Regularity Scales and Convergence of the Calabi Flow
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Abstract
We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kähler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.
Keywords
Calabi flow Backward regularity improvement Kähler geometry Donaldson conjectureMathematics Subject Classification
53C441 Introduction
Conjecture 1.1
 1.
The flow converges to a cscK metric on the same complex manifold (M, J).
 2.
The flow is asymptotic to a oneparameter family of extK metrics on the same complex manifold (M, J), evolving by diffeomorphisms.
 3.
The manifold does not admit an extK metric but the transformed flow \(J_t\) on \(\mathcal {J}\) converges to \(J'\). Furthermore, one can construct a destabilizing test configuration of (M, J) such that \((M, J')\) is the central fiber.
 4.
The transformed flow \(J_t\) on \(\mathcal {J}\) does not converge in smooth topology and singularities develop. However, one can still make sufficient sense of the limit of \(J_t\) to extract a scheme from it, and this scheme can be fitted in as the central fiber of a destabilizing test configuration.
Conjecture 1.1 has attracted a lot of attentions for the study of the Calabi flow. On the way to understand it, there are many important works. For example, Berman [3], He [36], and Streets [45] proved the convergence of the Calabi flow in various topologies, under different geometric conditions. Székelyhidi [48] constructed examples of global solutions of the Calabi flow which collapse at time infinity. A finitedimensional approximation approach to study the Calabi flow was developed in [32] by Fine.
Note that the global existence of the Calabi flows is a fundamental assumption in Conjecture 1.1. On Riemann surfaces, the global existence and the convergence of the Calabi flow have been proved by Chrusical [26], Chen [12], and Struwe [47]. However, much less is known in high dimension. It was conjectured by Chen [13] that every Calabi flow has global existence. This conjecture sounds to be too optimistic at the beginning. However, there are positive evidences for it. In [44], J. Streets proved the global existence of the minimizing movement flow, which can be regarded as weak Calabi flow solutions. Therefore, the global existence of the Calabi flow can be proved if one can fully improve the regularity of the minimizing movement flow, although there exist terrific analytic difficulties to achieve this. In general, Chen’s conjecture was only confirmed in particular cases. For example, if the underlying manifold is an Abelian surface and the initial metric is Tinvariant, Huang and Feng proved the global existence in [37].
Theorem 1.2
In the setup of Conjecture 1.1, we have \(\mathcal {LH}=\widetilde{\mathcal {LH}}=\mathcal {H}\) automatically. Therefore, Theorem 1.2 confirms the first two possibilities of Conjecture 1.1 in complex dimension 2.
Theorem 1.3
 There is a smooth family of diffeomorphisms \(\{\psi _s: s \in D \backslash \{0\} \}\) such that$$\begin{aligned}{}[\psi _s^* \omega ] = [\omega ], \quad \psi _s^* J_{s}= J_{1}, \quad \psi _1=Id. \end{aligned}$$

\([\omega ]\) is integral.

\((M^2, \omega , J_0)\) is a cscK surface.
Theorem 1.3 partially confirms the third possibility of Conjecture 1.1 in complex dimension 2, in the case that the \(C^{\infty }\)closure of the \(\mathcal {G}^{\mathbb {C}}\)leaf of \(J_1\) contains a cscK complex structure, for a polarized Kähler surface. Note that by the integral condition of \([\omega ]\) and reductivity of the automorphism groups of cscK complex manifolds, the construction of destabilizing test configurations follows from [29] directly.
Theorems 1.2 and 1.3 have high dimensional counterparts. However, in high dimension, due to the loss of scaling invariant property of the Calabi energy, we need some extra assumptions of scalar curvature to guarantee the convergence.
Theorem 1.4
Suppose \((M^n, \omega , J)\) is a compact extremal Kähler manifold.
Theorem 1.5
 There is a smooth family of diffeomorphisms \(\{\psi _s: s \in D \backslash \{0\} \}\) such that$$\begin{aligned}{}[\psi _s^* \omega ] = [\omega ], \quad \psi _s^* J_{s}= J_{1}, \quad \psi _1=Id. \end{aligned}$$

\([\omega ]\) is integral.

