This is the first of a series of three papers in the study of of constant scalar curvature Kähler metrics (cscK metrics), following a program outlined in [9]. In this paper, we focus on establishing a priori estimates for cscK metrics in compact Kähler manifold without boundary. Our estimates can be easily adapted to extremal Kähler metrics and for simplicity of presentations, we leave such an extension to the interested readers except to note that for extremal Kähler metrics, its scalar curvature is a priori bounded depending on Kähler class. In the subsequent two papers, we will use these estimates (and its generalizations) to study the Calabi-Donaldson theory on the geometry of extremal Kähler metrics, in particular, to establish the celebrated conjecture of Donaldson on geodesic stability as we as the well known properness conjecture relating the existence of cscK metrics with the properness of K energy functional.

In [9], the first named author advocates a new continuity path which links the cscK equation to certain second order elliptic equation, apparently inspired by the success of the classical continuity path for Kähler Einstein (KE) metrics and Donaldson’s continuity path for conical KE metrics. In general, apriori estimates are usually the prelude to the success of any continuity path aiming to obtain existence results of cscK metrics since openness is already established in [9].

Let us recall a conjecture made earlier by the first named author (c.f. [14]).

It is worthwhile to give a brief review of the history of this subject and hopefully, this will make it self-evident why this conjecture is interesting. A special case of constant scalar curvature Kähler metric is the well known KE metric which has been the main focus of Kähler geometry since the inception of the celebrated Calabi conjecture on Kähler Einstein metrics in 1950s. In 1958, E. Calabi published the fundamental $C^{3}$ estimate for Monge-Amp$\grave{\text{e}}$re equation [3] which later played a crucial role in Yau’s seminal resolution of Calabi conjecture [35] in 1976 when the first Chern class is either negative or zero (In negative case, T. Aubin has an independent proof) . This work of Yau is so influential that generations of experts in Kähler geometry afterwards largely followed the same route: Securing a $C^{0}$ estimate first, then move on to obtain $C^{2},\;C^{3}$ estimates etc. In the case of positive first Chern class, G. Tian proved Calabi conjecture in 1989 [32] for Fano surfaces when the automorphism group is reductive. It is well known that there are obstructions to the existence of KE metrics in Fano manifolds; around 1980s, Yau proposed a conjecture which relates the existence of Kähler Einstein metrics to the stability of underlying tangent bundles. This conjecture was settled in 2012 through a series of work CDS [11] [12] [13] and we refer interested readers to this set of papers for further references in the subject of KE metrics. The proof of CDS’s work is itself quite involved as it sits at the intersection of several different subjects: algebraic geometry, several complex variables, geometry analysis and metric differential geometry etc.

To move beyond CDS’s work on Kähler Einstein metrics, one direction is the study of the existence problem of cscK metrics which satisfy a 4th order PDE. The following is a conjecture which is a refinement of Calabi’s original idea that every Kähler class must have its own best, canonical representatives.

One conspicuous and memorable feature of CDS’s proof is the heavy use of Cheeger-Colding theory on manifold with Ricci curvature bounded from below. The apriori bound on Ricci curvature for KE metrics make such an application of Cheeger Colding theory seamlessly smooth and effective. However, if we want to attack this general conjecture, there will be a dauntingly high wall to climb since there is no a priori bound on Ricci curvatue. Therefore, the entire Cheeger-Colding theory needs to be re-developed if it is at all feasible. On the other hand, there is a second, less visible but perhaps even more significant feature of CDS’s proof is: The whole proof is designed for constant scalar curvature Kähler metrics and the use of algebraic criteria and Cheeger Colding theory is to conclude that the a $C^{0}$ bound holds for Kähler potential so that we can apply the apriori estimates for complex KE metrics developed by Calabi, Yau and others. Indeed, this is exactly how we make use of Cheeger Colding theory and stability condition in CDS’s proof to nail down a $C^{0}$ estimate on potential. Unfortunately, such an estimate is missing in this generality for a 4th order fully nonlinear equation. Indeed, as noted by other famous authors in the subject as well, the difficulty permeates the cscK theory are two folds: one cannot use maximal principle from PDE point of view and one can not have much control of metric from the bound of the scalar curvature.

In this paper, we want to tackle this challenge and we prove

As a consequence, we show that

The cscK metric equation can be re-written as a pair of coupled equations

Here $\Delta_{\varphi}$ denotes the Laplace operator defined by the Kähler form $\omega_{\varphi}:$

The following proposition might be well known to experts (c.f. [9]).

