On the extension and smoothing of the Calabi flow on complex tori

In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when $n=2$, the Calabi flow starting from a weak K\"ahler metric will become smooth immediately. It implies that in our settings, the weak minimizer of the Mabuchi energy is a smooth one.


Introduction
In geometrical flows, one of the important tools is the blowup analysis. For example, in Perelman's celebrated work, he uses the blowup analysis to understand the singularities under the Ricci flow in a 3-manifold [19,20,21]. To do this, he needs three tools: (1) Shi's estimates [22] in Ricci flow.
(2) Non-collapsing property of the Ricci flow. [19] (3) Classifications of the κ-solutions. [20] The blowup analysis plays an important role not only in geometrical flows, but also in the continuity methods. The most relevant work to our paper is Donaldson's celebrated proof of the Yau-Tian-Donaldson conjecture in toric surfaces [11,12,13,14]. In Donaldson's work, he also establishes the Shi-type estimates, the non-collapsing property and the classification of the limiting spaces.
In the Calabi flow, the first work using the blowup analysis is due to Chen-He [7,8] who study the Calabi flow on Fano toric surfaces. They showed that the Sobolev constant is uniformly bounded along the Calabi flow if the Calabi energy of the initial Kähler metric is less than an explicit constant. Thus to show the long time existence of the Calabi flow, they proved • Shi-type estimates in integral forms.
• Classification/non-existence of the limiting spaces. Inspired by the above work, the author proposed a project to study the Calabi flow in toric manifolds [17]. What we want to establish are the following: (1) Shi-type estimates. In [17], we classify the limiting spaces. Streets establishes the Shi-type estimates in [23]. Finally, joint with Feng [16], we prove that the Calabi flow exists for all time on C 2 /(Z 2 + iZ 2 ) with a 2-torus invariant initial metric.
All the above work require the classification of the limiting space. However, our following theorem suggests that the classification of the limiting spaces may not be necessary to prove the long time existence of the Calabi flow. Let us briefly explain the ideas here. Let ϕ(t), t ∈ [0, T ) be a sequence of relative Kähler potentials satisfying the Calabi flow equation. To show that ϕ(t) can be extended over T , by Chen-He's compactness theorem (Theorem 1.4 in [6]), one only needs to show that the Ricci curvature is uniformly controlled in [0, T ). The idea of the blowup analysis is that suppose the Ricci curvature is not uniformly bounded in [0, T ), then one can pick a sequence of time t i → T − , such that Let λ i = max x∈X |Rm|(t i , x). One then proceed to rescale the Calabi flow at t i by the factor λ i . So one obtain a sequence of the Calabi flow g (i) (t). One need to show that as i → ∞, g (i) (t) converges to a limiting Calabi flow g (∞) (t) in a noncompact Kähler manifold X ∞ . This step requires Shitype estimates and the non-collapsing property. The last step is to obtain a contradiction by studying the behavior of g (∞) (t) in X ∞ . This step usually requires the classification of the limiting spaces. Our observation is that in our settings, there exists an explicit constant C such that if the rescaling factor λ i ≥ C, then we can obtain a contradiction. Thus we completely avoid the study of the limiting Calabi flow g (∞) (t) and the limiting Kähler manifold X ∞ . Our main theorem reads as follows: Theorem 1.1. Let X = C n /(Z n + iZ n ). Let ϕ(t) ∈ H T , t ∈ [0, T ) be a one parameter family of relative Kähler potentials satisfying the Calabi flow equation. Suppose that there exists C E > 0 such that for any t ∈ [0, T ), the total energy X |Rm| n ω n (t) then there exists an explicit constant λ > 0 depending only on d(ϕ(0), 0), C E , n such that for any t ∈ [0, T ), Remark 1.2. Our theorem strengthens the results obtained in [16]: • For n ≥ 3 with the assumption that the total energy is controlled, we are not able to control the Riemann curvature at all in [16]. • For n = 2, we only be able to prove an inexplicit bound of the curvature long the Calabi flow in [16]. We are able to strengthen the results in [16] because we avoid going to the limiting spaces. In a forthcoming paper, this idea will be generalized to other cases. It seems that the only obstacle to prove the long time existence of the Calabi flow is the non-collapsing property of the Calabi flow. In the setting of our paper, the non-collapsing property only depends on the control of the total energy.
Let us discuss an application of our main theorem : the smoothing property of the Calabi flow. Given a weak Kähler metric, a natural question is that if we could smooth it and how we smooth it. There are many important work in this area. For example, Chen-Tian-Zhang use the Kähler Ricci flow to smooth a weak Kähler metric [9]. Similar to [9], we first need to define a unique weak Calabi flow starting from a weak Kähler metric. Then we show that this weak Calabi flow becomes smooth immediately. The existence of the weak Calabi flow follows from Streets' work in [24] which uses a general theory of Mayer [18]. Further development of the weak Calabi flow can be found on [25,2]. The remaining task for us is to show that the weak Calabi flow is a smooth one for t > 0.
Let S be the set of all smooth symplectic potentials and E be the completion of S in the sense of the Mabuchi distance. Our result is: . Suppose u 0 ∈ E and its Mabuchi energy is finite. Let u(t), t ≥ 0 be the weak Calabi flow starting from u 0 . Then for any t > 0, u(t) ∈ S.
Since the minimizer of the Mabuchi energy in E is a fixed point in the weak Calabi flow, we have an immediate corollary: Corollary 1.4. Any minimizer of the Mabuchi energy in E belongs to S. Remark 1.5. The regularity of weak minimizers of the Mabuchi energy in the general case has been proved by Berman-Darvas-Lu [3], assuming the existence of smooth cscK metrics. Their result partially confirms conjectures of Chen [4] and Darvas-Rubinstein [10]. It is also shown in [10] that the regularity of weak minimizers implies the properness of the Mabuchi energy [26,27,10].
In our proof, we do not use the fact that there exists cscK metrics in the Kähler class. Thus we expect that the Calabi flow can smooth the weak minimizers of the Mabuchi energy in general. As pointed out in [10], the regularity problem of the weak minimizers is the main obstacle in proving the existence of cscK metrics, assuming the Mabuchi energy is proper in the Kähler class.

