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February 2, 2007

Huisken on Uniformization, I

Posted by Urs Schreiber

Yesterday I heard a talk by Gerhard Huisken on Uniformization via the Heat equation.

A review of some ideas of the theory of Ricci flow, and how Perelman completed the proof of the Poincaré conjecture by using the dilaton field of string theory.

Here is my transcript, part I.

Part I of my transcript: from the Heat equation over mean curvature flow to Ricci flow.


We would like to understand the uniformization of metrics on closed Riemannian manifolds, i.e. the possibility of continuously deforming a given metric to a particularly homogeneous one.

As a warmup, consider the ordinary 1-dimensional heat equation ddtu=Δu \frac{d}{d t} u = \Delta u for a 1-parameter family of functions u:(x,t)u(t,x) u : (x,t) \mapsto u(t,x) on the real line.

It is noteworthy that this equation describes the gradient flow of the Dirichlet-Integral E(u):=12 |u| 2dx, E(u) := \frac{1}{2}\int_\mathbb{R} |u'|^2 \; d x \,, i.e. the fastest way to descrease this functional in the L 2L^2-norm.

This means that the heat equation has the effect of regularizing a bumpy function into one that varies less and less.

Another remarkable aspect is that the familiar solutions (now expressed more generally for the Laplace operator in nn-dimensions) u(x,t)=1(2πt) n/2exp(|x|4t) u(x,t) = \frac{1}{(2\pi t)^{n/2}} \exp \left( - \frac{|x|}{4t} \right) are self-similar in that for any λ>0\lambda \gt 0 we have u(x,t)=u(λx,λ 2t). u(x,t) = u(\lambda x, \lambda^2 t) \,.

We can express the regularization property of the heat equation quantitatively by the estimate sup B R/2|D mu(,t)|C(n,m)(1R 2+1t) m/2sup B R×[0,t]|u|. \mathrm{sup}_{B_{R/2}} \left| D^m u(\cdot,t) \right| \leq C(n,m) \left( \frac{1}{R^2} + \frac{1}{t} \right)^{m/2} \cdot \mathrm{sup}_{B_R\times [0,t]} |u| \,.

This says that the mm-spatial derivative D muD^m u of uu is bounded, over a ball of radius R/2R/2 by the given prefactor time the supremum of uu itself over time and over a ball of radius RR.

Furthermore, there is another way to look at the regularization property of the heat equation, namely by realizing it as the gradient flow of the entropy functional ulogudx \int u \mathrm{log}u \; d x but now with respect to the Wasserstein metric (this applies to u>0u \gt 0).

There is an estimate, called the Li-Yau Harnack estimate, which says that Δu|Dlogu| 2n2t. \Delta u \geq |D \mathrm{log}u|^2 - \frac{n}{2t} \,. This holds as stated for n\mathbb{R}^n and with slight modifications for arbitrary Riemannian manifolds whose curvature is bounded from below.

Estimates of this sort play an important role for understanding the following theory.

The regularization property of the heat equation has an analog in the following equation that describes curve shortening:

Consider any closed curve Γ(,0):S 1 2 \Gamma(\cdot,0) : S^1 \to \mathbb{R}^2 in the real plane and let tΓ(,t)t \mapsto \Gamma(\cdot,t) be a 1-parameter family of such curves satisfying the equation ddtΓ(p,t)=κ(p,t)=κ(p,t)ν(p,t), \frac{d}{dt} \Gamma(p,t) = \vec \kappa(p,t) = \kappa(p,t)\vec \nu(p,t) \,, where κ\vec\kappa is the second arc-length derivative of Γ\Gamma κ(p,t)=κ(p,t)ν(p,t)=dds t 2Γ(p,t), \vec \kappa(p,t) = \kappa(p,t) \vec \nu(p,t) = \frac{d}{d s_t^2} \Gamma(p,t) \,, ν(p,t)\vec \nu(p,t) is the unit normal vector to the curve at (p,t)(p,t) and κ(p,t)\kappa(p,t) is the spped of the curve at that point.

