Chủ đề này có 4 trả lời

[toilachinhtoi]

Hi everyone,
I want to discuss about Ricci flows and Perelman proof of Poincare conjecture. In fact, I am taking one course about this, so I don't know about many things, just depending on what I learn up to date in the class. Hence I am very grateful if you are joining for discussing, helping and answering.

I intend to address the topics in following orders:
_Recall about connections, differential forms, derivations of differential forms...
_The Ricci flows.
_(And if I can) Perelman proof.

[Kakalotta]

Good stuff. Let's start with Ricci flow directly.
It could be best to end up with the relationship w/ soliton and Moduli space.

[toilachinhtoi]

I hope that all of us will learn a lot from discussing.

About the material: My Prof suggests Topping's notes and Morgan-Tian.

Perelman’s Proof of the Poincar´e Conjecture
(following Morgan-Tian)

The Poincar´e Conjecture states that every simply connected, closed 3-manifold
is homeomorphic to the 3-sphere. The proof by Perelman is based on the Ricci flow
method developed by Hamilton (which is a geometric evolution equation). Originally,
Hamilton proved that a closed simply connected 3-manifold with positive Ricci curvature must be the 3-sphere by showing that the Ricci flow converges to a metric
of constant curvature. Key for this argument to work was that the Ricci flow on a
positively curved manifold exists for all positive time, which is not true in general.

Perelman’s many refinements create a Ricci flow with surgery which exists for
all time, regardless of curvature assumptions. It is probably fair to say that this
achievement promotes the theory of geometric partial differential equation to the tool
of choice to solve problems in Riemannian Geometry for many years to come.
The recent e-print by Morgan and Tian, available at http://www.arxiv.org/
pdf/math.DG/0607607, not only establishes the Poincar´e Conjecture following Perelman’s program but also gives sufficient background which makes us hope that a
generic Riemannian geometer has now a chance to understand the proof in its details.


We will try to understand the proof of the following results

Theorem (Hamilton 1982): Let http://dientuvietnam...etex.cgi?(M^3,g) is closed, http://dientuvietnam...mimetex.cgi?g_t on http://dientuvietnam...n/mimetex.cgi?M defined for all http://dientuvietnam...etex.cgi?g_0=g. Moreover, there exists a map
http://dientuvietnam.net/cgi-bin/mimetex.cgi?g_{\infty} is Riemann metric with positive constant
sectional curvature. So http://dientuvietnam.net/cgi-bin/mimetex.cgi?(\tilde{M},g_{\infty}) isometric to http://dientuvietnam...mimetex.cgi?S^3 with its usual metric.

Theorem (Perelman): http://dientuvietnam...mimetex.cgi?M^3 closed (no curvature assumption), http://dientuvietnam.net/cgi-bin/mimetex.cgi?\pi_1(M) is a free product of finite factors of finite groups. Then
http://dientuvietnam.net/cgi-bin/mimetex.cgi?M is diffeomorphic to a connected sum http://dientuvietnam.net/cgi-bin/mimetex.cgi?M_j is one of the following

-Space forms http://dientuvietnam.net/cgi-bin/mimetex.cgi?n.(p,t)-1)^np,t+n).

1) Ricci flows: A family of Riemannian metrics http://dientuvietnam.net/cgi-bin/mimetex.cgi?g_t on a manifold
http://dientuvietnam.net/cgi-bin/mimetex.cgi?M satisfying
http://dientuvietnam.net/cgi-bin/mimetex.cgi?\dfrac{d}{dt}g_t=-2Ricc(g_t).

Bài viết đã được chỉnh sửa nội dung bởi toilachinhtoi: 03-10-2006 - 08:07

[toilachinhtoi]

4) Einstein manifolds: http://dientuvietnam.net/cgi-bin/mimetex.cgi?n\not=2 then http://dientuvietnam...imetex.cgi?(M,g) is manifold iff http://dientuvietnam...n/mimetex.cgi?M is Einstein http://dientuvietnam...n/mimetex.cgi?M is http://dientuvietnam.net/cgi-bin/mimetex.cgi?t=\dfrac{1}{2\lambda} we have http://dientuvietnam...etex.cgi?g_t=0.

Question: Is this contradict to Hamilton's theorem?

5) Ricci solitons: http://dientuvietnam...etex.cgi?(M,g_t) a Ricci flow is Ricci solitons if there
exist diffeomorphisms http://dientuvietnam...n/mimetex.cgi?X be a vector field of http://dientuvietnam...etex.cgi?(M,g_0), and http://dientuvietnam...imetex.cgi?Y_t. Then http://dientuvietnam.net/cgi-bin/mimetex.cgi?(M,g_0).

Ex: Do computations for above assertations.

[Kakalotta]

Đây là các files video về Ricci flows, ở MSRI, mọi người có thể down load về để nghiên cứu.
http://www.msri.org/...1/show_semester