Acknowledgement
The author would like to thank Professor Pengfei Guan, Tarmás Darvas, Bing Wang, Chengjian Yao for stimulating discussions.

Introduction
be the set of relative Kähler potentials. Feng and Szekelyhidi [15] considered the subset H T which consists of all the torus invariant relative Kähler potentials, i.e., Thus for any ϕ ∈ H T , ϕ is a smooth, periodic function on R n with period [−1, 1] n such that is a smooth, strictly convex function on R n . Feng and Szekelyhidi [15] considered the Legendre transform of ψ: u. Let (x 1 , . . . , x n ) = ∇ψ(ξ 1 , . . . , ξ n ) be the dual coordinates. We have Then the dual coordinates of (x 1 + 2, x 2 , . . . , x n ) is (ξ 1 + 2, ξ 2 , . . . , ξ n ). Thus i is a smooth, strictly convex function. Then there exists a one-to-one correspondence between H T and S through the Legendre transform. We also denote E to be the completion of S in the sense of the Mabuchi distance.

Maximum domain of a special convex function
In this section, we want to understand if one can have a special (see definition below) convex function u on R n . If not, what is the maximum domain in R n one can have for u. Let us start with an analytic result: Then for any n, C > 0, there exists x 0 ∈ [1, ∞) depending only on C, M, n such that Proof. Suppose the conclusion is not true. Then there exists C, n > 0 such that for any x ∈ [1, ∞), we have Let us construct a sequence of positive constant R i , i = 0, 1, . . . such that R i = 2 i . By Hölder's inequality, we know that Let R > 0 be a positive constant and let It yields Let us analyze the left hand side of the inequality (1). Since u is a special convex function, for any x ∈ R n \B E (O, 1), we have u r > C 0 where r = d E (O, x). Combining with the fact that |Du| < M , we have Lemma 3.3. There exists a constant C 1 > 0 depending only on n, C 0 , M such that Proof. The proof of this lemma is elementary and we leave it to the interested readers.
As a corollary, we have where C 2 depends only on n, C 0 , M, C E .
Proof. It is easy to see that For the second term, we have Combining the above two terms, we obtain the conclusion. Now let us analyze the right hand side of the inequality (1). Let (r, θ = (θ 1 , . . . , θ n−1 )) be the spherical coordinates. Since 0 < D r u(r, θ) < M for any r > 0, θ ∈ S n−1 , for any 0 < r ≤ R 2M , we have u(r, θ) < R/2. Using the coordinate system as in Lemma 3 of [12] and Lemma 4.3 of [17] (without changing the value of u rr ), direct calculations show that: Let C 3 > 0 be a constant to be determined later. Then by Proposition (3.1), there exists R 0 > 1 such that Thus we have We can choose C 3 = 4C 2 (2M ) n+1 C 2 0 V ol(S n−1 ) + 1 to obtain a contradiction. By Proposition (3.1), we have the following theorem: Theorem 3.5. For a special function of type (M, C 0 , C E ). Its domain is contained in the Euclidean ball centered at O with radius

