Hamilton-Perelman's Proof of the Poincar\'e Conjecture and the Geometrization Conjecture

In this paper we survey the recent developments of the Ricci flows on complete noncompact K\"{a}hler manifolds and their applications in geometry.

In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.

We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S2xS1 or to some mem- ber of F. This result generalises G. Perelman's classification theorem for compact 3-manifolds of positive scalar curvature. The main tool is a...

This is the announcement of an alternative approach to the 3-dimensional Poincar\'e Conjecture, different from Perelman's big and spectacular breakthrough. No claim concerning the other parts of the Thurston Geometrization Conjecture, come with our purely 4-dimensional line of argument.

This is a personal view of some problems on minimal surfaces, Ricci flow, polyhedral geometric structures, Haken 4-manifolds, contact structures and Heegaard splittings, singular incompressible surfaces after the Hamilton-Perelman revolution.

We study the long time behaviour of Ricci flow with bubbling-off on a possibly noncompact $3$-manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston's hyperbolisation theorem for $3$-manifolds with toral boundary that generalizes Perelman's proof of the hyperbolisation conjecture in the closed case.

This paper is a survey on the structure of manifolds with a lower Ricci curvature bound.

In this paper, we prove Hamilton-Ivey estimates for the Ricci-Bourguignon flow on a compact manifold, with $n=3$ and $\rho<0$. As a consequence, we prove that compact ancient solutions have nonnegative sectional curvature for all negative $\rho$.

Motivated by the Hamilton's Ricci flow, we define the homogeneous flow of a parallelizable manifold and show the long time existence and uniqueness of its solutions on $[0,\infty).$ Using this flow, we outline a simple proof of the Poincare Conjecture.

Ricci flow on two dimensional surfaces is far simpler than in the higher dimensional cases. This presents an opportunity to obtain much more detailed and comprehensive results. We review the basic facts about this flow, including the original results by Hamilton and Chow concerning Ricci flow on compact surfaces. The rationale for this paper, however, is especially to survey recent work concerning this flow on open surfaces, including various classes of both complete and inco...