\((M^n, \omega , J_0)\) is a cscK manifold.
It is interesting to compare the Calabi flow and the Kähler Ricci flow on Fano manifolds at the current stage. For simplicity, we fix \([\omega ]=2\pi c_1(M, J)\). Modulo the pioneering work of H.D. Cao([9], global existence) and G. Perelman([43], scalar curvature bound), Theorems 1.4 and 1.5 basically says that the convergence of the Calabi flow can be as good as that for the Kähler Ricci flow on Fano manifolds, whenever some critical metrics are assumed to exist, in a broader sense. The Kähler Ricci flow version of Theorems 1.4 and 1.5 has been studied by Tian and Zhu in [51] and [52], based on Perelman’s fundamental estimate. A more general approach was developed by Székelyhidi and Collins in [27]. Our proof of Theorems 1.4 and 1.5 uses a general continuity method, see for example, TianZhu’s work [52] in the setting of Kähler Ricci flow. However, the continuity method does not work without regularity improvement properties. Therefore, it becomes a key step to obtain such regularity improvement properties, which is one of our major contributions in this paper. We prove Theorems 2.22 and 2.23 for the Calabi flow as the regularity improvement properties.
If the flows develop singularity at time infinity, then the behavior of the Calabi flow and the Kähler Ricci flow seems much different. Based on the fundamental work of Perelman, we know collapsing does not happen along the Kähler Ricci flow. In [23] and [24], it was proved by Chen and the second author that the Kähler Ricci flow will converge to a Kähler Ricci soliton flow on a QFano variety. A different approach was proposed in complex dimension 3 in [50], by Tian and Zhang. However, under the Calabi flow, Székelyhidi [48] has shown that collapsing may happen at time infinity, by constructing examples of global solutions of the Calabi flow on ruled surfaces. In this sense, the Calabi flow is much more complicated. Of course, this is not surprising since we do not specify the underlying Kähler class. A more fair comparison should be between the Calabi flow and the Kähler Ricci flow, in the same class \(2\pi c_1(M, J)\), of a given Fano manifold. However, few is known about the Calabi flow in this respect, except the underlying manifold is a toric Fano surface (c.f. [17]).
Theorem 1.6
Theorem 1.6 is nothing but the Calabi flow counterpart of the main theorems in [53] and [22]. The tools we used in the proof of Theorem 1.6 are motivated by the study of the analogue question of the Ricci flow by the second author in [22] and [53]. Actually, the methods in [53] and [22] were built in a quite general frame. It was expected to have its advantage in the study of the general geometric flows.
The paper is organized as follows. In Sect. 2, we develop two concepts—curvature scale and harmonic scale— to study geometric flows. Based on the analysis of these two scales under the Calabi flow, we show global backward regularity improvement estimates. In Sect. 3, we combine the regularity improvement estimates, the excellent behavior of the Calabi functional along the Calabi flow and the deformation techniques to prove Theorems 1.2–1.5. Moreover, we give some examples where Theorems 1.2–1.5 can be applied. In Sect. 4 we show Theorem 1.6 and in Sect. 5 we discuss some further research directions of Calabi flow.
2 Regularity Scales
2.1 Preliminaries
2.2 Estimates Based on Curvature Bound
The global high order regularity estimate of the Calabi flow was studied in [17], when Riemannian curvature and Sobolev constant are bounded uniformly. Taking advantage of the localization technique developed in [34] and [46], one can localize the estimate in [17].
Lemma 2.1
Proof
This follows from the same argument as Theorem 4.4 of [46] and the Sobolev embedding theorem. \(\square \)
Lemma 2.2
 (1)
\(\Phi \) is a local biholomorphic map from \({\hat{B}}(0, r_1)\) to its image.
 (2)
\(\Phi (0)=p\).
 (3)
\(\Phi ^*(g)(0)=g_{E}\), where \(g_E\) is the standard metric on \({\mathbb C}^n\).
 (4)
\(r_2^{1}g_E\le \Phi ^*g\le r_2 g_E\) in \({\hat{B}}(0, r_1)\).
Proof
This is only an application of Proposition 1.2 of TianYau [49]. Similar application can be found in [10]. \(\square \)
Theorem 2.3
Proof
By equation (2.1), we see that (2.6) follows from (2.5). We shall only prove (2.5).

