Calabi flow on toric varieties with bounded Sobolev constant, I

Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of the Calabi flow $\varphi(t)$ on $X$ is uniformly bounded in $[0, T)$ if the Sobolev constant of $\varphi(t)$ is uniformly bounded in $[0, T)$. We also show that if $(X, P)$ is uniform $K$-stable, then the modified Calabi flow converges exponentially fast to an extremal K\"ahler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper K\"ahler manifold.


Introduction
In [9,10], Chen and He study the Calabi flow on toric surfaces with bounded Sobolev constant. The study of Calabi flow with bounded Sobolev constant has been also elaborated by Li and Zheng [19]. Related work can be found in [20,21,12,11,23,24,25]. Interested readers are encouraged to read these papers and the references therein.
In this note, we study the Calabi flow on toric varieties with bounded Sobolev constant and on general Kähler manifolds which are quasi-proper. Our first result is: Theorem 1.1. Let X be a toric variety with Delzant polytope P . Let ϕ(t), 0 ≤ t < T < ∞ be a one parameter family toric invariant relative Kähler potentials satisfying the Calabi flow equation. Suppose that the Sobolev constant of ϕ(t) is uniformly bounded. Then where C is some constant independent of t.
Suppose (X, P ) is uniform K-stable, we would like to understand the global convergence of the (modified) Calabi flow. The following result can be compared to the global convergence results in [16,17,22]. Theorem 1.2. If (X, P ) is uniform K-stable, then the modified Calabi flow introduced in [18] will converge to an extremal Kähler metric exponentially fast if • The Calabi flow starts from a toric invariant Kähler metric.
• The Sobolev constant is uniformly bounded along the flow.
• The Ricci curvature is uniformly bounded along the flow.
For a general manifold, we have: Let (X, J, ω) be a quasi-proper Kähler manifold in the sense of Chen [7], i.e., there exists a small constant δ > 0 and a constant C such that the Mabuchi energy Then Acknowledgment: The results in this note are motivated by discussions with Vestislav Apostolov. The author would like to thank him for sharing his ideas. Thanks also go to Yiyan Xu and Kai Zheng for help discussions.

Notations and setup
2.1. Kähler geometry. Let (X, J, ω) be a Kähler manifold with complex dimension n, where locally g ij is a positive definite Hermitian matrix. The Kähler metric is (locally): The set of Kähler metrics can be identified with H/R, where We call ϕ ∈ H is a relative Kähler potential. The corresponding Kähler metric is

Its volume form is
where n is the complex dimension of X. Its Ricci and scalar curvature are: The Calabi flow [3,4] starting from ω ϕ is defined as: where R = X Rϕ ω n ϕ X ω n ϕ is a topological constant. The short time existence of the Calabi flow is established in [8]. Now we introduce three functionals I, J, D. The I and J functionals are introduced by Aubin [2]: for any ϕ ∈ H, We have The third functional is the D functional defined as follows: (c.f. [13])

Direct calculations show that
Thus D(t) = const along the Calabi flow.
2.2. Toric geometry. Let X be a toric manifold and ω be a toric invariant Kähler metric. We obtain the Delzant polytope P of X through the moment map. On P , we have the standard Lebesgue measure dµ. On each facet P i of P , the measure dσ equals to 1 | n i | times the standard Lebesgue measure, where n i is an inward normal vector associated to P i . Also for any vertex v of P , there exists exactly n facets P 1 , . . . , P n intersecting at v and ( n 1 , . . . , n n ) is a basis of Z n .
Suppose that P has d facets. Each facet P i can be represented by where l i (x) = x, n i + c i . A symplectic potential u of P satisfies the following Guillemin boundary conditions [15]: • u is a smooth, strictly convex function on P .
• The restriction of u to each facet of P is also a smooth, strictly convex function. • where f (x) is a smooth function onP . On the open orbit of the (C * ) n = R n × T n action of X, a toric invariant Kähler metric ω can be express as where ψ is a real, smooth, strictly convex function on R n and z i = ξ i + √ −1θ i , (ξ 1 , . . . , ξ n ) ∈ R n , (θ 1 , . . . , θ n ) ∈ T n . In fact, the Legendre dual of ψ is a symplectic potential u on P . Thus The following Proposition is due to Donaldson [14]: For any toric invariant ϕ ∈ H, we obtain a symplectic potential u ϕ through the Legendre dual of ψ + ϕ. The map from ϕ to u ϕ is one to one and onto.
Using the Legendre transform, Abreu [1] shows that the scalar curvature equation of a toric invariant metric ω ϕ on the open (C * )n orbit can be transformed to in the symplectic side. Thus the Calabi flow equation on P is

