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Xi-Ping Zhu

Publications and source records attributed to Xi-Ping Zhu.

34 records · Page 2Linked to original sources
arXiv2012

Yau's gradient estimates on Alexandrov spaces

In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

math.DG↗
arXiv2010

Ricci Curvature on Alexandrov spaces and Rigidity Theorems

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.

math.DG↗
arXiv2010

On a new definition of Ricci curvature on Alexandrov spaces

Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.

math.DG↗
arXiv2008

Complete classification of compact four-manifolds with positive isotropic curvature

In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $\mathbb{S}^4,$ or $\mathbb{R}\mathbb{P}^4$ or quotients of $\mathbb{S}^3\times \mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $\mathbb{S}^3\times \mathbb{R}$, or a connected sum of them.

math.DG↗
arXiv2006

Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.

math.DG↗
arXiv2006

Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelman's second paper on the Ricci flow.

math.DG↗
arXiv2005

Uniqueness of the Ricci Flow on Complete Noncompact Manifolds

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.

math.DG↗
arXiv2003

Sharp Dimension Estimates of Holomorphic Functions and Rigidity

Let $M^n$ be a complete noncompact K$\ddot{a}$hler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}$$_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $d$ on $M^n$. In this paper we prove that $$dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{[d]}(\mathbb{C}^n),$$ for all $d>0$, with equality for some positive integer $d$ if and only if $M^n$ is holomorphically isometric to $\mathbb{C}^n$. We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.

math.DG↗
arXiv2003

Ricci flow on compact Kähler manifolds of positive bisectional curvature

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact Kähler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of the uniform estimate does not rely on the exsitence of Kähler-Einstein metrics on $M^n$, but instead on the first author's Harnack inequality for the Kähler-Ricc flow, and a very recent local injectivity radius estimate of Perelman for the Ricci flow.

math.DG↗
arXiv2002

On Complete Noncompact Kähler Manifolds with Positive Bisectional Curvature

We prove that a complete noncompact Kähler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to {\bf C}$^{2}$.

math.DG↗
arXiv2002

A Uniformization Theorem Of Complete Noncompact 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 noncompact complex two dimensional Kähler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then $M$ is biholomorphic to $\C^2$. During the proof, we also discover an interesting gap phenomenon which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.

math.DG↗
arXiv2002

Positively Curved Complete Noncompact Kähler Manifolds

In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature and with finite analytic Chern number $c_{1}(M)^{2}$ is biholomorphic to ${\C}^2$.

math.DG↗
arXiv2002

Volume Growth and Curvature Decay of Positively Curved Kähler manifolds

In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex dimension). The second one states that the curvature of a complete noncompact Kähler manifold with positive bisectional curvature decays at least linearly in the average sense. The third result is concerned with the relation between the volume growth and the curvature decay. We prove that the curvature decay of a complete noncompact Kähler manifold with nonnegative curvature operator and with the maximal volume growth is precisely quadratic in certain average sense.

math.DG↗