\(\displaystyle \lim _{i\rightarrow +\infty } \sup _{M_i \times [1,0]} \mathrm{Rm}_{\tilde{g}_i}=0\).

\(\displaystyle \sup _{M_i \times [1,0]} f_l(\cdot , \cdot , \tilde{g}_i(t)) \le 1\).

\(\displaystyle f_l(x_i, 0, \tilde{g}_i)=1\).
The proof of Theorem 2.3 follows the same line as that in [46] by J. Streets, we do not claim the originality of the result. We include the proof here for the convenience of the readers and to show the application of the local biholomorphic map \(\Phi \), which will be repeatedly used in the remainder part of this subsection. Actually, by delicately using interpolation inequalities, the constants in Theorem 2.3 can be made explicit.
Note that for a given Calabi flow, \(S \equiv 0\) implies that all the high derivatives of S vanish. Therefore, for the Calabi flows with uniformly bounded Riemannian curvature and very small scalar curvature S, it is expected that the high derivatives of S are very small. In the remainder part of this subsection, we will justify this observation. Similar estimates for the Ricci flow were given by Theorem 3.2 of [53] and Lemma 2.1 of [22] by the parabolic Moser iteration. However, since the Calabi flow is a fourthorder parabolic equation, the parabolic Moser iteration in the case of the Ricci flow does not work any more. Here we use a different method to overcome this difficulty.
To estimate the higher derivatives of the curvature, we need the interpolation inequalities of Hamilton in [35]:
Lemma 2.4
Combining Lemma 2.4 with Sobolev embedding theorem, we have the following result.
Lemma 2.5
Proof
We next show some local estimates on the derivatives of the scalar curvature.
Lemma 2.6
Proof
The following proposition is a weak version of the corresponding result for the Ricci flow in [22, 53].
Proposition 2.7
Proof
2.3 From Metric Equivalence to Curvature Bound
If we regard curvature as the 4th order derivative of Kähler potential function, then Theorem 2.3 can be roughly understood as from \(C^4\)estimate to \(C^l\)estimate, for each \(l \ge 5\). In this subsection, we shall set up the estimate from \(C^2\) to \(C^4\) for the Calabi flow family.
Lemma 2.8

\(Q_g(0)=1\) and \(Q_g(t)\le 2\) for any \(t\in [1, 0]\).

\(Q_h(t)\le 2\) for any \(t\in [1, 0]\).

