We prove that on a Kähler manifold admitting an extremal metric $ω$ and for any Kähler potential $φ_0$ close to $ω$, the Calabi flow starting at $φ_0$ exists for all time and the modified Calabi flow starting at $φ_0$ will always be close to $ω$. 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 $ω$ exponentially fast.
In this note we give a characterization of Kaehler metrics which are both Calabi extremal and Kaehler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.
We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains to the case of complex dimension 2, where it plays a key role in construction of Bach-flat metrics on appropriate 4-manifolds.
In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.
We apply a local differential geometric framework from Kähler toric geometry to (re)construct Calabi's extremal Kähler metrics on $\bbC\bbP^n$ blown-up at a point from data on the moment polytope.
In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi-Yau manifolds and a compact metric space in the Gromov-Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat Kähler metrics on Calabi-Yau manifolds along a smoothing is establi…
In this paper we extend recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of constant scalar Kähler metric on a compact Kähler manifold to Calabi's extremal metric. Our argument follows \cite{CC3} and there are no new a prior estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector $V$ if and only if the modified Mabuchi energy is proper, modulo the action the subgroup in the identity component of automorphism group which commutes with the flow of $V$. We introduce two essen…
We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.
We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.
This paper is a survey of some recent progress on the study of Calabi's extremal Kähler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then turn to the question of what one expects when no extremal metric exists.
Using the short time existence of the Calabi flow, we prove that any extremal Kaehler metric on a product toric variety is a product extremal Kaehler metric.
In this short note, we prove that a Calabi extremal Kaehler-Ricci soliton on a compact toric Kaehler manifold is Einstein. This solves for the class of toric manifolds a general problem stated by the authors that they solved only under some curvature assumptions.
In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an example where the modified K energy is bounded from below but not proper.
We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.
Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of the Calabi flow $φ(t)$ on $X$ is uniformly bounded in $[0, T)$ if the Sobolev constant of $φ(t)$ 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 Kähler 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 Kähler manifold.
From Calabi's extremal metrics to scalar-flat Kähler cones
Université du Québec à Montréal（蒙特利尔大学）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；University of Alberta（阿尔伯塔大学）
Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $ \int_{\partial P} u ~ d σ< C_1, $ then there exists a constant $C_2$ depending only on $C_1$ and $P$ such that $\max_P u < C_2$. As an application, we show that if $(X,P)$ is analytic uniform $K$-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Riemannian curvature is uniformly bounded along the Calabi flow. Also we provide a proof of a conjecture of Donaldson. Finally,…
We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with the conformal extremal metric on the regular $2n$-gon. The hexagon was considered by Calabi. The region covered by the maximal number $n$ of geodesics bands extends over most of the surface and exhibits positive curvature. As $n\to \infty$ the metric, away from the boundary, approaches the well-known round extremal metric on $\mathbb{RP}_2$. We extend Calabi'…
Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $\int_{\partial P} u ~ d σ< C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$. Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson's $M$-condition holds for $u$. As an application, we show that if $(X,P)$ is (analytic) relative $K$-stable, then the modified Calabi flow converges t…
We study the stability of compact pseudo-Kähler manifolds, i.e. compact complex manifolds $X$ endowed with a symplectic form compatible with the complex structure of $X$. When the corresponding metric is positive-definite, $X$ is Kähler and any sufficiently small deformation of $X$ admits a Kähler metric by a well-known result of Kodaira and Spencer. We prove that compact pseudo-Kähler surfaces are also stable, but we show that stability fails in every complex dimension $n\geq 3$. Similar results are obtained for compact neutral Kähler and neutral Calabi-Yau manifolds. Finally, motivated by a …
Let $(M,g)$ be a compact Kähler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the Kähler form $ω_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell Kähler metric if the conformal metric $\tilde g = f^{-2}g$ has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kähler manifold…
We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising sequence.
Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is derived. A construction, due to Calabi, of a 1-parameter family of extremal metrics of non-constant scalar curvature is recast very simply …
We introduce a new geometric flow of Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. The flow depends on the choice of a background metric, it always reduces to a scalar equation and preserves some special classes of Hermitian structures, as balanced and Gauduchon metrics. We show that the flow has always a unique short-time solution and we provide a stability result when the background metric is Kaehler-Einstein with nonpositive scalar curvature.
We prove the longtime existence and convergence of the Calabi flow on toric Fano surfaces in a large family of Kahler classes where the class has positive extremal Hamiltonian potential and the initial Calabi energy is bounded by some constant. This is an extension of our previous work. We use the toric condition in a more essential way to rule out bubbles.
In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach-Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein-Maxwell equations in general relativity. Inspired by the work of C. LeBrun on Einstein-Maxwell equations on compact Kaehler surfaces, we give a variational characterization of solutions to Bach-Merkulov equations as critical points of the Weyl functional. We also show that extremal Kaehler metrics are solutions to these equations, although, contrary to the Einstein-Maxwel…
In this paper, we study rotationally symmetric extremal Kähler metrics on $\mathbb C^n$ ($n\geq 2$) and $\mathbb C^2 \backslash\{0\}$. We present a classification of such metrics based on the zeros of the polynomial appearing in Calabi's Extremal Equation. As applications, we prove that there are no $U(n)$ invariant complete extremal Kähler metrics on $\mathbb C^n$ with positive bisectional curvature, and we give a smooth extension lemma for $U(n)$ invariant extremal Kähler metrics on $\mathbb{C}^n\backslash\{0\}$. We retrieve known examples of smooth or singular extremal Kähler metrics on Hir…
We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics arise as transversal Kahler structures of Sasaki metrics compatible with the same CR structure and having commuting Sasaki--Reeb vector fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kahler metric, illuminating and uniting several explicit constructions in Kahler and Sasaki geometry. It also…
We establish a regularity result for the metric on any 4-dimensional extremal Kähler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity $L^2(|\Riem|)$ in a surrounding ball is sufficiently small compared to the pointwise norm of its scalar curvature. Consequently sequences of 4-dimensional extremal Kähler metrics with uniformly bounded Calabi energies and scalar curvature have convergent subsequences in the Gromov-Hausdorff topology. Gromov-Hausdorff limits are length spaces with the structure of …
Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.
