Holomorphic Bisectional Curvature Research Articles

A Matrix Li–Yau–Hamilton Estimate for the Green Function on Kähler Manifolds

Abstract We prove a matrix Li–Yau–Hamilton inequality for the Green function on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. This estimate is an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on Kähler manifolds. It is also the complex counterpart of the matrix estimate on Riemannian manifolds obtained previously by the author.

ON THE EMBEDDING OF LEVI-FLAT HYPERSURFACES IN THE COMPLEX PROJECTIVE PLANE (AND AN APPENDIX WITH L´ASZL´O LEMPERT)

Let L be a hypothetical smooth Levi flat hypersurface in CP2 and r the signed distance to L by means of the Fubini-Study metric g. Denote Lru = cru the second order elliptic equation for the infinitesimal Levi-flat deformations of L, where cr = dbJbr + br ∧ Jbr, br = ιXrdγr, Xr = gradgr/ ∥gradgr∥2 g, γr is the restriction of dcr to L and db is the differentiation along the leafs of the Levi foliation. Then −cr ≥ H as leaf-wise (1, 1)-forms, where H is the holomorphic bisectional curvature of CP2. We give also an example of a Levi-flat manifold L of dimension 3 verifying that there exists a (1, 0)-form α on L such that ∂α is a K¨ahler form on every leaf of the Levi foliation, but L is not embeddable in CP2.

Curvatures of strongly convex Kähler-Finsler manifolds

In this article, we study curvatures on a strongly convex (weakly) Kähler-Finsler manifold. First, we prove that the holomorphic sectional curvature is just half of the flag curvature in a holomorphic plane section on a strongly convex weakly Kähler-Finsler manifold. Second, we compare curvatures associated to the Rund connection with curvatures associated to the Chern-Finsler connection or the complex Berward connection on a strongly convex Kähler-Finsler manifold. Finally, we discuss relationships between flag curvatures and holomorphic bisectional curvatures, and compare two kinds of S-curvatures on a strongly convex Kähler-Finsler manifold.

Bergman metric on the symmetrized bidisk and its consequences

On the symmetrized bidisk [Formula: see text] with the Bergman metric, the holomorphic sectional curvature is negatively pinched and the holomorphic bisectional curvature is not. The consequences of invariant metrics are provided.

Curvature of the base manifold of a Monge–Ampère fibration and its existence

In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge–Ampère equation, which is called a Monge–Ampère fibration. There exist two canonical types of generalized Weil–Petersson metrics on the base complex manifold of the fibration. For the second generalized Weil–Petersson metric, we obtain an explicit curvature formula and prove that the holomorphic bisectional curvature is non-positive, the holomorphic sectional curvature, the Ricci curvature, and the scalar curvature are all bounded from above by a negative constant. For a holomorphic vector bundle over a compact Kähler manifold, we prove that it admits a projectively flat Hermitian structure if and only if the associated projective bundle fibration is a Monge–Ampère fibration. In general, we can prove that a relative Kähler fibration is Monge–Ampère if and only if an associated infinite rank Higgs bundle is Higgs-flat. We also discuss some typical examples of Monge–Ampère fibrations.

Construction of negatively curved complete intersections

Using the Donaldson–Auroux theory, we construct complete intersections in complex projective manifolds, which are negatively curved in various ways. In particular, we prove the existence of compact simply connected Kähler manifolds with negative holomorphic bisectional curvature. We also construct hyperbolic hypersurfaces, and we obtain bounds for their Kobayashi hyperbolic metric.

On projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. Specifically, we prove that $X$ admits a locally trivial morphism $\phi:X \to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite \'etale cover $A \to Y$ by an abelian variety $A$, by combining the author's previous work with the theory of holomorphic foliations. Moreover, we show that the universal cover of $X$ is biholomorphic and isometric to the product $\mathbb{C}^m \times F$ of the universal cover $\mathbb{C}^m$ of $Y$ with a flat metric and the rationally connected fiber $F$ with a K\"ahler metric whose holomorphic sectional curvature is semi-positive.

On Lightlike Submersion of Radical Transversal Lightlike Submanifolds of a Kaehler Manifold

The aim of present study is to introduce the idea of a lightlike submersion onto an indefinite almost Hermitian manifold from a radical transversal lightlike submanifold of an indefinite Kaehler manifold. We show that if an indefinite almost Hermitian manifold B admits a lightlike submersion φ : K → B of a radical transversal lightlike submanifold K of an indefinite Kaehler manifold then B must be an indefinite Kaehler manifold. We also establish the relation between the holomorphic sectional and holomorphic bisectional curvature of Κ and B.