\(e^{\epsilon } g(0)\le h(0)\le e^{\epsilon }g(0)\).
Proof
 (1)
\(Q_{g_i}(0)=1\) and \(Q_{g_i}(t)\le 2\) for any \(t\in [1, 0]\).
 (2)
\(Q_{h_i}(t)\le 2\) for any \(t\in [1, 0]\), and \(\log Q_{h_i}(0)\ge \delta _0\).
 (3)
\(e^{\epsilon _i} g_i(0)\le h_i(0)\le e^{\epsilon _i}g_i(0)\).
As a direct corollary, we have the next result.
Lemma 2.9
There exists a constant \(\epsilon _0=\epsilon _0(n)\) with the following properties.
Proof
2.4 Curvature Scale
Inspired by Theorem 2.3, we define a concept “curvature scale,” to study improving regularity property of the Calabi flow.
Definition 2.10
Lemma 2.11
Proof
By the definition of curvature scale, there exists time \(t \in [1, 0]\) such that \(Q_g(t)=1\) and we denote by \(t_0\) the maximal time \(t\in [1, 0]\) with this property. There are several cases to consider:
Lemma 2.12
Proof
A direct corollary of the above results is
Lemma 2.13
Proof
Lemma 2.14
 (1)
\(Q(0)=1\), where \(Q=Q_g\);
 (2)
\(Q(t) \le 2\) for all \(t \in [1,0]\);
 (3)
\(\log Q(t) \le \log 2\) for every \(t \in [0,K]\) and \(\log Q(K)=\log 2\).
Proof
The next result shows that under the scalar curvature conditions, if the curvature at some time is large enough, then the curvature before that can be controlled.
Proposition 2.15
 (1)
the scalar curvature \(S(t)\le 1\) for every \(t\in [1, 0]\);
 (2)Q(0) is big enough, i.e.,where \(\epsilon \) and A are constants given by Lemma 2.14, Q is \(Q_g\).$$\begin{aligned} Q(0)> \max \left\{ 2^{\frac{3\alpha }{\alpha }} A^{\frac{1}{\alpha }}, 2^{\frac{2(1\alpha )}{\alpha }} \left( \frac{A}{\epsilon }\right) ^{\frac{1}{\alpha }} \right\} , \end{aligned}$$
Proof
Claim 2.16
Proof
Case 1. \(\displaystyle \sup _{s \in [t_0Q(t_0)^{2}, t_0]} Q(s) = 2 Q(t_0)\).
Case 2. \(\displaystyle Q(t_0)=\frac{2}{\sqrt{Q(0)^{2}+t_0}}\).
We will show that both cases will never happen if \(t_0 <\bar{t}\).
Using Proposition 2.15, we can estimate the curvature scale when the curvature at some time is not large.
Proposition 2.17

