinery, referring to [17] and references therein for more details.

Let

$$\mathrm{PSH}_{\omega}:=\{\varphi\in L^{1}(\omega^{n}):\varphi\text{ is upper semi-continuous and }\omega_{\varphi}=\omega+\sqrt{-1}\partial\bar{\partial}\varphi\geq 0\}.$$

Following Guedj–Zeriahi [23, Definition 1.1] we define the subset of full mass potentials:

$$\mathcal{E}_{\omega}:=\big{\{}\varphi\in\mathrm{PSH}_{\omega}:\lim_{j\to-\infty}\int_{\{\varphi\leq j\}}(\omega+\sqrt{-1}\partial\bar{\partial}\max\{\varphi,j\})^{n}=0\big{\}}.$$

For each $\varphi\in\mathcal{E}_{\omega}$, define $\omega^{n}_{\varphi}:=\lim_{j\to-\infty}\textbf{1}_{\{\varphi>j\}}(\omega+\sqrt{-1}\partial\bar{\partial}\max\{\varphi,j\})^{n}$. The measure $(\omega+\sqrt{-1}\partial\bar{\partial}\max\{\varphi,j\})^{n}$ is defined by the work of Beford–Taylor [3] since $\max\{\varphi,j\}$ is bounded. Consequently, $\varphi\in\mathcal{E}_{\omega}$ if and only if $\int_{X}\omega^{n}_{\varphi}=\int_{X}\omega^{n}$, justifying the name of $\mathcal{E}_{\omega}$.

Next, define a further subset, the space of finite $1$-energy potentials:

$$\mathcal{E}_{\omega}^{1}:=\big{\{}\varphi\in\mathcal{E}_{\omega}:\int_{X}|\varphi|\omega^{n}_{\varphi}<\infty\big{\}}.$$

Consider the following weak Finsler metric on $\mathcal{H}_{\omega}$ [18]:

$$||\xi||_{\varphi}:=\frac{1}{V}\int_{X}|\xi|\omega^{n}_{\varphi},\ \xi\in T_{\varphi}\mathcal{H}_{\omega}=C^{\infty}(X,\mathbb{R}).$$

We denote by $d_{1}$ the associated pseudo-metric and recall the following result characterizing the $d_{1}$-metric completion of $\mathcal{H}_{\omega}$ [18, Theorem 3.5]:

Let us now recall several properties of the $d_{1}$ metric that will be used in this paper.

One can extend the functionals $Ent_{\omega},I_{\omega},J_{\omega},E_{\omega}$ and $\mathcal{J}^{\chi}_{\omega}$ to the space $\mathcal{E}^{1}_{\omega}$.

We need the following compactness result going back to [5] (see [20, Theorem 5.6] for a convenient formulation for our context).

For any $u,v\in\mathcal{E}^{1}_{\omega}$, one can also define

$$I(u,v):=\frac{1}{V}\int_{X}(v-u)(\omega^{n}_{u}-\omega^{n}_{v}),$$

$$E(u,v):=E_{\omega}(v)-E_{\omega}(u),$$

and

$$J(u,v):=\frac{1}{V}\int_{X}(v-u)\omega^{n}_{u}-E(u,v).$$

Recall that (see [17, Proposition 3.40])

(3.1)

$$|E(u,v)|\leq d_{1}(u,v).$$

The next quasi-triangle inquality is proved in [5, Theorem 1.8].

We will need the following convergence criterion, which is a simple consequence of [5, Proposition 2.3].

We will frequently use the space

$$\mathcal{H}_{0}:=\{\varphi\in\mathcal{H}_{\omega}|E_{\omega}(\varphi)=0\}.$$

Recall that,

$$G:=Aut_{0}(X),$$

the connected component of complex Lie group of holomorphic automorphisms of $X$, naturally acts on $\mathcal{H}_{0}$.

Finally, we recall that $G$ acts on $\mathcal{H}_{0}$ isometrically.

Then define (as in [20])

$$d_{1,G}(u,v):=\inf_{f\in G}d_{1}(u,f.v).$$

In this part, we prove Theorem 1.4 and Theorem 1.5.

We begin by introducing the following analytic threshold.

(4.1)

$$\gamma(X,\{\omega\}):=\sup\bigg{\{}\gamma\in\mathbb{R}\bigg{|}\inf_{\varphi\in\mathcal{H}_{\omega}}(K_{\omega}(\varphi)-\gamma(I_{\omega}-J_{\omega})(\varphi))>-\infty\bigg{\}},$$

In this part we derive some a priori estimates for the iteration sequence (1.2). By Theorem 1.4 we can find some $\tau>0$ so that the iteration carries on forever. Up to scaling the Kähler class, we assume without loss of generality that $\tau=1$. Taking trace of (1.2) we then have

$$R(\omega_{i+1})=\bar{R}-n+\mathrm{tr}_{\omega_{i+1}}\omega_{i},\ \omega_{0}=\omega.$$

Write

$$\omega_{i}=\omega+\sqrt{-1}\partial\bar{\partial}u_{i},u_{i}\in\mathcal{H}_{0}.$$

Also, let $F_{i}\in C^{\infty}(X,\mathbb{R})$ be such that

(5.1)

$$(\omega+\sqrt{-1}\partial\bar{\partial}u_{i})^{n}=e^{F_{i}}\omega^{n}.$$

Then

(5.2)

$$\Delta_{\omega_{i}}F_{i}=\mathrm{tr}_{\omega_{i}}(\operatorname{Ric}(\omega)-\omega_{i-1})+n-\bar{R}.$$

In other words,

$$\Delta_{\omega_{i}}(F_{i}+u_{i-1})=\mathrm{tr}_{\omega_{i}}(\operatorname{Ric}(\omega)-\omega)+n-\bar{R}.$$

Therefore, we are in the situation considered in [13, §3].