We define K-stability of a polarized Sasakian manifold relative to a maximal torus of automorphisms. The existence of a Sasaki-extremal metric in the polarization is shown to imply that the polarization is K-semistable. Computing this invariant for the deformation to the normal cone gives an extention of the Lichnerowicz obstruction, due to Gauntlett, Martelli, Sparks, and Yau, to an obstruction of Sasaki-extremal metrics. We use this to give a list of examples of Sasakian manifolds whose Sasaki cone contains no extremal representatives. These give the first examples of Sasaki cones of dimensi…
Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics on each factor, provided that the integral Futaki invariants of the polarized manifold vanish or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau about the splitting of a Kaehler-Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kaehler metrics.
We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact moduli space, are proven to be K-stable although it is well known that they are \textit{not} necessarily asymptotically (semi)stable. As a consequence, we have orbifold counterexamples, to the folklore conjecture "K-stability implies asymptotic stability". They have Kahler-Einstein (orbifold) metrics so the result of Donaldson does not hold for orbifolds.
In this paper, improving a preceding work, we obtain asymptotic polybalanced kernels associated to extremal Kaehler metrics on polarized algebraic manifolds. As a corollary, we have a stronger asymptotic relative Chow-polystability for extremal Kaehler polarized algebraic manifolds. Finally, related to the Yau-Tian-Donaldson Conjecture for extremal Kaehler metrics, we shall discuss the difference between strong relative K-stability and relative K-stability.
We prove that the existence of extremal metrics implies asymptotically relative Chow stability. An application of this is the uniqueness, up to automorphisms, of extremal metrics in any polarization.
We obtain a structure theorem for the group of holomorphic automorphisms of a conformally Kähler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the Kähler-Einstein, cscK, and extremal Kähler cases. Combined with previous results of LeBrun, Apostolov-Maschler and Futaki-Ono, this completes the classification of the conformally Kähler, Einstein--Maxwell metrics on $\mathbb{CP}^1 \times \mathbb{CP}^1$. We also use our result in order to introduce a (relative) Mabuchi energy in the more general context of $(K, q, a)$-extremal Kähler metrics in a …
In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polarisation class. A variant for a complete extremal metric on the complement of a smooth divisor is also given. On toric surfaces we prove a Jordan-Holder type theorem for decomposing semistable surfaces into stable pieces. On a ruled surface we compute the infimum of the Calabi funct…
The Einstein-Maxwell equations on a smooth compact 4-manifold are reformulated as a purely Riemannian variational problem analogous to Calabi's variational problem for extremal Kahler metrics. Next, Seiberg-Witten theory is used to show that these two problems are in fact intimately related. Extremal Kahler metrics are then used to probe the limits of Seiberg-Witten curvature estimates. The article then concludes with a brief survey of some recent results on extremal Kahler metrics.
The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau's conjecture, which originated from Calabi's conjecture, asks whether ``stability'' and ``existence of extremal metrics'' for polarized manifolds are equivalent. In this note, the recent progress of this subject, by Donaldson, Tian and our grou…
In this paper, by generalizing the concept of balanced metrics, we shall show that Donaldson's asymptotic approximation of balanced metrics for constant scalar curvature cases can be generalized to extremal Kaehler cases.
We first give a precise statement on the short time existence of the Calabi flow and prove a stability result: any metric near a constant scalar curvature metric will flow to this cscK metric exponentially fast. Secondly, we prove that a compactness theorem in space of the kahler metrics given unifrom 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-sigularity result about a weak constant scalar curveture metric in a punctured ball. One of the main difficulties is that the metr…
We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on Kähler surface we show that any "maximal bubble" has to be a scalar flat ALE Kähler metric. In some certain classes on toric Fano surface, the Sobolev constant is a priori bounded along the Calabi flow with small Calabi energy. Also we can show in certain case no maximal 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 r…
An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive type implies uniform K-stability and we point out the existence of a formal solution of the Abreu equation for any angle along the invariant divisor. Applying the recent result of Chen--Cheng [10] and He [27], we conclude that the existence of a compatible extremal toric almost K…
We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metrics on complex projective space $\mathbb{C}P^{n}$ is exhibited. We derive an analogous bound in the case when the metric is extremal and a detailed study is made of the accuracy of the bound in the case of Calabi's extremal metrics on $\mathbb{C}P^{2}\sharp -\mathbb{C}P^{2}$.
We study the existence of weighted extremal Kähler metrics in the sense of Apostolov-Calderbank-Gauduchon-Legendre and Lahdili on the total space of an admissible projective bundle over a Hodge Kähler manifold of constant scalar curvature. Admissible projective bundles have been defined by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, and they include the projective line bundles (Hwang-Singer) and their blow-downs (Koiso-Sakane), thus providing a most general setting for extending the existence theory for extremal Kähler metrics pioneered by a seminal construction of Calabi. We obtain a ge…
We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.
Conformally Kähler, Einstein-Maxwell metrics and $f$-extremal metrics are generalization of canonical metrics in Kähler geometry. We introduce uniform K-stability for toric Kähler manifolds, and show that uniform K-stability is necessary condition for the existence of $f$-extremal metrics on toric manifolds. Furthermore, we show that uniform K-stability is equivalent to properness of relative K-energy.