the scalar curvature \(S(t) \le 1\) for all \(t\in [2, 0]\).
 the curvature tensor satisfies$$\begin{aligned} Q(0) \le \max \left\{ 2^{\frac{3\alpha }{\alpha }} A^{\frac{1}{\alpha }}, 2^{\frac{2(1\alpha )}{\alpha }} \left( \frac{A}{\epsilon }\right) ^{\frac{1}{\alpha }} \right\} . \end{aligned}$$
Proof
2.5 Harmonic Scale
In Propositions 2.15 and 2.17, we prove the “stability” of the curvature scale under the scalar bound condition. However, this condition is in general not available. We observe that Calabi energy is scaling invariant for complex dimension 2. For this particular dimension, scalar bound condition can more or less be replaced by Calabi energy small, whenever collapsing does not happen. For the purpose to rule out collapsing, we need a more delicate scale, which is the harmonic scale introduced in this subsection.
Definition 2.18
For the harmonic radius, Anderson–Cheeger showed the following result.
Lemma 2.19
Next, we introduce the harmonic scale which will be used in the convergence of Calabi flow on Kähler surfaces.
Definition 2.20
Lemma 2.21
There is a universal small constant \(\epsilon \) with the following properties.
Proof
2.6 Backward Regularity Improvement
We now can summarize the main results in Sect. 2 as the following backward regularity improvement theorems.
Theorem 2.22
There is a \(\delta =\delta (T_0, B, c_0)\) with the following properties.
Proof
Note that \(H_{g(T)}(M)\) is uniformly bounded from below. Therefore, up to rescaling, we can apply Lemma 2.21 to obtain a \(\delta \) such that \(H_{g(t)}(M)\) is uniformly bounded from below whenever \(t \in [T2\delta , T]\). Then the statement follows from the application of Theorem 2.3.
Theorem 2.23
There is a \(\delta =\delta (n,T_0,A,B)\) with the following properties.
The proof of Theorem 2.23 is nothing but a mild application of Propositions 2.15 and 2.17, with loss of accuracy. The reason for developing Proposition 2.15 and Proposition 2.17 with more precise statement is for the later use in Sect. 4, where we study the blowup rate of Riemannian curvature tensors. Note that Theorem 2.23 deals with collapsing case also. If we add a noncollapsing condition at time T, then the scalar curvature bound in Theorem 2.23 can be replaced by a uniform bound of \({ \left\ S \right\ }_{L^p}\) for some \(p>n\). This was pointed out by Donaldson [31].
Remark 2.24
All the quantities in Theorems 2.22 and 2.23 are geometric quantities, and consequently are invariant under the action of diffeomorphisms. Therefore, if we transform the Calabi flow by diffeomorphisms, then all the estimates in Theorems 2.22 and 2.23 still hold. In particular, they hold for the modified Calabi flow(c.f. Definition 3.1) and the complex structure Calabi flow(c.f. equation (1.3)).
3 Convergence of the Calabi Flow
3.1 Deformation of the Modified Calabi Flow Around extK Metrics
In this subsection, we fix the underlying complex manifold and evolve the Calabi flow in a fixed Kähler class.
Definition 3.1
Equation (3.3) is called the modified Calabi flow equation. Correspondingly, the functional \(Ca(\omega _{\varphi })Ca(\omega )\) is called the modified Calabi energy.
The space \(\mathcal {H}\) (c.f. equation (1.4)) has an infinitely dimensional Riemannian symmetric space structure, as described by Donaldson [28], Mabuchi [40], and Semmes [41]. Every two metrics \(\omega _{\varphi _1}, \omega _{\varphi _2}\) can be connected by a weak \(C^{1,1}\)geodesic, by the result of Chen [11]. Therefore, \(\mathcal {H}\) has a metric induced from the geodesic distance d, which plays an important role in the study of the Calabi flow. For example, the Calabi flow decreases the geodesic distance in \(\mathcal {H}\)(c.f. [7]). Furthermore, by the invariance of geodesic distance up to automorphism action, the modified Calabi flow also decreases the geodesic distance. However, d is too weak for the purpose of improving regularity. Even if we know that \(d(\omega _{\varphi }, \omega )\) is very small, we cannot obtain too much information of \(\omega _{\varphi }\). For the convenience of improving regularity, we introduce an auxiliary function \(\hat{d}\) on \(\mathcal {H}\).
Definition 3.2
Lemma 3.3
For each \(\epsilon >0\), there is a \(\delta =\delta (\omega , \epsilon )\) with the following property.
Proof
There is a modified Calabi flow version of the shorttime existence theorem of ChenHe(c.f. [15]).
Lemma 3.4
There is a \(\xi _0=\xi _0(\omega )\) with the following properties.
Now we fix \(\xi _0\) in Lemma 3.4 and define \(\delta _0=\delta _0(\omega , \xi _0)\) as in Lemma 3.3.
Lemma 3.5
Suppose \(\eta _0< \xi _0\) is small enough such that \( Ca(\omega _{\omega _\varphi })Ca(\omega )<\delta _0\) for every \(\varphi \in \mathcal {H}\) satisfying \({ \left\ \omega _{\varphi }\omega \right\ }_{C^{k,\frac{1}{2}}}<\eta _0\). Then the modified Calabi flow starting from \(\omega _{\varphi }\) has global existence whenever \({ \left\ \omega _{\varphi }\omega \right\ }_{C^{k,\frac{1}{2}}}<\eta _0\).
Proof
Clearly, the flow exists on [0, 1] and \(\hat{d}(\omega _{\varphi (1)})<\xi _0\). We continue to use induction to show that the flow exists on [0, N] for each positive integer N and \(\hat{d}(\omega _{\varphi (N)})<\xi _0\).
Theorem 3.6
Suppose \(\varphi \in \mathcal {H}\) satisfying \({ \left\ \omega _{\varphi }\omega \right\ }_{C^{k,\frac{1}{2}}}<\eta _0\). Then the modified Calabi flow starting from \(\omega _{\varphi }\) converges to \(\varrho ^*\omega \) for some \(\varrho \in \mathrm {Aut}_0(M,J)\), in the smooth topology of Kähler potentials.
Proof
First, let us show the convergence in distance topology.
Second, we improve the convergence topology from distance topology in (3.6) to smooth topology.
The convergence in Theorem 3.6 could be as precise as “exponential” if we have further conditions on \(\omega \) or \(\omega _{\varphi }\).
Theorem 3.7

\(\omega \) is a cscK metric.