Following [9], Proposition 2.1, we outline some key arguments for this proposition: since $g_{\varphi}$ is quasi-isometric, then Equation (1.2) is uniformly elliptic with a bounded right hand side. Therefore, by De Giorgi-Nash-Moser theory([21], Theorem 8.22), $[F]_{C^{\alpha}(M,g)}$ is uniformly bounded for some $0<\alpha<1$. Substituting this into Equation (1.1), it becomes a complex Monge-Amp$\grave{\text{e}}$re equation with $C^{\alpha}$ bound on the right hand side. Following theory of Caffarelli, Evans-Krylov(see [34] for details on extension to complex setting), we know $[\varphi]_{C^{2,\alpha^{\prime}}(M,g)}$ is uniformly bounded, for each $\alpha^{\prime}<\alpha$. This means (1.2) is uniformly elliptic with coefficients in $C^{\alpha^{\prime}}$. Hence we may apply Schauder theory([21], Theorem 6.2) to conclude an estimate for $||F||_{2,\alpha^{\prime}}$. Now we can go back to (1.1). Differentiating the equation, we can conlude $\varphi$ is bounded in $C^{4,\alpha^{\prime}}$. Hence we may bootstrap this way and get estimates for all higher derivatives.

In this short argument, it is obvious that the crucial assumption is that the metric in question is quasi-isometric. The hard challenge is to prove a priori that the metric $\omega_{\varphi}$ is quasic isometric. However, there is not much room for improvement at least locally, following the well known example of Pogorelov on real Monge-Amp$\grave{\text{e}}$re equation. In [23], W.Y. He adapted the construction of Pogorelov’s example to complex setting and obtained a complete solution to

in $\mathbb{C}^{n}$ which is not $C^{2}.\;$ Thus, for this conjecture to be true, the global nature of compact Kähler manifold must come into play in a crucial way.

Theorem 1.1 can be expanded into a more detailed version. The constants $C$ in the theorem below can change from line to line. More generally, throughout this paper, the “C” without subscript may change from line to line, while if there is subscript, then it is some fixed constant.

Some remarks are in order:

1. (1)

   The strength of statement is roughly in increasing order. The equivalence of (1) and (6) gives Theorem 1.1.

2. (2)

   From (5) to (6), this is exactly Proposition 1.1, since this assumption implies $\frac{1}{C^{\prime}}\omega_{0}\leq\omega_{\varphi}\leq C^{\prime}\omega_{0}$. All other estimates are new.

3. (3)

   Here is the flow line of our proof:

   $$(1)\stackrel{{\scriptstyle section\,5}}{{\Longrightarrow}}(2)+(4)\;\;{\rm and}\;\;(3)\stackrel{{\scriptstyle section\,2}}{{\Longleftrightarrow}}(4)\stackrel{{\scriptstyle section\,3}}{{\Longrightarrow}}(5)\stackrel{{\scriptstyle section\,4}}{{\Longrightarrow}}(6).$$

Now we present technical theorems which lead to this main theorem. Indeed, these technical theorems are interesting in its own right and may be used in other applications.

We also show that one can estimate the upper bound of $F$ directly in terms of gradient bound of $\varphi$. This result is not directly needed for our main result, but can be of independent interest.

For second order estimate, Chen-He[14] establish an a priori bound on $n+\Delta\varphi$ in terms of $|\nabla F|_{L^{p}}(p>2n)$ via integral estimate, in absense of (1.2). Inspired by this paper [14] and utilizing the additional equation (1.2), we are able to obtain a $W^{2,p}$ estimate for any $p>0$, using only $||F||_{0}$. Theorem 1.5 is used essentially in this estimate.

If we can prove an upper bound for $F$, then the following theorem becomes very interesting.

It is interesting to compare this result with second derivative estimates for complex Monge-Amp$\grave{e}$re equations. In [26], the authors obtained $C^{2,\alpha}$ estimates for complex Monge-Amp$\grave{e}$re equation, depending on $C^{1,\beta}$ bound of the solution (with $\beta$ close enough to 1) and $C^{\alpha}$ bound of the right hand side. In [23], the authors obtained $W^{3,p}$ bound of solution to complex Monge-Amp$\grave{e}$re depending only on $C^{0}$ bound of the solution and $W^{1,p}$ bound of right hand side for $p>2n$. In this result, we are not assuming any regularity of the right hand side $e^{F}$, but assumes quite strong bound ($W^{2,p}$ for $p$ large) as a price to pay, and the second equation (6.2) needs to be used in an essential way.

