Positive Bisectional Curvature Research Articles

Diameter rigidity for Kähler manifolds with positive bisectional curvature

We prove that a Kähler manifold with positive bisectional curvature and maximal diameter is isometric to complex projective space with the Fubini-Study metric.

Complete open Kähler manifolds with nonnegative bisectional curvature and non-maximal volume growth

It is still an open problem that a complete open Kahler manifold with positive bisectional curvature is Stein. This paper partially resolves the problem by putting a restriction to volume growth condition. The partial solution here improves the observation in B. Wong and Q.-S. Zhang (Asian J Math 7(3):341, 2003). The improvement is based on assuming a weaker volume growth condition that is not sufficiently maximal.

Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact Kähler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress. In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C1n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, C1n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on Kähler manifolds with minimal volume growth.

Diameter rigidity for Kähler manifolds with positive bisectional curvature

Let \(M^n\) be a compact Kähler manifold with bisectional curvature bounded from below by 1. If \(diam(M) = \pi / \sqrt{2}\) and \(vol(M)> vol({\mathbb {C}}{\mathbb {P}}^n)/ 2^n\), we prove that M is biholomorphically isometric to \({\mathbb {C}}{\mathbb {P}}^n\) with the standard Fubini-Study metric.

Remarks on shrinking gradient Kähler–Ricci solitons with positive bisectional curvature

In this short note, using an argument by Munteanu and Wang, we provide an alternative proof of the fact first obtained by Lei Ni that shrinking gradient Kähler–Ricci solitons with positive bisectional curvature must be compact.

Fano manifolds with nef tangent bundle and large Picard number

Abstract: We study Fano manifolds with nef tangent bundle and large Picard number.Key words: Fano manifold; nef tangent bundle; homogeneous manifold; large Picardnumber.1. Introduction. The classical Gauss-Bonnet Theorem implies that the only compactRiemann surface with positive curvature is theRiemann sphere. In the higher dimensional case, theFrankel conjecture claims that a compact Ka¨hlermanifold with positive bisectional curvature is theprojective space. This conjecture was solved byMori [12] and Siu-Yau [20], independently. Mori’sproof is purely algebraic and he obtained a moregeneral result. In fact, he solved the Hartshorneconjecture, which says that the projective space isthe only projective manifold with ample tangentbundle [12]. After that, in complex geometry, Mokproved the generalized Frankel conjecture on com-pact Ka¨hler manifolds with semipositive bisectionalcurvature [11]. As a generalization of their works,complex projective manifolds with nef tangentbundle have been studied by many authors (forinstance, see [14]). By the result of Demailly,Peternell and Schneider [4], the study can bereduced to the case of Fano manifolds. Let us recallthe following conjecture posed by Campana andPeternell.Conjecture 1.1 ([2]). Any Fano manifoldwith nef tangent bundle is rational homogeneous.This conjecture holds if the dimension is atmost four [6], and this is also true for five-foldswhose Picard number greater than one [21]. Re-cently Kanemitsu [9] proved the above conjecturefor five-folds of Picard number one. In this paper,we will generalize a result of [21] to the higherdimensional case. Our main result isTheorem 1.2. Let X beaFanomanifoldwith nef tangent bundle. Let m be the dimension, nthe Picard number and i

Holomorphic curvatures of twistor spaces

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities

Abstract We consider a subset S of the complex Lie algebra 𝔰𝔬(n, ℂ) and the cone C(S) of curvature operators which are nonnegative on S. We show that C(S) defines a Ricci flow invariant curvature condition if S is invariant under AdSO(n, ℂ). The analogue for Kähler curvature operators holds as well. Although the proof is very simple and short, it recovers all previously known invariant nonnegativity conditions. As an application we reprove that a compact Kähler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to ℂℙn. Moreover, the methods can also be applied to prove Harnack inequalities.

Some results on compact Kähler surfaces with non-positive bisectional curvature

In 1961 L. Frankel conjectured that any compact Kahler manifold of positive bisectional curvature is biholomorphic to a projective space. This was solved by S. Mori in 1979 and Siu and Yau in 1980. N. Mok in 1988 gave a full solution to the generalized Frankel’s conjecture and proposed a dual problem. The results in this paper are attempts to give modified solutions to this problem in complex dimension two.

A note on compact Kähler-Ricci flow with positive bisectional curvature

A note on compact Kähler-Ricci flow with positive bisectional curvature

The Kähler–Ricci flow with positive bisectional curvature

We show that the K\"ahler-Ricci flow on a manifold with positive first Chern class converges to a K\"ahler-Einstein metric assuming positive bisectional curvature and certain stability conditions.

On the volume growth of Kählerian manifolds with positive curvature near infinity

In this paper, the authors proved that the order of volume growth of Kählerian manifolds with positive bisectional curvature near infinity is at least half of the real dimension (i.e., the complex dimension).

Ricci flow on Kähler-Einstein manifolds

This is the continuation of our earlier article [10]. For any Kähler-Einstein surfaces with positive scalar curvature, if the initial metric has positive bisectional curvature, then we have proved (see [10]) that the Kähler-Ricci flow converges exponentially to a unique Kähler-Einstein metric in the end. This partially answers a long-standing problem in Ricci flow: On a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has positive bisectional curvature? In this article we give a complete affirmative answer to this problem

A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature

In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete non-compact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have maximal volume growth, then M is biholomorphic to C 2. This gives a partial affirmative answer to the well-known conjecture of Yau [41] on uniformization theorem. During the proof, we also verify an interesting gap phenomenon, predicted by Yau in [42], which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

Rank and k‐nullity of contact manifolds

We prove that the dimension of the 1‐nullity distribution N(1) on a closed Sasakian manifold M of rank l is at least equal to 2l − 1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k‐contact manifolds which are not Sasakian. On a closed, 2n + 1‐dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n + 1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold M is isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k‐nullity, Weinstein conjecture, and minimal unit vector fields.

Ricci flow on compact Kähler manifolds of positive bisectional curvature

This Note announces a new proof of the uniform estimate on the curvature of metric solutions to the Ricci flow on a compact Kähler manifold with positive bisectional curvature. This proof does not pre-suppose the existence of a Kähler–Einstein metric on the manifold, unlike the recent work of XiuXiong Chen and Gang Tian. It is based on the Harnack inequality for the Ricci–Kähler flow (see Invent. Math. 10 (1992) 247–263), and also on an estimation of the injectivity radius for the Ricci flow, obtained recently by Perelman. To cite this article: H.-D. Cao et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).

On complete noncompact Kähler manifolds with positive bisectional curvature

We prove that a complete noncompact Kahler manifold Mnof positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of Cn and we show that the manifold is topologically R2n. In particular, when Mn is a Kahler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to C2.

Theorems of Barth-Lefschetz Type on Kähler Manifolds with Partially Positive Curvature

The author obtains the theorems of Barth-Lefschetz type on Kahler manifolds with partially positive bisectional curvature without the assumption of nonnegative bisectional curvature. Some applications of the results to holomorphic mappings are given.

Ricci flow on Kähler-Einstein surfaces

In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the K\"ahler 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 K\"ahler-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 the following theorem: Let $M$ be a K\"ahler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler Ricci flow will converge exponentially fast to a K\"ahler-Einstein metric with constant bisectional curvature. Such a result holds for K\"ahler-Einstein orbifolds.

An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties

An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties