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A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow
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
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,
The Curve Shortening Problem
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
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
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
Lectures on Mean Curvature Flows
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
Complete Riemannian manifolds with pointwise pinched curvature
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
Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains
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
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.
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