On the Applications of Bochner-Kodaira-Morrey-Kohn Identity

This paper is devoted to studying some applications of the Bochner-Kodaira-Morrey-Kohn identity. For this study, we define a condition which is called (Hq) condition which is related to the Levi form on the complex manifold. Under the (Hq) condition and combining with the basic Bochner-Kodaira-Morrey-Kohn identity, we study the L2 ∂ Cauchy problems on domains in ℂn, Kähler manifold and in projective space. Also, we study this problem on a piecewise smooth strongly pseudoconvex domain in a complex manifold. Furthermore, the weighted L2 ∂ Cauchy problem is studied under the same condition in a Kähler manifold with semi-positive holomorphic bisectional curvature. On the other hand, we study the global regularity and the L2 theory for the ∂-operator with mixed boundary conditions on an annulus domain in a Stein manifold between an inner domain which satisfy (Hn−q−1) and an outer domain which satisfy (Hq).

Comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces

We give an analogue of triangle comparison for Kähler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov–Hausdorff limits. We discuss tangent cones and singular Kähler spaces.

On the Dehn functions of Kähler groups

We address the problem of which functions can arise as Dehn functions of Kähler groups. We explain why there are examples of Kähler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a Kähler group which has Dehn function bounded below by a cubic function and above by $n^6$. As a consequence we obtain that for a compact Kähler manifold having non-positive holomorphic bisectional curvature does not imply having quadratic Dehn function.

Canonical complex extensions of Kähler manifolds

Given a complex manifold $X$, any K\"ahler class defines an affine bundle over $X$, and any K\"ahler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact K\"ahler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Sz\H{o}ke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non-negative holomorphic bisectional curvature and big tangent bundle.

RC-POSITIVITY, VANISHING THEOREMS AND RIGIDITY OF HOLOMORPHIC MAPS

AbstractLet $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

Some results on normal GCR-lightlike submanifolds of indefinite nearly Kaehler manifolds

The purpose of this paper is to study normal [Formula: see text]-lightlike submanifolds of indefinite nearly Kaehler manifolds. We find some necessary and sufficient conditions for an isometrically immersed [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold to be a normal [Formula: see text]-lightlike submanifold. Further, we derive a characterization theorem for holomorphic bisectional curvature of a normal [Formula: see text]-lightlike submanifold of an indefinite nearly Kaehler manifold.

On the Kähler-Einstein metric at strictly pseudoconvex points

ABSTRACTWe prove a boundary regularity result for the complete Kähler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary points, and we deduce the asymptotic behavior of their holomorphic bisectional curvatures near such points. Finally, we prove that the same asymptotic behavior holds at boundary points at which the squeezing function tends to one.

An extension of a theorem of Paul Yang on negatively pinched curvature

We prove that there is no complete Hermitian metric h with bounded torsion on polydisc $$D^{n}$$ such that: the holomorphic bisectional curvature of h is bounded above by a negative constant and the second Ricci curvature of h is bounded below by another negative constant. On the other hand, we also present some interesting examples of domains, which are neither biholomorphic to product nor to homogeneous manifolds, that do not admit such Hermitian metrics.

On the curvature of Einstein–Hermitian surfaces

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein–Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini–Study metric up to rescaling. This result relaxes the Kähler condition in Berger’s theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.

Comparison Theorems and Their Applications on Kähler Finsler Manifolds

We obtain complex Hessian and Laplacian comparison theorems on Kahler Finsler manifolds with holomorphic bisectional curvature bounded from below. As applications, we prove Bishop type volume comparison theorem and Myers type theorem for Kahler Finsler manifolds. Moreover, we also obtain some comparison theorems for the first eigenvalue of Finsler–Laplacian and give a sharp upper estimate of the bottom of the spectrum for Kahler Finsler manifolds whose holomorphic bisectional curvature is bounded from below by $$-\,1$$.

Biharmonic holomorphic maps into Kähler manifolds

We study biharmonic holomorphic maps from an almost Hermitian manifold into a Kähler manifold. First, by a simple observation of the curvature term in the biharmonic equation, we establish non-existence results of biharmonic holomorphic maps into Kähler manifolds with non-positive holomorphic bisectional curvature, which extend the similar results of biharmonic maps between Riemannian manifolds. Second, by applying the second variation formula of biharmonic maps, we prove a non-existence result of stable biharmonic holomorphic maps into complex projective space.

Longtime existence of the Kähler-Ricci flow on ℂⁿ

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on C n \mathbb {C}^n without assuming the initial metric has bounded curvature, thus extending results in an earlier work of the authors. We prove the existence of a longtime bounded curvature solution emerging from any complete U ( n ) U(n) -invariant Kähler metric with non-negative holomorphic bisectional curvature, and that the solution converges as t → ∞ t\to \infty to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on C n \mathbb {C}^n which are not necessarily U ( n ) U(n) -invariant.