Kähler-Einstein Metrics and the Calabi Conjecture

We study the construction and classification of weakly Bochnerflat (WBF) metrics (i.e., Kähler metrics with coclosed Bochner tensor) on compact complex manifolds.A Kähler metric is WBF if and only if its 'normalized' Ricci form is a hamiltonian 2-form: such 2-forms were introduced and studied in previous papers in the series.It follows that WBF Kähler metrics are extremal.We construct many new examples of WBF metrics on projective bundles and obtain a classification of compact WBF Kähler 6-manifolds, extending work by the first three authors on weakly selfdual Kähler 4-manifolds.The constructions are independent of previous papers in the series, but the classification relies on the classification of compact Kähler manifolds with a hamiltonian 2-form [3].

We prove that if $$M^n(n\ge 4)$$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $$R_0>\sigma _{n}K_{\max }$$ , where $$\sigma _n\in (\frac{1}{4},1)$$ is an explicit positive constant depending only on $$n$$ , then $$M$$ must be isometric to a spherical space form. Moreover, we prove that if an $$n(\ge {\!\!4})$$ -dimensional compact Einstein manifold satisfies $$K_{\min }\ge \eta _n R_0,$$ where $$\eta _n\in (\frac{1}{4},1)$$ is an explicit positive constant, then $$M$$ is locally symmetric. It should be emphasized that the pinching constant $$\eta _n$$ is optimal when $$n$$ is even. We then obtain some rigidity theorems for Einstein manifolds under $$(n-2)$$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $$M$$ is an $$n(\ge {\!\!4})$$ -dimensional compact Einstein submanifold in the simply connected space form $$F^{N}(c)$$ with constant curvature $$c\ge 0$$ , and the normalized scalar curvature $$R_0$$ of $$M$$ satisfies $$R_0>\frac{A_n}{A_n+4n-8}(c+H^2),$$ where $$A_n=n^3-5n^2+8n$$ , and $$H$$ is the mean curvature of $$M$$ , then $$M$$ is isometric to a standard $$n$$ -sphere.

Two well-known questions in differential geometry are “Does every compact manifold of dimension greater than four admit an Einstein metric?” and “Does an Einstein metric of a negative scalar curvature exist on a sphere?” We demonstrate that these questions are related: For everyn≥5 the existence of metrics for which the deviation from being Einstein is arbitrarily small on every compact manifold of dimensionn (or even on every smooth homology sphere of dimensionn) implies the existence of metrics of negative Ricci curvature on the sphereS n for which the deviation from being Einstein is arbitrarily small. Furthermore, assuming either a version of the Palais-Smale condition or the plausible looking existence of an algorithm deciding when a given metric on a compact manifold is close to an Einstein metric, we show for anyn≥5 that: 1) If everyn-dimensional smooth homology sphere admits an Einstein metric thenS n admits infinitely many Einstein structures of volume one and of negative scalar curvature; 2) If every compactn-dimensional manifold admits an Einstein metric then every compactn-dimensional manifold admits infinitely many distinct Einstein structures of volume one and of negative scalar curvature.

Let L ¯ i →X i be a holomorphic line bundle over a compact complex manifold for i=1,2. Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S=S 1 ×S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant (ℂ * ) n i -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X i are (generalized) flag varieties and L ¯ i negative ample line bundles over X i . When H 1 (X 1 ;ℝ)=0 and c 1 (L ¯ 1 )∈H 2 (X 1 ;ℂ) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for H q (S;𝒪 S ) when X i are projective manifolds, L ¯ i ∨ are very ample and the cone over X i with respect to the projective imbedding defined by L ¯ i ∨ are Cohen-Macaulay. We obtain applications to the Picard group of S. When X i =G i /P i where P i are maximal parabolic subgroups and S is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on S is purely transcendental over ℂ.

