The Calabi flow on Kähler Surfaces with bounded Sobolev constant (I)

Abstract

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...

A Concentration‐Collapse Decomposition for L2 Flow Singularities

“…The main tool exploited in these arguments is the local smoothing of curvature derivatives in L 2 , combined with local covering arguments using the exponential map. These L 2 smoothing estimates for Calabi flow are exhibited in [14] .4) one can localize these estimates. Implicitly the evolution equations for curvature and derivatives depend on the complex structure J as well as the metric g, but this presents no difficulty as one can simply pull J back as well in defining the solutions to Calabi flow on local covers.…”

“…Also, one would like to generalize previous low-energy results for the Calabi flow on surfaces [14,15] to a more general setting. However, the techniques used to prove Theorem 1.21 do not immediately extend to prove a corresponding gap theorem for Kähler manifolds with small Calabi energy and a nearly Euclidean lower bound on the volume of sufficiently small balls.…”

Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

“…This in particular implies that Ca(φ(t i − 1))− Ca(φ(t i + 1)) converges uniformly to zero. By the parabolic curvature estimates in [6] we know the path…”

The pseudo-Calabi flow

“…A critical point of this Calabi flow is precisely a cscK metric. This flow has been actively studied in recent years; cf., e.g., [12], [16], [17], [18], [24], [58]. However, the remaining technical di‰culties are still daunting simply because it is a 4th order flow.…”