Calabi Flow Research Articles

The global existence and convergence of the Calabi flow on [formula omitted

In this note, we study the long time existence of the Calabi flow on X=Cn/Zn+iZn. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldsonʼs estimates and Streetsʼ regularity theorem. Next we show that the curvature is uniformly bounded along the Calabi flow on X when the dimension is 2, partially confirming Chenʼs conjecture. Moreover, we show that the Calabi flow exponentially converges to the flat Kähler metric for arbitrary dimension if the curvature is uniformly bounded, partially confirming Donaldsonʼs conjecture.

Stability of Calabi flow near an extremal metric

We prove that on a K\"ahler manifold admitting an extremal metric $\omega$ and for any K\"ahler potential $\varphi_0$ close to $\omega$, the Calabi flow starting at $\varphi_0$ exists for all time and the modified Calabi flow starting at $\varphi_0$ will always be close to $\omega$. Furthermore, when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group, the modified Calabi flow converges to an extremal metric near $\omega$ exponentially fast.

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

We consider the formation of singularities along the Calabi flow by assuming the uniformly bounded Sobolev constants. On Kahler surfaces we prove that if curvature tensor is not uniformly bounded, then one can form a singular model called deepest bubble; such deepest bubble has to be a scalar flat ALE Kahler metric. In certain Kahler classes on toric Fano surfaces, the Sobolev constants are a priori bounded along the Calabi flow with small Calabi energy. We can also show in certain cases no deepest bubble can form along the flow. It follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently. To illustrate our results more clearly, we focus on an example on \({\mathbb{CP}^2}\) blown up three points at generic position. Our result also implies existence of constant scalar curvature metrics on \({\mathbb{CP}^2}\) blown up three points at generic position in the Kahler classes where the exceptional divisors have the same area.

Local Solution and Extension to the Calabi Flow

We consider the local solution of the Calabi flow for rough initial data. In particular, we prove that for any smooth metric, there is a C α neighborhood such that the Calabi flow has a short time solution for any C α metric in the neighborhood. We also prove that on a compact Kähler surface, if the evolving metrics of the Calabi flow are all L ∞ equivalent, then the Calabi flow exists for all time and converges to an extremal metric subsequently.

On the extension of Calabi flow on toric varieties

Inspired by recent study of Donaldson on constant scalar curvature metrics on toric complex surfaces, we study obstructions to the extension of the Calabi flow on a polarized toric variety. Under some technical assumptions, we prove that the Calabi flow can be extended for all time.

Calabi flow and projective embeddings

Let X ⊂ ℂℙN be a smooth subvariety. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X which attempts to deform the given embedding into a balanced one. If L → X is an ample line bundle, considering embeddings via H 0(Lk) gives a sequence of balancing flows. We prove that, provided these flows are started at appropriate points, they converge to Calabi flow for as long as it exists. This result is the parabolic analogue of Donaldson's theorem relating balanced embeddings to metrics with constant scalar curvature [12]. In our proof we combine Donaldson's techniques with an asymptotic result of Liu and Ma [17] which, as we explain, describes the asymptotic behavior of the derivative of the map FS o Hilb whose fixed points are balanced metrics.

Optimal test-configurations for toric varieties

On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists for all time on a toric variety then it minimizes the Calabi functional. In this case the infimum of the Calabi functional is given by the supremum of the normalized Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson.

On the Calabi Flow

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.

Non-Commutative Ricci and Calabi Flows

Starting from an improved version of the bicomplex structure associated the continual Lie algebra with non-commutative base algebra, we obtain dynamical systems resulting from the bicomplex conditions. General expressions for conserved currents associated to a continual Lie algebra bicomplex are found explicitly in two first orders. The Moyal-product counterparts for two-dimensional Ricci and Calabi flow equations depending on non-commutative variables are introduced.

The 2-Dimensional Calabi Flow

AbstractIn this paper, based on a Harnack-type estimate and a local Sobolev constant bounded for the Calabi flow on closed surfaces, we extend author’s previous results and show the long-time existence and convergence of solutions of 2-dimensional Calabi flow on closed surfaces. Then we establish the uniformization theorem for closed surfaces.

