In this paper, we study the Calabi flow on a polarized Kähler manifold and some related problems. We first give a precise statement on the short time existence of the Calabi flow for any c 3,α ( M ) initial Kähler potential. As an application, we prove a stability result: any metric near a constant scalar curvature Kähler (CscK) metric will flow to a nearby CscK metric exponentially fast. Secondly, we prove that a compactness theorem in the space of the Kähler metrics given uniform Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-singularity result about a weak constant scalar curvature metric in a punctured disc.

Abstract. We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in R 2n , we show that the parabolic equation (1.1) has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time t = 0. In particular, under the mean curvature flow (1.2) the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as t → ∞. Our assumption on the Lipschitz norm is equivalent to the assumption that the underlying Larangian potential u is uniformly convex with its Hessian bounded in L ∞ . We apply this result to prove a Bernstein type theorem for translating solitons, namely that if such an entire Lagrangian graph is a smooth translating soliton, then it must be a flat plane. We also prove convergence of the evolving graphs under additional conditions. See Theorem 1.1 and Theorem 1.3 for details.

In this paper we extend a recent breakthrough of Chen and Cheng on the existence of a constant scalar Kähler metric on a compact Kähler manifold to Calabi’s extremal metric. There are no new a priori estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector V V if and only if the modified Mabuchi energy is proper, modulo the action of the subgroup in the identity component of the automorphism group which commutes with the flow of V V . We introduce two essentially equivalent notions, called reductive properness and reduced properness. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy.

Kähler surface is uniquely determined by its underlying geometric structure. However, a proof seems out of reach at this point.Recently Chen-LeBrun-Weber [10] studied the existence of extK metrics in some Kähler classes on M ∼ CP 2 ♯2CP 2 ; M can be obtained by CP 2 blown up at two different points. They constructed an Einstein metric on M ∼ CP 2 ♯2CP 2 which is conformal to an extK metric in a particular Kähler class. The strategy in [10] is to use continuous deformation of extK metrics and a weak compactness theorem for extK metrics [12]. A possible bubble in the deformation turns out to be a scalar flat ALE Kähler surface. A key step is to eliminate the formation of such a bubble by using the smallness of the Calabi energy. Theorem 1.1 makes it possible for us to adopt the strategy in [10], albeit new difficulty arises in our case. For instance the bound of scalar curvature plays a key role in [10] while along the Calabi flow, such a bound is not known. Also we need to bound the Sobolev constants along the Calabi flow. Note that this is a non-collapsing (or local non-collapsing) result. In general it is hard to obtain non-collapsing result (or local non-collapsing) along a geometric evolution equation, such as Perelman's non-local collapsing result along the Ricci flow [30]. On Kähler surfaces with c 1 > 0, we can actually bound the Sobolev constants along the Calabi flow geometrically. Combining all these together, we can prove that the Calabi flow exists for all time and converges to an extK metric in many classes on toric surfaces with c 1 > 0. For simplicity, we only present a particular example of our results.We consider M ∼ CP 2 ♯3CP 2 . M can be obtained by CP 2 blown up at three generic points. This is known as a del Pezzo surface with tori symmetry. Its automorphism group contains a compact two-torus T 2 . A metric which is invariant under the action of T 2 is called a toric metric. Calabi showed that any extK metric admits the maximal symmetry that the manifold allows [4]; namely an extK metric is invariant under the identity component of a maximal compact subgroup of the automorphism group of the manifold. This implies that any extK metric on M has to be toric. Let H be a hyperplane in CP 2 . After blown up, we still use H to denote the corresponding hypersurface on M and E i , i = 1, 2, 3 to denote the exceptional divisors. For simplicity, we use [H], [E i ] to denote the homology classes and their Poincaré dual-the cohomology classes. A Kähler class on M can be described as [ω] x,y,z = 3) are invariant under the action of Z 3 , which acts on exceptional divisors as a cyclic group. Note that the positivity condition of a Kähler class requires that x ∈ (0, 3/2) and M degenerates to CP 2 when x = 0, 3/2. We will give a full detailed discussion for x = 1/2. We can also prove similar results for any x ∈ (0, 3/2) using the same method. We will give a rough discussion and state our theorem for x ∈ (0, 3/2) in appendix. Theorem 1.2. Let [ω] = 3H − 1/2([E 1 ]+ [E 2 ]+ [E 3 ]) on M . Suppose the initial me...

We consider the complex Monge-Ampère equation on a compact Kähler manifold (M, g) when the right hand side F has rather weak regularity. In particular we prove that estimate of △φ and the gradient estimate hold when F is in W 1,p0 for any p 0 > 2n. As an application, we show that there exists a classical solution in W 3,p0 for the complex Monge-Ampère equation when F is in W 1,p0 .