Nonnegative 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.

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.

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.

A Liouville Property of Holomorphic Maps

We prove a Liouville property of holomorphic maps from a complete Kähler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kähler manifold with a certain assumption on the sectional curvature.

On Kähler structures over symmetric products of a Riemann surface

Given a positive integer n n and a compact connected Riemann surface X X , we prove that the symmetric product S n ( X ) S^n(X) admits a Kähler form of nonnegative holomorphic bisectional curvature if and only if genus ( X ) ≤ 1 \text {genus}(X)\, \leq \, 1 . If n n is greater than or equal to the gonality of X X , we prove that S n ( X ) S^n(X) does not admit any Kähler form of nonpositive holomorphic sectional curvature. In particular, if X X is hyperelliptic, then S n ( X ) S^n(X) admits a Kähler form of nonpositive holomorphic sectional curvature if and only if n = 1 ≤ genus ( X ) n\,=\,1\, \leq \, \text {genus}(X) .

Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds

A Łojasiewicz-type estimate is a powerful tool in studying the rigidity properties of the harmonic map heat flow. Topping proved such an estimate using the Riesz potential method, and established various uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to $\mathbb{S}^{2}$ (J. Differential Geom. 45 (1997), 593–610). In this note, using an inequality due to Sobolev, we will derive the same estimate for maps from $\mathbb{S}^{2}$ to a compact Kahler manifold N with nonnegative holomorphic bisectional curvature, and use it to establish the uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to N, which generalizes Topping’s result.

THE DEGREE OF HOLOMORPHIC APPROXIMATION ON A TOTALLY REAL SET

AbstractLet E be a totally real set on a Stein open set Ω on a complete noncompact Kähler manifold (M,g) with nonnegative holomorphic bisectional curvature such that (Ω,g) has bounded geometry at E. Then every function f in a Cp class with compact support on Ω and $\overline \partial $-flat on E up to order p−1,p≥2 (respectively, in a Gevrey class of order s>1, with compact support on Ω and $\overline \partial $-flat on E up to infinite order) can be approximated on compacts subsets of E by holomorphic functions fk on Ω with degree of approximation equal k−p/2 (respectively, exp (−c(s)k1/2(s−1)) ).

Complex Hessian Equation on Kähler Manifold

In this paper, complex Hessian equation over Kähler manifold was studied. Under the condition that the underline Kähler manifold has non-negative holomorphic bisectional curvature, the existence and regularity of the solution was proved.

Volume Growth Estimates of Manifolds with Nonnegative Curvature Outside a Compact Set

In this article, using the properties of Busemann functions, the authors prove that the order of volume growth of Kähler manifolds with certain nonnegative holomorphic bisectional curvature and sectional curvature is at least half of the real dimension. The authors also give a brief proof of a generalized Yau's theorem.

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincare–Lelong equation on a class of Kahler manifolds with nonnegative holomorphic bisectional curvature, \(\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)\) for some a > 67(n + 4)2. We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincare inequality.

On the Complex Structure of Kähler Manifolds with Nonnegative Curvature

We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative and bounded holomorphic bi-sectional curvature and maximal volume growth is biholomorphic to complex Euclidean space Cn. We also show that the volume growth condition can be removed if we assume the Kähler manifold has average quadratic scalar curvature decay and positive curvature operator.

Sharp dimension estimates of holomorphic functions and rigidity

Let M n be a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature. Denote by O d (M n ) the space of holomorphic functions of polynomial growth of degree at most d on M n . In this paper we prove that dimcod(M n ) ≤ dim c O [d] (C n ), for all d > 0, with equality for some positive integer d if and only if M n is holomorphically isometric to C n . We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

Plurisubharmonic functions and the Kähler-Ricci flow

In this paper, a sharp linear trace Li-Yau-Hamilton inequality for Kähler-Ricci flow is proved. The new inequality extends the previous trace Harnack inequality obtained by H.-D. Cao. We also establish sharp gradient estimates for the positive solution of the time-dependent heat equation for some cases. Finally, we apply this new linear trace Li-Yau-Hamilton inequality to study the Liouville properties of the plurisubharmonic functions on complete Kähler manifolds with bounded nonnegative holomorphic bisectional curvature.

Poisson Equation, Poincaré-Lelong Equation and Curvature Decay on Complete Kähler Manifolds

In the first part of this work, the Poisson equation on complete noncompact manifolds with nonnegative Ricci curvature is studied. Sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained. Sharp estimates on the solutions are also derived. In the second part, these results are applied to the study of curvature decay on complete Kähler manifolds. In particular, the Poincaré-Lelong equation on complete noncompact Kähler manifolds with nonnegative holomorphic bisectional curvature is studied. Several applications are then derived, which include the Steinness of the complete Kähler manifolds with nonnegative curvature and the flatness of a class of complete Kähler manifolds satisfying a curvature pinching condition. Liouville type results for plurisubharmonic functions are also obtained.

The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature

Soit (X, h) une variete de Kahler compacte a n dimensions de courbure bisectionnelle holomorphe non negative et soit (X~, h~) son espace de recouvrement universel. Alors il existe des entiers non negatifs k, N 1 ,…, N l et des espaces symetriques hermitiens compacts ineductibles M 1 ,…, M k de rang ≥2, tels que (X~, h~) est isometriquement bi-holomorphe a (C k , g 0 )×(P Nu 1, θ 1 )×…×(P Nu l, θ l )×(M 1 , g 1 )×…×(M p , g p ), ou g 0 est la metrique euclidienne sur C k , g 1 ,…, g p sont des metriques canoniques sur M 1 , …, M p , et θ i , 1≤i≤l, est une metrique de Kahler sur P Nu i portant une courbure bisectionnelle holomorphe non negative

A note on Kähler manifolds of nonnegative holomorphic bisectional curvature

A note on Kähler manifolds of nonnegative holomorphic bisectional curvature