The algebraic structure of geometric flows in two dimensions

There is a common description of different intrinsic geometric flows in two dimensions using Toda field equations associated to continual Lie algebras that incorporate the deformation variable t into their system. The Ricci flow admits zero curvature formulation in terms of an infinite dimensional algebra with Cartan operator d/dt. Likewise, the Calabi flow arises as Toda field equation associated to a supercontinual algebra with odd Cartan operator d/d \theta - \theta d/dt. Thus, taking the square root of the Cartan operator allows to connect the two distinct classes of geometric deformations of second and fourth order, respectively. The algebra is also used to construct formal solutions of the Calabi flow in terms of free fields by Backlund transformations, as for the Ricci flow. Some applications of the present framework to the general class of Robinson-Trautman metrics that describe spherical gravitational radiation in vacuum in four space-time dimensions are also discussed. Further iteration of the algorithm allows to construct an infinite hierarchy of higher order geometric flows, which are integrable in two dimensions and they admit immediate generalization to Kahler manifolds in all dimensions. These flows provide examples of more general deformations introduced by Calabi that preserve the Kahler class and minimize the quadratic curvature functional for extremal metrics.

Extremal Metrics for Quadratic Functional of Scalar Curvature on Closed 3-Manifolds

In this paper, we first show the global existence of the three-dimensionalCalabi flow on any closed 3-manifold with an arbitrary background metric g0. Second, we show the asymptotic convergence of a subsequence ofsolutions of the Calabi flow on a closed 3-manifold with Yamabe constant Q 0, up to conformal transformations. With itsapplication, we prove the existence of extremal metrics for quadraticfunctional of scalar curvature on a closed 3-manifold which is served asan extension of the Yamabe problem on closed manifolds. Moreover, theexistence of extremal metrics on complete noncompact 3-manifolds willdiscuss elsewhere.

On the existence of extremal metrics on complete noncompact 3-manifolds

In this paper, the authors consider the 3-dimensional local Calabi flow on noncompact 3-manifolds with a complete metric of nonpositive scalar curvature and negative somewhere. Then long-time existence and asymptotic convergence of a subsequence of solutions of the flow are claimed. For its applications, it is proved that every noncompact 3-manifold admits a complete extremal metric. In particular, there exists a complete metric on a noncompact 3-manifold, a counterexample to the Yamabe problem, which is conformal to a complete extremal metric of non-constant scalar curvature.

On the existence of nontrivial extremal metrics on complete noncompact surfaces

In this paper, we consider the 2-dimensional local Calabi flow on a complete noncompact surface $(\Sigma ,g_{0})$ . Then, based on the Harnack-type estimate, we show the long-time existence and asymptotic convergence of a subsequence of solutions of such a flow on $(\Sigma ,g_{0})$ with $R_{0}\leq 0$ and $R_{0}$ bounded from above by a negative constant on a ball. For its applications, this will lead to the existence of extremal metrics on a complete noncompact surface of finite topological type. In particular, there exists an extremal metric of nonconstant Gaussian curvature on $\Sigma =\mathbf{H}^2$ or $\mathbf{R}^2.$

Global existence and convergence of solutions of the Calabi flow on Einstein 4-manifolds

In this paper, firstly, we show the Bondi-mass type estimate of solutions of Calabi flow on closed 4-manifolds. Secondly, in our applications, we obtain the long time existence on closed 4-manifolds. In particular, we are able to show the asymptotic convergence of a subsequence of solutions of the Calabi flow on closed Einstein 4-manifolds.

Calabi flow in Riemann surfaces revisited: a new point of view

In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we prove the long term existence and convergence of the Calabi flow. Thus give a new proof to Chruscial's theorem. The set of simple ideas of global integral estimates and concentration compactness should have further implications in other heat flow problems.

Compactness theorems and the Calabi flow on Kähler surfaces with stable tangent bundle

Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact Kahler surfaces with stable tangent bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime existence and asymptotic convergence for solutions of the Calabi flow on compact Kahler surfaces admitting no nonzero holomorphic tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up.

Global existence and convergence of solutions of Calabi flow on surfaces of genus $h\geq 2$

In this paper, based on a kind of Harnack estimate for the Calabi flow on surfaces, we show the longtime existence and convergence of solutions of 2-dimensional Calabi flow on surfaces $(\Sigma ,g_{0})$ of genus $h \geq 2$ with any arbitrary background metric $g_{0}$.

Cα-compactness and the Calabi flow on Kähler surfaces with negative scalar curvature.

Cα-compactness and the Calabi flow on Kähler surfaces with negative scalar curvature.

The Calabi flow with small initial energy

We show that on Kahler manifolds M with c_1(M)=0 the Calabi flow converges to a constant scalar curvature metric if the initial Calabi energy is sufficiently small. We prove a similar result on manifolds with c_1(M)<0 if the Kahler class is close to the canonical class.