Theorem 1.8 is reminiscent to a renowned problem in $\mathbb{C}^{n}$ which goes back to S. T. Yau, E. Calabi: whether global solution of Calabi Yau metric in $\mathbb{C}^{n}$ must be Euclidean metric or not? This problem is disapproved by a nontrivial construction of Calabi Yau metric in $\mathbb{C}^{2}$ by C. LeBrun. Perhaps one need to strengthen the assumption by assuming it is asymptotically Euclidean at $\infty.\;$ This is made known to be true by G. Tian in dimension $n=2\>$ and conjectured to be true in all dimensions. While we prepare this paper, it is now known through a surprising result of Y. Li in dimension 3 [27] and then Conlon-Rochon [16], G. Szekelyhidi [31] in all dimensions that this fails in general. This exciting new development makes statement like Corollary 1.5 below more interesting. This corollary offers a different point of view: If we control asymptotical growth of the underlying metrics, then the rigidity result still hold for scalar flat Kähler metrics (in particular Calabi Yau metrics) in $\mathbb{C}^{n}.\;$

We will prove this corollary in section 6, using a similar argument as Proposition 6.1 We observe that this theorem covers the well-known Calabi Yau metric equation

as a special case. One interesting question is, what is the smallest number $p$ for which this corollary still holds? In section 6, we also show that when $n=2$, for a solution $\varphi$ of cscK in a domain of $\mathbb{C}^{n}$, if $|\nabla\varphi|$ is locally bounded, then the volume ratio ${\omega_{\varphi}^{n}\over\omega^{n}}$ is also bounded from above locally. It is not clear to us if this estimate can be generalized to higher dimensions.

Finally we would like to explain the organization of this paper:

In section 2, we prove Proposition 1.4, Theorem 1.5 and Theorem 1.6.

In section 3, we prove Theorem 1.7 by iteration, which is a crucial step towards the main result.

In section 4, we use iteration again to improve $L^{p}$ bound of $n+\Delta\varphi$ for $p<\infty$ to an $L^{\infty}$ bound of $n+\Delta\varphi$, proving Theorem 1.8. This estimate requires a bound for $||n+\Delta\varphi||_{L^{p}}$ for some $p>p_{n}$ ¹¹ 1 Here $p_{n}\leq(3n-3)(4n+1)$. But this most likely is not sharp. , depending on $||F||_{0}$. The key ingredient is a calculation for $\Delta_{\varphi}(|\nabla_{\varphi}F|^{2})$. Combining the results in section 2, 3, 4 as well as Proposition 2.1 gives estimate for all higher derivatives in terms of $||\varphi||_{0}$ and $||F||_{0}$.

In Section 2-4, we always assume $|\varphi|$ is a priori bounded. This assumption is removed in Section 5 where we prove Theorem 1.3 and Theorem 1.4. From these two results, we get estimate for $||F||_{0}$ and $||\varphi||_{0}$ depending only on entropy bound of $F$. The key ingredient is the use of $\alpha$-invariant and the construction of a new test function. On the other hand, if we start with a bound for $||\varphi||_{0}$, and use the convexity of $K$-energy along $C^{1,1}$ geodesics, it is relatively easy to get an entropy bound of $F$, hence all higher estimates.

In section 6, we obtain some interior estimates for cscK in a bounded domain of $\mathbb{C}^{n}$. Such estimates are not directly needed for our main results but may be of independent interest.

Acknowledgment In the Fall of 1997, Sir Simon invited the first named author to join him in exploring the space of Kähler potentials, and Sir Simon has been remarkably generous about sharing his time and ideas ever since. It is therefore a deep pleasure to dedicate this paper to Sir Simon Donaldson, in celebration of his 60th birthday, and in acknowledgment of the far-reaching influence of his profound mathematical ideas, which have changed the landscape of mathematics so much.

The main theorem in this section is to prove that the first derivative of $\varphi$ is pointwisely controlled by volume ratio $e^{F}$ from above, assuming a bound for $||\varphi||_{0}$. Conversely, the bound for $||\nabla\varphi||_{0}$ can in turn control $e^{F}.\;$ However, this control is much weaker since it is of global nature.

First we show that a $C^{0}$ bound for $\varphi$ implies a lower bound for $F$.

This proposition first appeared in [24]. However, for the convenience of the reader, we include a proof here.

Next we move on to estimate the upper bound of $F$ in terms of $\displaystyle\max_{M}|\nabla\varphi|$.

Conversely, we have the following key estimate, which will be needed when we do the $W^{2,p}$ estimates of $\varphi$.