\(\omega _{\varphi }\) is Ginvariant, where G is a maximal compact subgroup of \(Aut_0(M,J)\).
3.2 Convergence of Kähler Potentials
The key of this subsection is the following regularity improvement properties.
Proposition 3.8
Proof
The solution of the modified flow (3.3) and the Calabi flow differs only by an action of \(\varrho \), which the automorphism generated by Re(X) from time \(t_01\) to \(t_0\), where X is defined in (3.1). Since X is a fixed holomorphic vector field. It follows that \({ \left\ \varrho \right\ }_{C^{l,\frac{1}{2}}}\) is uniformly bounded by \(A_l\) for each positive integer l. Therefore, in order to show (3.8) for modified Calabi flow, it suffices to prove it for unmodified Calabi flow.
Clearly, the Riemannian geometry of \(\omega _{\varphi (t_0)}\) is uniformly bounded, by shrinking \(\eta _0\) if necessary. We also have \( Ca(\omega _{\varphi (t_01)})Ca(\omega _{\varphi (t_0)}) <\epsilon \). Therefore, Theorem 2.22 applies and we obtain uniform \(\nabla _\varphi ^l Rm(\omega _\varphi )_{g_\varphi }\) bound for each nonnegative integer l. Due to the bound of curvature derivatives, one can obtain the metric equivalence for a fixed time period before \(t_0\), say on \([t_0\frac{1}{4}, t_0]\). Then we see that \(\omega _{\varphi (t_0\frac{1}{4})}\) is uniformly equivalent to \(\omega \) and has uniformly bounded Ricci curvature. This forces that \(\omega _{\varphi (t_0\frac{1}{4})}\) has uniform \(C^{1,\frac{1}{2}}\)norm, due to Theorem 5.1 of ChenHe [15]. Consequently, (3.8) follows from the smoothing property of the Calabi flow(c.f. the proof of Theorem 3.3 of [15]). \(\square \)
Proposition 3.9
Proposition 3.9 is the high dimension correspondence of Proposition 3.8. The proof of Proposition 3.9 is almost the same as that of Proposition 3.8, except that we use Theorem 2.23 to improve regularity, instead of Theorem 2.22.
Now we are ready to prove our main theorem for the extremal Kähler metrics.
Proof of Theorem 1.2
We continue to show that I is also closed. Without loss of generality, we can assume that \([0,\bar{s}) \subset I\) and it suffices to show that \(\bar{s} \in I\).
For each \(s \in [0,\bar{s})\), let \(T_s\) be the first time such that \(\hat{d}(s,t)=0.5 \eta _0\). By the convergence assumption and the continuity of \(\hat{d}\), each \(T_s\) is a bounded number.
Claim 3.10
The bound of \(T_s\) is uniform, i.e., \(\displaystyle \sup _{s \in [0,\bar{s})} T_s < \infty .\)
Theorem 1.4 can be proved almost verbatim, except replacing Propsotion 3.8 by Proposition 3.9.
3.3 Convergence of Complex Structures
In this subsection, we regard the Calabi flow as the flow of the complex structures on a given symplectic manifold \((M,\omega )\). We also assume this symplectic manifold has a cscK complex structure \(J_0\). Note that the uniqueness theorem of extK metrics in a given Kähler class of a fixed complex manifold plays an important role in the convergence of potential Calabi flow. Similar uniqueness theorem will play the same role in the convergence of complex structure Calabi flow. Actually, by the celebrated work of ChenSun(c.f. Theorem 1.3 of [20]), on each \(C^{\infty }\)closure of a \(\mathcal {G}^{\mathbb {C}}\)leaf of a smooth structure J, there is at most one cscK complex structure(i.e., the metric determined by \(\omega \) and J is cscK), if it exists.