Controlling max ϕ
Let u(t, x) be a sequence of symplectic potentials satisfying the Calabi flow equation on P . Our first lemma is where C(P, T, u 0 ) is a constant depending on P, T and u(0, x).
Proof. Since the Calabi flow decreases the distance [5], for any t ∈ [0, T ), we have By the triangle inequality, we have Similarly, Thus, It is easy to see that there exists a constant C(P, T, u 0 ) depending on P, T and u(0, x) such that u(t, x) L 2 < C(P, T, u 0 ), An immediate corollary is: where C 1 (P, T, u 0 ) is some constant depending on P, T and u(0, x).
Since the L 2 norm of u(t, x) is bounded by C(P, T, u 0 ), our next lemma shows that min x∈P u(x) is bounded from below by some constant C 2 (P, T, u 0 ) which depends on u(0, x), P and T . Proof. To simplify our notation, we write u(t, x) as u(x). Let o be the barycenter of P . By shrinking the vertices by 1 2 around o, we get P 1 2 . Our first observation is that there exists a constant C 3 (P, T, u 0 ) depending on P, T and u(0, x) such that for any x ∈ P 1 2 , we have u(x) < C 3 (P, T, u 0 ). This is because: • u(x) is a convex function.
• For any x ∈ P 1 2 , any hyperplane l passing through x will cut P as P 1 and P 2 . There exists a positive constant C 4 (P ) depending on P such that the Euclidean volume of P 1 and P 2 are both greater than C 4 (P ). Let x 0 ∈P be a point such that u(x 0 ) = min x∈P u(x). Then there exists a constant C 5 (P ) depending on P such that the follwing holds: • At least one facet of P 1 2 , say Q ⊂ ∂P 1 2 such that the Euclidean distance between x 0 and Q is greater than C 5 (P ). By shrinking the vertices of Q by 1 2 around x 0 , we obtain Q 1 2 . We denotẽ P as the convex hull of x 0 and Q 1 2 . Then for any point x ∈P , we have Since we know it is clear that there exists a constant C 2 (P, T, u 0 ) depending on P , T and u(0, x) such that min x∈P u(x, t) > C 2 (P, T, u 0 ).
Translating our result to the complex side, we have the following proposition: Proposition 3.4. There exists a constant C 3 (P, T, u 0 ) depending on P, T and u(0, x) such that for any t < T we have max z∈X ϕ(t, z) < C 3 (P, T, u 0 ).
In the toric case, we can have a stronger result: Lemma 3.6. For any toric invariant Kähler metric Let the Legendre dual of ψ, ψ + ϕ be u, u ϕ respectively. We have It implies that On the other hand, let x = ∇ψ(ξ), ξ ∈ R n . Similarly we have Then we have As a consequence, we have Corollary 3.7. There exists a constant C 4 (P, T, u 0 ) depending on P, T and u(0, x) such that for any t < T we have max z∈X ϕ(t, z) > C 4 (P, T, u 0 ).

L 1 -norm
It is easy to see that for any t < T , we have In fact, it is well known that the bound of X |ϕ| ω n can be derived from the bound of max z∈X ϕ(z). The arguments go as follows: Let z 0 ∈ X be the point where ϕ(z) reaches its maximum. By the Green's formula, we have Since ω ϕ = ω + √ −1∂∂ϕ > 0, taking trace with respect to ω, we have n + △ ω ϕ > 0.
Thus we obtain where C(ω) is a constant depending on ω.
• If ϕ(z 0 ) ≤ 0, then it is straightforward to see inequality (1). Our next lemma shows that we can also bound the L 1 -norm of ϕ(t) with respect to ω(t).
We obtain the bound of I(ϕ) which also gives us the bound of X ϕ ω n ϕ .
Hence we obtain the inequality (2).
Proof. Notice that The Sobolev inequality shows that It is clear that If the complex dimension of X is 2, then we are done. If n > 2, then we let f = |ϕ| Repeating the steps, we get the conclusion.
Once we have the L 2 estimate, we could get the L ∞ estimate by using the De Giorgi-Nash-Moser iteration: Proposition 5.2. For a smooth relative Kähler potential ϕ, where C n, C s , max z∈X ϕ(z), L 2 ωϕ (ϕ) is a constant depending on n, the Sobolev constant of ω ϕ , max z∈X ϕ(z) and L 2 ωϕ (ϕ)).
Thus for any p ≥ 1, Thus by the Sobolev inequality, we have Then By iterations, we get the conclusion.

Proofs of the theorems
Proof of Theorem (1.1). By Proposition (3.4) and Corollary (3.7), we uniformly control max z∈X ϕ(t, z) for any t ∈ [0, T ). Then Lemma (4.1) and Lemma (4.2) provide us the uniform L 1 -norm bound of ϕ(t) with respect to ω and ω(t) respectively. Hence Proposition (5.1) gives us the L 2 -norm of ϕ(t) with respect to ω(t) uniformly and Proposition (5.2) gives us the L ∞ -norm of ϕ(t) uniformly.

Global convergence.
Proof of Theorem (1.2). Let us fix t 0 > 0 and write ϕ(t) as ϕ 0 and u(t) as u 0 . Let u be the normalized symplectic potential of u 0 at some point x 0 ∈ P . Since (X, P ) is uniformly K-stable, we have P u dµ < C where C is some constant independent of t by Proposition 5.1.8 and Lemma 5.1.3 in [14].
We shall consider the corresponding Kähler potential ψ of u under the Legendre transformation and the relative Kähler potential ϕ = ψ − ψ ω . As in Proposition (3.4) and Corollary (3.7), we obtain the upper and lower bound of max z∈X ϕ(z). By using the arguments of Lemma (3.6), we obtain the bounds of ϕ>0 ϕ ω n ϕ , ϕ<0 −ϕ ω n .
Lemma 2.1 in [26] provides the bound of J(ϕ). Thus we obtain the bound of X |ϕ| ω n ϕ .
As before, we obtain the L ∞ estimate of ϕ. The rest of the proof is identical to the proof of Theorem 1.6 of [17].

General case.
Proof of Theorem (1.3). By the proof of Theorem 1.4 in [7], we obtain uniform upper bounds of max z∈X ϕ(t, z) and X |ϕ| ω n .
Then Lemma (4.2) provide us the L 1 -norm bound of ϕ(t) with respect to ω(t) uniformly. Hence Proposition (5.1) gives us the L 2 -norm of ϕ(t) with respect to ω(t) uniformly and Proposition (5.2) gives us the L ∞ -norm of ϕ(t) uniformly.