Complete Noncompact Kahler Manifold Research Articles

Chern–Ricci flows on noncompact complex manifolds

In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that any complete noncollapsed Kahler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kahler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume. Combining this result with the result of Chau-Tam, we give another proof that a complete noncompact Kahler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to the complex Euclidean space. This last result has already been proved by Gang Liu recently using other methods. This last result is partial confirmation of a uniformization conjecture of Yau.

Rigidity Theorems of Complete Kähler-Einstein Manifolds and Complex Space Forms

We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kahler manifolds. We derive some elliptic differential inequalities from Weitzenb¨ock formulas for the traceless Ricci tensor of Kahler manifolds with constant scalar curvature and the Bochner tensor of Kahler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L∞ pinching results are established to characterize Kahler-Einstein manifolds among Kahler manifolds with constant scalar curvature and complex space forms among Kahler-Einstein manifolds. Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact Kahler manifolds and noncompact Kahler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge, these kinds of results have not been reported.

Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture

We study the uniformization conjecture of Yau by using the GromovHausdorff convergence. As a consequence, we confirm Yau’s finite generation conjecture. More precisely, on a complete noncompact Kahler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. During the course of the proof, we prove if M is a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, then M is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on Kahler manifolds with nonnegative bisectional curvature.

Global Kähler–Ricci flow on complete non-compact manifolds

In this paper, we study the global Kahler–Ricci flow on a complete non-compact Kahler manifold. We prove the following result. Assume that \((M,g_0)\) is a complete non-compact Kahler manifold such that there is a potential function f of the Ricci tensor, i.e., $$\begin{aligned} R_{i{\bar{j}}}(g_0)=f_{i{\bar{j}}}. \end{aligned}$$ Assume that the quantity \(|f|_{C^0}+|\nabla _{g_0}f|_{C^0}\) is finite and the \(L^2\) Sobolev inequality holds true on \((M,g_0)\). Then the Kahler–Ricci flow with the initial metric \(g_0\) either blows up at finite time or infinite time to Ricci-flat metric or exists globally with Ricci-flat limit at infinite time. Related results are also discussed.

On a modified parabolic complex Monge–Ampère equation with applications

We study a modified parabolic complex Monge–Ampere type equation on a complete non-compact Kahler manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form Ω and initial Kahler metric g0 on M, the modified Kahler–Ricci flow g′ = −Ric + Ω has a long time solution converging to a complete Kahler metric such that Ric = Ω, which extends the result in Cao (Invent Math 81:359–372, 1985) to non-compact manifolds. We will also obtain a long time existence result for the Kahler–Ricci flow which generalizes a result (Chau et al. in Can J Math, to appear).

A sharp estimate for the bottom of the spectrum of the Laplacian on Kähler manifolds

On a complete noncompact Kahler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by m2 if the Ricci curvature is bounded from below by −2(m+1). Then we show that if this upper bound is achieved then either the manifold is connected at infinity or it has two ends and in this case it is diffeomorphic to the product of the real line with a compact manifold and we determine the metric.

A Remark on Steinness*

In this paper, the authors prove that if M n is a complete noncompact Kahler manifold with a pole p, and its holomorphic bisectional curvature is asymptotically non-negative to p, then it is a Stein manifold.

A survey of the Kähler-Ricci Flow and Yau’s Uniformization Conjecture

Yau’s uniformization conjecture states: a complete noncompact Kahler manifold with positive holomorphic bisectional curvature is biholomorphic to C. The Kahler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the Kahler-Ricci flow with focus on its application to uniformization. Other interesting methods and results related to the study of Yau’s conjecture are also discussed.

A uniqueness result of Kähler Ricci flow with an application

In this paper, we will study the problem of uniqueness of Kahler Ricci flow on some complete noncompact Kahler manifolds and the convergence of the flow on C with the initial metric constructed by Wu and Zheng.

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.

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.

Plurisubharmonic Functions and the Structure of Complete Kähler Manifolds with Nonnegative Curvature

In this paper, we study global properties of continuous plurisubharmonic functions on complete noncompact Kahler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds. We prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation. Optimal Liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established. The results are then applied to prove several structure theorems on complete noncompact Kahler manifolds with nonnegative bisectional or sectional curvature.

Poisson equation and Hermitian-Einstein metrics on vector bundles over complete noncompact Kahler manifolds

Poisson equation and Hermitian-Einstein metrics on vector bundles over complete noncompact Kahler manifolds

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kahler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ℂn. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].

On Yau’s uniformization conjecture

Let $M^n$ be a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $\mathbb{C}^n$. This confirms Yau's uniformization conjecture when M has maximal volume growth.