Let \(g_0\) be the metric compatible with \(\omega \) and \(J_0\). Then it is clear that \(g_0\) is cscK and therefore smooth metric. We can choose coordinate system of M such that \(g,J,\omega \) are all smooth in each coordinate chart. We equip the tangent bundle and cotangent bundle and their tensor products with the natural metrics induced from \(g_0\). Clearly, if a diffeomorphism \(\varphi \) preserves both \(\omega \) and \(J_0\), then it preserves \(g_0\) and therefore locates in \(\mathrm {ISO}(M, g_0)\), which is compact Lie group. Therefore, each \(\varphi \) is smooth and has a priori bound of \(C^{l,\frac{1}{2}}\)norm for each positive integer l, whenever \(\varphi \) is regarded as a smooth section of the bundle \(M \times M \rightarrow M\), equipped with natural metric induced from \(g_0\). There is an almost version of this property. In other words, if \(\varphi \) preserves \(\omega \) and the \(C^{k,\frac{1}{2}}\)norm of \(\varphi ^* J\) is very close to \(J_0\), then \(\varphi \) has a priori bound of \(C^{k+1,\frac{1}{2}}\)norm. This is basically because of the improving regularity property of isometry(c.f. [8]).
Proposition 3.11
Proof
Lemma 3.12
Proof
Regard \(\varphi \) as an isometry from \((M,\omega ,J)\) to \((M,\omega , \varphi _{*} J)\). Then the proof boils down to Proposition 3.11. Note that \(\varphi _* J\) is the push forward of J, which is the same as \((\varphi ^{1})^* J\), and the default metric we take \(C^{k,\frac{1}{2}}\)norm is the metric \(g_0\). \(\square \)
Lemma 3.13
For each smooth complex structure J compatible with \(\omega \), \(\tilde{d}(J)\) is achieved by a diffeomorphism \(\varphi \in C^{k+1,\frac{1}{2}}(M,M)\) and \(\varphi ^* \omega =\omega \).
Lemma 3.14
\(\tilde{d}\) is a continuous function on the moduli space of complex structures, equipped with \(C^{k,\frac{1}{2}}\)topology.
Proof
Therefore, \(\tilde{d}\) is continuous at \(J_A\) by combining the above two inequalities. Since \(J_A\) is chosen arbitrarily in the moduli space of complex structures, we finish the proof. \(\square \)
Lemma 3.15
Suppose \(J_A=J_s(t)\) for some \(s \in D\) and \(t \ge 0\). There is a constant \(\delta >0\) such that if \(\tilde{d}(J_A)<\delta \), then the Calabi flow starting from \(J_A\) has global existence and converges to \(\psi ^*(J_0)\) for some \(\psi \in \mathrm {Symp}(M, \omega )\).
Proof
By Theorem 5.3 of [20], there is a small \(\delta \) such that every Calabi flow starting from the \(\delta \)neighborhood of \(J_0\), in \(C^{k,\frac{1}{2}}\)topology, will converge to some cscK \(J'\). Note that the Calabi flow solution always stays in the \(\mathcal {G}^{\mathbb {C}}\)leaf of \(J_1\), hence \(J'\) is in the \(C^{\infty }\)closure of \(J_1\). Also, on the other hand, according to the conditions of \(J_s\), we see that \(J_0\) is also in the \(C^{\infty }\)closure of \(J_1\). Therefore, one can apply the uniqueness degeneration theorem, Theorem 1.3 of [20] to obtain that \((M, \omega , J_0)\) and \((M, \omega , J')\) are isomorphic. Namely, \((\omega , J')=\eta ^*(\omega , J_0)\) for some diffeomorphism \(\eta \).
With these preparation, we are ready to prove Theorem 1.3.
Proof of Theorem 1.3
Claim 3.16
Suppose \([0,\bar{s}) \subset I\) for some \(\bar{s} \in (0,1]\), then \(\bar{s} \in I\).
We argue by contradiction.