In this paper we extend the Cheeger–Colding–Tian theory to the conic Kahler–Einstein metrics. In general, there are no smooth approximations of a family of conic Kahler–Einstein metrics with Ricci curvature uniformly bounded from below. So we have to deal with the technical issues to extend the original arguments.

Weak almost contact metric manifolds (i.e., the complex structure is replaced by a nonsingular skew-symmetric tensor), defined by the author and R. Wolak, allow a new look at the classical theory and find novel applications in mathematics and physics. An important case of these manifolds, which is locally a twisted product, is a weak $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -Kenmotsu manifold defined by the author and D.S. Patra. In the paper, the concept of the $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> -Ricci tensor is adapted to weak almost contact manifolds, the interaction of the $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> - $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> -Ricci soliton with the weak $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -Kenmotsu structure is studied and new characteristics of Einstein metrics are obtained.

A Weyl manifold is a conformal manifold equipped with a torsion free connection preserving the conformal structure, called a Weyl connection. It is said to be Einstein-Weyl if the symmetric tracefree part of the Ricci tensor of this connection vanishes. In particular, if the connection is the Levi-Civita connection of a compatible Riemannian metric, then this metric is Einstein. Such an approach has two immediate advantages: firstly, the homothety invariance of the Einstein condition is made explicit by focusing on the connection rather than the metric; and secondly, not every Weyl connection is a Levi-Civita connection, and so Einstein-Weyl manifolds provide a natural generalisation of Einstein geometry. The simplest examples of this generalisation are the locally conformally Einstein manifolds. A Weyl connection on a conformal manifold is said to be closed if it is locally the Levi-Civita connection of a compatible metric; but it need not be a global metric connection unless the manifold is simply connected. Closed EinsteinWeyl structures are then locally (but not necessarily globally) Einstein, and provide an interpretation of the Einstein condition which is perhaps more appropriate for multiply connected manifolds. For example, S1×Sn−1 admits flat Weyl structures, which are therefore closed Einstein-Weyl. These closed structures arise naturally in complex and quaternionic geometry. Einstein-Weyl geometry not only provides a different way of viewing Einstein manifolds, but also a broader setting in which to look for and study them. For instance, few compact Einstein manifolds with positive scalar curvature and continuous isometries are known to have Einstein deformations, yet we shall see that it is precisely under these two conditions that nontrivial Einstein-Weyl deformations can be shown to exist, at least infinitesimally. The Einstein-Weyl condition is particularly interesting in three dimensions, where the only Einstein manifolds are the spaces of constant curvature. In contrast, three dimensional Einstein-Weyl geometry is extremely rich [16, 68, 72], and has an equivalent formulation in twistor theory [34] which provides a tool for constructing selfdual four dimensional geometries. In section 10, we shall discuss a construction relating Einstein-Weyl 3-manifolds and hyperKahler 4-manifolds [40, 29, 50, 79]. Twistor methods also yield complete selfdual Einstein metrics of negative scalar curvature with prescribed conformal infinity [48, 35]. An important special case of this construction is the case of an Einstein-Weyl conformal infinity [34, 61]. Although Einstein-Weyl manifolds can be studied, along with Einstein manifolds, in a Riemannian framework, the natural context is Weyl geometry [23]. We

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kahler metric onG/T does not have volume-preserving geodesic symmetries. From the Kahler/de Rham decomposition of a compact homogeneous Kahler manifold [8], our result extends to the invariant Kahler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kahler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.

We establish the unique complete Kähler–Einstein metric with negative scalar curvature on a broad class of complete Kähler manifolds, including those manifolds whose covering space can be biholomorphically embedded into a Kähler manifold with holomorphic sectional curvature bounded above by a negative constant. We further present several new examples of complete noncompact Kähler–Einstein manifolds, generated by the results.

