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A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

- H. Cao, Xiping Zhu
- Mathematics
- 1 June 2006

In this paper, we give a complete proof of the Poincare and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof… Expand

Uniqueness of the Ricci flow on complete noncompact manifolds

- Binglong Chen, Xiping Zhu
- Mathematics
- 21 May 2005

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,… Expand

The Curve Shortening Problem

- K. Chou, Xiping Zhu
- Mathematics
- 6 March 2001

BASIC RESULTS Short Time Existence Facts from Parabolic Theory Evolution of Geometric Quantities INVARIANT SOLUTIONS FOR THE CURVE SHORTENING FLOW Travelling Waves Spirals The Support Function of a… Expand

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

- Binglong Chen, Xiping Zhu
- Mathematics
- 23 April 2005

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… Expand

Ricci Curvature on Alexandrov spaces and Rigidity Theorems

- Huichun Zhang, Xiping Zhu
- Mathematics
- 16 December 2009

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… Expand

A perturbation result on positive entire solutions of a semilinear elliptic equation

- Xiping Zhu
- Mathematics
- 1 August 1991

Lectures on Mean Curvature Flows

- Xiping Zhu
- Mathematics
- 1 September 2002

The curve shortening flow for convex curves The short time existence and the evolution equation of curvatures Contraction of convex hypersurfaces Monotonicity and self-similar solutions Evolution of… Expand

Complete Riemannian manifolds with pointwise pinched curvature

- Binglong Chen, Xiping Zhu
- Mathematics
- 1 May 2000

Abstract.An analogous Bonnet-Myers theorem is obtained for a complete and positively curved n-dimensional (n≥3) Riemannian manifold Mn. We prove that if n≥4 and the curvature operator of Mn is… Expand

Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains

- Xiping Zhu, Huan-Song Zhou
- Mathematics
- 1990

By using the concentration-compactness method of Lions [14, 16] and the mountain pass theorem of Ambrosetti and Rabinowitz [3], through a careful inspection of the energy balance for some sequence of… Expand

Yau's gradient estimates on Alexandrov spaces

- Huichun Zhang, Xiping Zhu
- Mathematics
- 20 December 2010

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.

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