There is a sequence of \(s_i\) such that \(s_i \rightarrow \bar{s}\) and \(T_{s_i} \rightarrow \infty \).

\(\displaystyle \sup _{s \in [\bar{s}\xi ,\bar{s})}T_s <A\) for some constant A.
The proof of Theorem 1.5 is almost the same as Theorem 1.3. The only difference is that we use Theorem 2.23 to improve regularity, rather than Theorem 2.22.
3.4 Examples
In this subsection, we will give higher dimension examples with global existence. On such examples, our results and methods developed in previous sections can be applied.
Example 3.17
Example 3.18
4 Behavior of the Calabi Flow at Possible Finite Singularities
All our previous discussion in this paper is based on the global existence of the Calabi flow, and there do exist some nontrivial examples of global existent Calabi flow. However, it is not clear whether the global existence of the Calabi flow holds in general. Suppose the Calabi flow starting from \(\omega _{\varphi }\) fails to have global existence. Then there must be a maximal existence time T. By the work of ChenHe [15], we see that Ricci curvature must blowup at time T. In this subsection, we will study the behavior of more geometric quantities at the first singular time T. For simplicity of notations, we often let the singular time T to be 0. Recall that P, Q, R are defined in equation (1.9).
Proposition 4.1
Proof

\(Q_{g_i}(0)=1\),

\(Q_{g_i}(t)\le 1\) for all \(t\in [1, 0]\),

\(Q_{g_i}\Big (Q(s_i)^2(s_{i+1}s_i)\Big )=2\).
Since the Riemannian curvature tensor blows up at the singular time along the Calabi flow, Proposition 4.1 directly implies the following result.
Corollary 4.2
Next, we would like to estimate Q(t) near the singular time of the Calabi flow. Analogous results for Ricci flow are proved by the maximum principle (cf. for example, Lemma 8.7 of [25]). Here we show similar results for the Calabi flow using the higher order curvature estimates.
Lemma 4.3
There exists a constant \(\delta _0=\delta _0(n)>0\) with the following properties.
Proof
The next result gives an upper bound of Q near the singular time with the assumption on P.
Lemma 4.4
Proof
We now prove Theorem 1.6.
Proof of Theorem 1.6
After rescaling the flow, we can assume that \(T\ge 1\). It is clear that (1.12) follows from the combination of inequality (1.11) and the definition of typeI singularity(c.f. [39]), i.e., \(\displaystyle \limsup _{t \rightarrow 0} Q^2t<\infty \). Therefore, we only need to show (1.10) and (1.11), which will be dealt with separately.
1. Proof of inequality (1.10):
2. Proof of inequality (1.11):
5 Further Study
The methods and results in this paper can be generalized in the following ways.
1. The method we developed in this paper reduces the convergence of the flow to three important steps: uniqueness of critical metrics, regularity improvement, and good behavior of some functional along the flow. Theorems 1.3 and 1.5 can be proved for minimizing extK metrics in a general class, assuming a uniqueness theorem of minimizing extK metric in a fixed \(\mathcal {G}^{\mathbb {C}}\)leaf’s \(C^{\infty }\)closure, or a generalization of Theorem 1.3 of [20]. This will be discussed in a subsequent paper.
2. The fourth possibility of Donaldson’s conjectural picture seems to be extremely difficult. By the example of G.Székelyhidi, the flow singularity at time infinity could be very complicated. However, if we assume the underlying Kähler class to be \(c_1(M,J)\) which has definite sign or zero, then the limit should be a normal variety and could be used to construct a destabilizing test configuration.
3. Our deformation method could be applied to a more general situation. In Sect. 3, we deformed the complex structures and the metrics within a given Kähler class. Actually, even the underlying Kähler classes can be deformed. The general deformations will be discussed in a separate paper.
Notes
Acknowledgements
The authors would like to thank Professor Xiuxiong Chen, Simon Donaldson, Weiyong He, Claude LeBrun and Song Sun for insightful discussions. Haozhao Li and Kai Zheng would like to express their deepest gratitude to Professor Weiyue Ding for his support, guidance and encouragement during the project. Part of this work was done while Haozhao Li was visiting MIT and he wishes to thank MIT for their generous hospitality. H. Li: Supported by NSFC Grant No. 11671370. B. Wang: Supported by NSF Grant DMS1312836. K. Zheng: Supported by the EPSRC on a Programme Grant entitled “Singularities of Geometric Partial Differential Equations” Reference Number EP/K00865X/1.
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