Modified Blowup analysis
We will prove Theorem (1.1) in this section. In [16], we obtained the following results: where Du(t) denotes the Euclidean derivative of u(t).
Then following the analysis in Section 4 of [16], one obtain that there exists C 0 > 0 depending only on n, M, C E such that for any x outside B E (O, 1), Let v(x) =ũ(0, x). Thus v(x) is a special convex function of type (M, C 0 , C E ) defined on a domain [−λ, λ] n . Theorem (3.5) gives a contradiction if we choose λ = 2 + R 0 C 0 . The second case is t 0 < 1. Similar to the above case, let us rescale the flow by λ √ 2t 0 . Without loss of generality, let us assume λ > 1, so we obtain a new flow defined in [−2, 0]. If for any t ∈ [−1, 0], max |Rm|(t) < 2, then we can apply the argument in the first case to obtain a contradiction. If there exists some t ∈ [−1, 0] such that max |Rm|(t) ≥ 2, then the usual point-picking techniques allow us to obtain a flow in [−1, 0]. Applying the arguments in the first case, we can also obtain a contradiction.

Smoothing property
Let E be the completion of S in the sense of the Mabuchi distance. In fact E is the L 2 -completion of S. For any f ∈ E, it is a periodic function on R n with period P = [−1, 1] n . Moreover, i is a convex function. Thus (D 2 u) exists almost everywhere on R n which implies that (D 2 f ) exists almost everywhere on R n . Following the ideas of [28,30], we consider the mollification f h of f , i.e., x − y h dy for some nonnegative function η supported on the unit ball B 1 (0) and satisfying R n η = 1. It is easy to see that f h (x) + 1 2 n i=1 x 2 i is a smooth, strictly convex function on R n . For any point x ∈ R n that (D 2 f )(x) exists, we have [29]. Let us consider a sequence of smooth, strictly convex function on R n : where r is some function from N to N such that r(i) ≥ i for all i ∈ N.
The following lemma is crucial for us: Proof. The first statement is easy to prove: Then dominated convergence theorem shows that lim m→∞ P log det(D 2 v m ) dx = P log det(D 2 u) dx.

Now for every
Again, v j,m (x) is a smooth, strictly convex function on R n . In fact, log det(D 2 v j,m (x)) ≥ −n log m, log det(D 2 v m (x)) ≥ −n log m.
Following the proof of Lemma 2.2 in [30], one concludes that Thus for each m, one can choose j = r(m) ≥ m such that It is easy to see that with our choice of r : N → N, we have i . It is easy to see that there exists C 1 , C 2 > 0 such that for any m ∈ N, t > 0, Then for any fixed t > 0, Theorem (1.1) provides uniform control of the curvature of u m (t, x). Thus for any fixed t > 0, k ∈ N, one has uniform control of C k norm of u m (t, x). Let u(t) be the weak Calabi flow starting from u 0 . Then by the fact that the (weak) Calabi flow decreases the Mabuchi distance [5,24,18], one has d(u m (t), u(t)) ≤ d(u m , u 0 ).
We conclude that u(t) is a smooth function. Then f (x, y) is a periodic function on R 2 (up to the second order) with period [−1, 1] 2 . It is easy to see that u(x, y) = f (x, y) + 1 2 (x 2 + y 2 ) in E\S and its Mabuchi energy is: