In the present paper, we prove a stability theorem for the Kaehler Ricci flow near the infimum of the functional E_1 under the assumption that the initial metric has Ricci > -1 and |Riem| bounded. At present stage, our main theorem still need a topological assumption (1.2) which we hope to be removed in subsequent papers. The underlying moral is: if a Kaehler metric is sufficiently closed to a Kaehler Einstein metric, then the Kaehler Ricci flow converges to it. The present work should be viewed as a first step in a more ambitious program of deriving the existence of Kaehler Einstein metrics with an arbitrary energy level, provided that this energy functional has a uniform lower bound in this Kaehler class.
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.
We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.
We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.
We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to obtain a long time existence of the Kaehler-Ricci flow. In the second part of the paper we develope a so-called "moment" type estimate for the Kaehler-Ricci flow, through which we prove that the flow preserves several properties. In the last part we study the convergence of the Kaehler-Ricci flow.
In this paper, we announce the following results: Let M be a Kaehler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the Kähler-Ricci flow converges exponentially fast to a Kaehler-Einstein metric with constant bisectional curvature.
It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.
In this paper, we construct a set of new functionals of Ricci curvature on any Kaehler manifolds which are invariant under holomorphic transfermations in Kaehler Einstein manifolds and essentially decreasing under the Kaehler Ricci flow. Moreover, if the initial metric has non-negative bisectional curvature, using Tian's inequality, we can prove that each of the functionals has uniform lower bound along the flow which gives a set of integral estimates on curvature. Using this set of integral estimates, we are able to show the following theorem: Let M be a Kaehler-Einstein surface with positive…
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 apply the results from the article Cahen, Schwachhöfer: Special symplectic connections, to the case of Bochner-Kaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in $su(n,1)$. The relation between Sasaki and Bochner-Kaehler metrics in cone and transveral metrics constructions is discussed. The connection of the special symplectic and Weyl connections is outlined. The duality between the Ricci-type and Bochner-Kaehler metrics is shown.
In this paper, authors have established Chen's inequalities for the submanifolds of quaternionic Kaehler manifolds characterized by Ricci quarter-symmetric metric connection. Other than these inequalities, we have also derived generalized normalized Casorati curvature inequalities for the same submanifolds.
Let $g(t)$ with $t\in [0,T)$ be a complete solution to the Kaehler-Ricci flow: $\frac{d}{dt}g_{i\bar j}=-R_{i\bar j}$ where $T$ may be $\infty$. In this article, we show that the curvatures of $g(t)$ is uniformly bounded if the solution $g(t)$ is uniformly equivalet. This result is stronger than the main result in Šešum \cite{sesum} within the category of Kähler-Ricci flow.
For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.
We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conje…
We show a connection between the linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow and the monotonicity formula for the positive currents. As an application of the linear trace Li-Yau-Hamilton stated in this paper and the one proved by Chow-Hamiltonm, we give another proof on the classification of the limit solutions of the Kaehler-Ricci (Ricci) flow.
We study stability of non-compact gradient Kaehler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kaehler potential of the soliton will converge to the original soliton under Kaehler-Ricci flow as time tends to infinity. To obtain this result, we construct appropriate barriers and introduce an L^p-norm that decays for these barriers with non-negative Ricci curvature.
This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kaehler-Ricci solitonic metric if and only if it is Fano.
We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kaehler metrics constructed by M. B. Stenzel on the cotangent bundle of a compact, rank one, globally symmetric space.
In this paper we investigate the balanced condition (in the sense of Donaldson) and the existence of an Englis expansion for the LeBrun's metrics on $C^2$. Our first result shows that a LeBrun's metric on $C^2$ is never balanced unless it is the flat metric. The second one shows that an Englis expansion of the Rawnsley's function associated to a LeBrun's metric always exists, while the coefficient $a_3$ of the expansion vanishes if and only if the LeBrun's metric is indeed the flat one.
We study the convergence of a modified Kaeher-Ricci flow defined by Zhou Zhang. We show that the flow converges to a singular metric when the limit class is degenerate. This proves a conjecture of Zhang.
Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebraic possibility in this representation space can in fact be geometrically realized by a left-invariant Kaehler-Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.
In this paper we provide a local construction of a Sasakian manifold given a Kähler manifold. Obatined in this way manifold we call Sasakian lift of Kähler base. Almost contact metric structure is determined by the operation of the lift of vector fields - idea similar to lifts in Ehresmann connections. We show that Sasakian lift inherits geometry very close to its Kähler base. In some sense geometry of the lift is in analogy with geometry of hypersurface in Kähler manifold. There are obtained structure equations between corresponding Levi-Civita connections, curvatures and Ricci tensors of the…
Let M be an n-dimensional Kähler manifold with numerically effective Ricci class. In this note we prove that, if the first Betti number b_1(M)=2n, then M is biholomorphic to the complex torus T^n_C.
We determine an explicit expression for the Ricci tensor of a K-manifold, that is of a compact Kaehler manifold M with vanishing first Betti number, on which a semisimple group G of biholomorphic isometries acts with an orbit of codimension one. We also prove that the Kaehler form and the Ricci form of M are uniquely determined by two special curves with values in g = Lie(G), say Z, Z': R \to g = Lie(G) and we show how the curve Z' is determined by the curve Z. These results are used in another work with F. Podesta', where new examples of non-homogeneous compact Kaehler-Einstein manifolds with…
We exhibit families of non trivial (i.e. not Kaehler-Einstein) radial Kaehler-Ricci solitons (KRS), both complete and not complete, which can be Kaehler immersed into infinite dimensional complex space forms. This result shows that the triviality of a KRS induced by a finite dimensional complex space form proved in [12] does not hold when the ambient space is allowed to be infinite dimensional. Moreover, we show that the radial potential of a radial KRS induced by a non-elliptic complex space form is necessarily defined at the origin.
The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density).
We consider the Kaehler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kaehler manifold X. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in terms of cohomological data on X. We also give a sufficient condition for the singularity, if there is one, to be type-II.
Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs $(J,ω)$ are parametrized up to complex isomorphism (where $J$ is a complex structure and $ω$ is a symplectic structure). Such structure gives rise to a pseudo Riemannian metric $g$ for which $J$ is parallel. It is proved that most of these complex homogeneous spaces admit a pseudo Kähler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular Kähler Lie algebras are flat in dimension four. Ot…
We study Hamiltonian dynamics of gradient Kaehler-Ricci solitons that arise as limits of dilations of singularities of the Ricci flow on compact Kaehler manifolds. Our main result is that the underlying spaces of such gradient solitons must be Stein manifolds. Moreover, on all most all energy surfaces of the potential function f of such a soliton, the Hamiltonian vector field V_f of f, with respect to the Kaehler form of the gradient soliton metric, admits a periodic orbit. The latter should be impotant in the study of singularities of the Ricci flow on compact Kaehler manifolds in light of th…
We study G-invariant Kaehler metrics on G^C from the Hamiltonian point of view. As an application we show that there exist (GxG)-invariant Ricci-flat Kaehler metrics on G^C for any compact semisimple Lie group G.
A classical result of D. McDuff asserts that a simply-connected complete Kaehler manifold $(M,g,ω)$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $Ψ: M\rightarrow R^{2n}$ (where $n$ is the complex dimension of $M$), satisfying the following property (proved by E. Ciriza): the image $Ψ(T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $Ψ(p)=0$, is a complex linear subspace of $C^n \simeq R^{2n}$. The aim of this paper is to exhibit, for all positive integers $n$, examples of $n$-dimensional compl…
The intent of this short note is to provide context for and an independent proof of the discovery of Klaus Kroencke that complex projective space with its canonical Fubini--Study metric is dynamically unstable under Ricci flow in all complex dimensions N>1. The unstable perturbation is not Kaehler. This provides a counterexample to a well known conjecture widely attributed to Hamilton. Moreover, it shows that the expected stability of the subspace of Kaehler metrics under Ricci flow, another conjecture believed by several experts, needs to be interpreted in a more nuanced way than some may hav…
We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms and C-totally real submanifolds of Sasakian space forms.
In this note we show that any real exact G-invariant (1,1)-form is the Ricci form of a Kaehler metric on the complexification of an irreducible compact symmetric space G/K.
Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized Kähler-Ricci flow on a compact Kähler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2πc_{1}(M)$. If there is a constant $C$ independent of $t$ such that $$ \int_{M}|Rm(g(t))|^{n}dv_{t}\leq C,$$ then, for any $t_{k}\to \infty$, a subsequence of $(M, g(t_{k}))$ converges to a compact orbifold $(X, h)$ with only finite many singular points $\{q_{j}\}$ in the Gromov-Hausdorff sense, where $h$ is a Kähler metric on $X\backslash \{q_{j}\}$ satisfying the Kähler-Ricci soliton equation, i.e. there is a smooth function $…
In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds.
We study the almost Kaehler geometry of adjoint orbits of non-compact real semisimple Lie groups endowed with the Kirillov-Kostant-Souriau symplectic form and a canonically defined almost complex structure. We give explicit formulas for the Chern-Ricci form, the Hermitian scalar curvature and the Nijenhuis tensor in terms of root data. We also discuss when the Chern-Ricci form is a multiple of the symplectic form, and when compact quotients of these orbits are of Kaehler type.
In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost Kähler - Ricci geometric evolution equations. The se…
We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter ~ e^{-t/2}. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with …
Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 Φ$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.
Let (M,J) be a compact complex 2-manifold which which admits a Kaehler metric for which the integral of the scalar curvature is non-negative. Also suppose that M does not admit a Ricci-flat Kähler metric. Then if M is blown up at sufficiently many points, the resulting complex surface admits Kaehler metrics with scalar curvature identically equal to zero.
Given a compact constant scalar curvature Kaehler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kaehler Ricci-flat resolution, we find sufficient conditions on the position of the singular points to ensure the existence of a global constant scalar curvature Kaehler desingularization. This generalizes the results previously obtained by the first author with F. Pacard. A series of explicit examples is discussed.
If a smooth compact 4-manifold M admits a Kaehler-Einstein metric g of positive scalar curvature, Gursky showed that its conformal class [g] is an absolute minimizer of the Weyl functional among all conformal classes with positive Yamabe constant. Here we prove that, with the same hypotheses, [g] also minimizes of the Weyl functional on a different open set of conformal classes, most of which have negative Yamabe constant. An analogous minimization result is then proved for Einstein metrics g which are Hermitian, but not Kaehler.
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…
We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled S…
We prove a general result about the short time existence and uniqueness of second order geometric flows transverse to a Riemannian foliation on a compact manifold. Our result includes some flows already existing in literature, as the transverse Ricci flow, the Sasaki-Ricci flow and the Sasaki J-flow and motivates the study of other evolution equations. We also introduce a transverse version of the Kaehler-Ricci flow adapting some classical results to the foliated case.
We address some aspects of four dimensional chiral N=1 supersymmetric theories on which the scalar manifold is described by Kähler geometry and can further be viewed as Kähler-Ricci soliton generating a one-parameter family of Kähler geometries. All couplings and solutions, namely the BPS domain walls and their supersymmetric Lorentz invariant vacua turn out to be evolved with respect to the flow parameter related to the soliton. Two models are discussed, namely N=1 theory on Kähler-Einstein manifold and U(n) symmetric Kähler-Ricci soliton with positive definite metric. In the first case we fi…
We show that, in toric Kaehler geometry, the sign of the Ricci curvature corresponds exactly to convexity properties of the volume functional. We also discuss analogous relationships in the more general context of quasi-homogeneous manifolds, and existence results for minimal Lagrangian submanifolds.
We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced.
In this paper, we show that the uniform L^4-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular transverse Fano Sasakian (2n+1)-manifold M. When M is dimension up to seven and the space of leaves of the characteristic foliation is well-formed, we first show that any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton on the limit space which is a S^1-orbibundle over the unique singular Kaehler-Ricci soliton on a normal projective variety with codimension two orbifold singularitie…
As seen in the works of Calabi, Cheng-Yau and Loftin, affine sphere equations have a close relationship with Kaehler-Einstein metrics. The main purpose of this note is to show that an equation analogous to those of hyperbolic affine spheres arises naturally from Kaehler-Einstein metrics on Einstein toric surfaces. The case for the remaining toric surfaces with Kaehler-Ricci solitons will also be discussed.
We introduce the notion of a hamiltonian 2-form on a Kaehler manifold and obtain a complete local classification. This notion appears to play a pivotal role in several aspects of Kaehler geometry. In particular, on any Kaehler manifold with co-closed Bochner tensor, the (suitably normalized) Ricci form is hamiltonian, and this leads to an explicit description of these Kaehler metrics, which we call weakly Bochner-flat. Hamiltonian 2-forms draw attention to a general construction of Kaehler metrics which interpolates between toric geometry and Calabi-like constructions of metrics on (projective…
In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the Kähler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functionals in case of Kaehler manifolds. On Kähler-Einstein manifold with positive scalar curvature, if the initial metric has positive bisectional curvature, we prove that these functionals have a uniform lower bound, via the effective use of Tian's inequality. Consequently, we prove …
Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\in \RR$, we give a sharp estimate of the upper bound of $ρ(x)=\dis(x, \partial M)$, in terms of the mean curvature bound of the boundary. When $\partial M$ is compact, the upper bound is achieved if and only if $M$ is isometric to a disk in space form. A Kaehler version of estimation is also proved. Moreover we prove a Laplace comparison theorem for distance function to the boundary of Kaehler manifold and also estimate the first eigenvalue of th…
An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric $g_F$. This paper contains two results. In the first one we prove that if $g_F$ is an extremal Kaehler metric then $(D_F, g_F)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field $X$ on $D_F$ such that $(g_F, X)$ is a Kaehler-Ricci soliton.
In this paper, we generalize Chen-Tian energy functionals to Kähler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of Kähler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu's holomorphic invariant.