A Cartan manifold is a smooth manifold M whose slit cotangent bundle 0 * M T is endowed with a regular Hamiltonian K which is positively homogeneous of degree 2 in momenta. The Hamiltonian K defines a (pseudo)-Riemannian metric ij g in the vertical bundle over 0 * M T and using it, a Sasaki type metric on 0 * M T is constructed. A natural almost complex structure is also defined by K on 0 * M T in such a way that pairing it with the Sasaki type metric an almost Kahler structure is obtained. In this paper we deform ij g to a pseudo-Riemannian metric ij G and we define a corresponding almost complex Kahler structure. We determine the Levi-Civita connection of G and compute all the components of its curvature. Then we prove that if the structure

Given a minimal Lagrangian submanifold L in a negative Kaehler--Einstein manifold M, we show that any small Kaehler--Einstein perturbation of M induces a deformation of L which is minimal Lagrangian with respect to the new structure. This provides a new source of examples of minimal Lagrangians. More generally, the same is true for the larger class of totally real J-minimal submanifolds in Kaehler manifolds with negative definite Ricci curvature.

This chapter focuses on Einstein metrics which are Hermitian with respect to some integrable complex structure on a compact complex surface. These will, for the sake of brevity, sometimes be referred to as Einstein Hermitian metrics, so it is worth warning the reader that these are not a priori Hermite-Einstein in the sense of the theory of holomorphic vector bundles. The chapter explores a local result concerning the conformal curvature of Einstein Hermitian metrics. The identity component of the isometry group of an extremal Kahler metric is a maximal compact subgroup of the identity component of the complex automorphism group; and since the maximal compact is unique up to conjugation, a suitable change of homogeneous coordinates will make the extremal Kahler metric g invariant under the torus action. An extremal Kahler metric in the anti-canonical class would have to be a Kahler-Einstein metric because the relevant Futaki invariant vanishes, and so coincide with the metric of Siu.

OF THE DISSERTATION Some parabolic and elliptic problems in complex Riemannian geometry by Bin Guo Dissertation Director: Professor Jian Song This dissertation consists of three parts, the first one is on the blow-up behavior of Kahler Ricci flow on CP blown-up at one point, and the second one on the convergence of Kahler Ricci flow on minimal projective manifolds of general type, and the last one is on the existence of canonical conical Kahler metrics on toric manifolds. In the first part, we consider the Ricci flow on CP blown-up at one point starting with any rotationally symmetric Kahler metric. We show that if the total volume does not go to zero at the singular time, then any parabolic blow-up limit of the Ricci flow along the exceptional divisor is a non-compact complete shrinking Kahler Ricci soliton with rotational symmetry on Cn blown-up at one point, hence the FIK soliton constructed in [36]. In the second part, we consider the Kahler Ricci flow on a smooth minimal model of general type, following the ideas of Song ([76, 77]), we show that if the Ricci curvature is uniformly bounded below along the Kahler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the flow is homeomorphic to the canonical model of the manifold. Moreover, we will give a purely analytic proof of a recent result of Tosatti-Zhang ([102]) that if the canonical line bundle KX is big and nef, but not

It is a basic tenet in complex geometry that {\it negative} curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while {\it positive} curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifold $M$ with a Kahler metric with positive total scalar curvature is uniruled, which is equivalent to every point of $M$ being contained in a rational curve. We also prove that if $M$ possesses a Kahler metric of total scalar curvature equal to zero, then either $M$ is uniruled or its canonical line bundle is torsion. The proof of the latter theorem is partially based on the observation that if $M$ is not uniruled, then the total scalar curvatures of all Kahler metrics on $M$ must have the same sign, which is either zero or negative.

Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold Z is equipped with a canonical holomorphic foliation F which is equivariant with respect to the (C*)^m-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi-Eckmann manifolds, and their deformations. We construct transversely Kaehler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kaehler submanifold (or, more generally, a Fujiki class C subvariety) in such a moment-angle manifold is contained in a leaf of the foliation F. For a generic moment-angle manifold Z in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension. This implies, in particular, that the algebraic dimension of Z is zero.