# Ricci flow on two dimensional sphere

I want to visualize Ricci flow solution on the following sphere


Let $r> 0$

$L = \{ (x cos \theta, x sin \theta, x) | r < x < R \}$

$S$ : $(z-\sqrt{2} r)^2 + x^2 + y^2 = r^2$

$T$ : $(z- \sqrt{2} R)^2 + x^2 + y^2 = R^2$

If $R$ is sufficiently large, then we have a two dimensional sphere $U$ enclosed by $L$, $S$, and $T$.

The Ricci flow solution on $U$ shrinks fastly around region covered by $S$, but the region covered by $L$ remains unchanged, since the Gaussian curvature is 0.

This confuses me. Where is wrong ?

MOTIVATION : I want to know the Ricci flow on a orbifold $O$ which is smooth except one point. In generally, is there a solution on $O$ ? Around a singuler point, the curvature is very large so that Ricci flow solution on $O$ shrinks to the singular point fastly.

If we consider normalized Ricci flow on $O$, the solution goes to a "canonical" orbifold ?

What I say is that if $O$ is a two sphere with exactly one point singularity, then the solution goes to $lim_{r \rightarrow 0} U$

Anything related with my opinion is welcome.

• The induced metric on the sphere from the formulas you wrote is not $C^\infty$ smooth. The Ricci flow will immediately smooth it out, but the "cylinder" region will immediately not be flat anymore, so it will change under the flow. – YangMills Jun 5 '12 at 4:06
• For Ricci flow on orbifolds, look at this arxiv.org/pdf/1101.3733 – YangMills Jun 5 '12 at 4:08
• You really should post your second question separately as a new question, rather than adding it here. – Deane Yang Jun 7 '12 at 16:32
• @Yang : I am sorry. I will separate my question. – Hee Kwon Lee Jun 8 '12 at 10:22

Ricci flow is an intrinsic flow, so (unless you were to somehow recast it as an extrinsic flow) it doesn't make sense to talk about what happens to surfaces in $\mathbb{R}^3$ under Ricci flow.

• If the surface is positively curved, there is a unique embedding, so there is a completely canonical way to translate the flow into an extrinsic flow. – Igor Rivin Jun 8 '12 at 19:42
• That's what I was getting at when I wrote "unless you..." It's not quite as simple as you say since the embedding is only unique up to isometry (and as you point out, requires positive curvature). My basic point is that you can't ask about happens in some given situation when you haven't properly set up the situation. – Dan Lee Jun 8 '12 at 22:49

I believe that the Ricci flow on orbifolds was first studied by Hamilton in the paper "Three-Orbifolds with Positive Ricci Curvature", answering a question of Thurston. This paper, originally written in the early eighties, is published in the "Collected Papers on Ricci Flow" book edited by H.-D. Cao, etal (2003). It may well have been forgotten if it were not for Craig Hodgson still having a copy of the paper after many years (Hamilton no longer had a copy!).

The Ricci flow on 2-orbifolds was originally studied by Lang-Fang Wu, who considered the case of a positively curved initial metric on a closed 2-orbifold with positive Euler characteristic, proving global existence and convergence to a shrinking gradient Ricci soliton metric after adjustment by diffeomorphisms. That is, the flow is the constant area normalization of: $\partial g_{ij}/\partial t = -2(R_{ij}+\nabla_i \nabla_j f)$, where $f$ is the potential of the curvature. This is a candidate for canonical metric on the orbifold. Note that if the Euler characteristic $\chi$ is nonpositive, then the 2-orbifold must be good and one can apply Hamilton's surfaces theorem to the global smooth cover. When $\chi >0$, there can be bad orbifolds with one or two singular (cone) points. In joint work, Lang-Fang Wu extended her work to arbitrary initial metrics when $\chi >0$.

Most of the analysis of the Ricci flow on surfaces carries over to the case of 2-orbifolds since the notion of smoothness is global by passing to local finite covers; e.g., the maximum principle and integration by parts (this is an observation of Hamilton). The exception is the injectivity radius (noncollapsing) estimate. This was resolved in the work of Wu by an area estimate.

Besides the paper by Bruce Kleiner and John Lott mentioned by YangMills, there is also a paper by Bing-Long Chen and Xiping Zhu on 4-orbifolds with positive isotropic curvature, extending work and answering a conjecture of Hamilton.

There is also work of Hao Yin and of Mazzeo, Rubinstein and Sesum on the Ricci flow on surfaces with conical singularities related to work of Troyanov on surfaces with conical singularities.

I'm not familiar with the general literature in this area, so the above list of works is far from complete. E.g., besides other works in the Riemannian case, notably works on the Kaehler--Ricci flow for singular metrics.