The Ricci flow on surfaces

Hamilton, Richard S.

doi:10.1090/conm/071/954419

Cited by 1,017 publications

(797 citation statements)

References 0 publications

Supporting

Mentioning

784

Contrasting

Unclassified

Order By: Relevance

“…The Ricci flow and the soliton phenomenon gave a new proof for the uniformization theorem on compact surfaces and orbifolds without boundary ([C, Hal,W2,CW]). Understanding the solitons may provide insights toward studying the Ricci flow on higher-dimensional Kahler manifolds.…”

Section: ; Dtmentioning

confidence: 99%

A new result for the porous medium equation derived from the Ricci flow

Wu¹

1993

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract.Given K2, with a "good" complete metric, we show that the unique solution of the Ricci flow approaches a soliton at time infinity. Solitons are solutions of the Ricci flow, which move only by diffeomorphism. The Ricci flow on R2 is the limiting case of the porous medium equation when m is zero. The results in the Ricci flow may therefore be interpreted as sufficient conditions on the initial data, which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a "soliton-solution".

show abstract

Section: ; Dtmentioning

confidence: 99%

A new result for the porous medium equation derived from the Ricci flow

Wu¹

1993

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The next lemma provides some kind of semi-concavity estimate that has a large number of parallels in the theory of quasilinear parabolic equations, but also of Hamilton-Jacobi equations [1,16,22]. …”

Section: Proof I) This Easily Results From Comparison Of U With the mentioning

confidence: 99%

Infinite-Time Quenching in a Fast Diffusion Equation with Strong Absorption

Winkler

2009

Nonlinear differ. equ. appl.

View full text Add to dashboard Cite

Abstract. This work is concerned with the fast diffusion equation ut = Δu m − u κ in R n , where 0 < m < 1 and κ < 1. A global positive solution is said to quench regularly in infinite time if u(x k , t k ) → 0 for some bounded sequence (x k ) k∈N and some t k → ∞, and if sup (x,t)∈K×(0,∞) u(x, t) < ∞ for all compact K ⊂⊂ R n . It is shown that such regular quenching in infinite time occurs for a large class of initial data if κ > m, whereas it is impossible in one space dimension when κ < −m and the solution is radially symmetric and nondecreasing for x > 0. Mathematics Subject Classification (2000). 35K55, 35K65, 35B40.

show abstract

“…[17]), Hamilton [12] and Chow [5] (see also Theorem 5.1 in [6]) proved the following theorem concerning the 2-dimensional case: Proposition 1. Let g 0 be any Riemannian metric on M .…”

Section: Ricci and Yamabe Flowmentioning

confidence: 99%

Stochastic Particle Approximations for the Ricci Flow on Surfaces and the Yamabe Flow

Philipowski

2011

Potential Anal

View full text Add to dashboard Cite

We present stochastic particle approximations for the normalized Ricci flow on surfaces and for the non-normalized Yamabe flow on manifolds of arbitrary dimension.Keywords: Ricci flow on surfaces, Yamabe flow, stochastic interacting particle system. AMS subject classification: 53C44, 82C22. Ricci and Yamabe flowThanks to Perelman's seminal work on the geometrization and hence the Poincaré conjecture [19,20,21] the Ricci flow has attracted worldwide attention and led to many new developments (see e.g. [17]). The importance of this subject has been underlined by the award of the Fields medal to Perelman. There is now a strong interest in understanding the microscopic structure of the Ricci flow.The evolution of a Riemannian metric g = g t on a connected d-dimensional closed manifold M under the (normalized) Ricci flow is described by the partial differential equationHere Ric is the Ricci curvature andR the average scalar curvature of M , i.e.R := 1 vol(M ) M R, where R is the scalar curvature (all quantities taken with respect to g t ).In dimension d = 2 we have Ric = 1 2 Rg, so that in this case (1) 2. As t → ∞, g t converges in any C k -norm to a smooth metric g ∞ of constant curvature.

show abstract

The Ricci flow on surfaces

Cited by 1,017 publications

References 0 publications

A new result for the porous medium equation derived from the Ricci flow

A new result for the porous medium equation derived from the Ricci flow

Infinite-Time Quenching in a Fast Diffusion Equation with Strong Absorption

Stochastic Particle Approximations for the Ricci Flow on Surfaces and the Yamabe Flow

Contact Info

Product

Resources

About