This is a quasi-linear differential equation (quasi since the arc-length depends on time). It describes again a gradient flow (with respect to the L 2L^2-norm), now simply of the length of the curve E(Γ)= S 1ds. E(\Gamma) = \int_{S^1} d s \,.

It is a fact that under this flow, an embedded curve remains embedded. Meaning that a curve which doesn’t intersect itself to start with will never intersect itself in the future.

As an example, consider a curve which is initially a circle of radius R 0R_0 centered at x 0x_0 in 2\mathbb{R}^2, Γ 0(S 1)=S R 0 1(x 0). \Gamma_0(S^1) = S^1_{R_0}(x_0) \,. Then under the above flow it will remain circular and centered at x 0x_0 Γ t(S 1)=S R t 1(x 0) \Gamma_t(S^1) = S^1_{R_t}(x_0) but shrink in radius according to R t=R 0 22t. R_t = \sqrt{R_0^2 - 2t} \,. This goes on until t=T:=R 0 2/2 t = T := R_0^2/2 at which point the curve has collapes to a (“round”) point and the flow equation diverges.

The interesting thing is that any embedded curve inside such a circular curve will also shrink, and will never be overtaken by the outer circular curve – hence will also shrink to a point – but always to a “round” point, i.e. no matter how wiggly it was to start with, it will always completely unwind to a nice circular curve just before collapsing to a point.

Theorem (Grayson, Gage-Hamilton): If Γ 0(S 1)\Gamma_0(S^1) is an embedded curve then Γ t(S 1)\Gamma_t(S^1) contracts smoothly to a (“round”) point in finite time.

The idea of the proof is this: one analyzes all possible ways that the extrinsic curvature |κ| |\vec \kappa| \to \infty can become singular for tTt \to T. One uses the fact that all self-similar solutions are exact circles as in the above example and concludes that hence all singularities must be of this shape, too.

Now we go to higher dimensions, but still consider embedded geometries.

For an nn-dimensional something embedded in n+1\mathbb{R}^{n+1} F 0:M n n+1 F_0 : M^n \to \mathbb{R}^{n+1} we define a flow by the quasi-linear parabolic system ddtF(p,t)=H(p,t)=(λ 1++λ n)(p,t)ν(p.t):=Δ tF(p,t). \frac{d}{dt}F(p,t) = \vec H(p,t) = (\lambda_1 + \cdots + \lambda_n)(p,t) \cdot \vec\nu(p.t) := \Delta_t F(p,t) \,. Here the λ i\lambda_i are the principal extrinsic curvatures, i.e. the nn eigenvalues of the the second fundamental form of the hypersurface.

This flow is, once again, a gradient flow, now for the nn-dimensional “area”: E(F)= M ndμ. E(F) = \int_{M^n} d\mu \,.

Again, one can study the shrinking solutions of this flow. Those of curvature of definite signs are the nn-spheres and the nn-cyclinder.

The intrisic analog of this is Hamilton’s Ricci-Flow (from 1982). This is a a manifold with a family of metrics tg(t)t \mapsto g(t) that flow according to the equation ddtg ij=rR ij(g), \frac{d}{dt}g_{ij} = -r R_{ij}(g) \,, where R ijR_{ij} are the components of the Ricci curvature tensor.

We can see how this is close to the extrinsic setup considered above by noticing that for any choice of local coordianetes we have an expansion R ij=Δg ij+. R_{ij} = \Delta g_{ij} + \cdots \,. The Ricci tensor hence indeed plays the role of the Laplace operator, but now in a diffeomorphism invariant context.

This diffeomorphism invariance of the Ricci flow is one of its main beauties, but is also what makes handling it more subtle.

The other big problem is the understanding and handling of the singularities of this Ricci flow.

(The main insight by Perelman is, roughly (as far as I understood) that by adjoining the dilaton field to the gravitational (= metric) field one is able to handle a dynamical re-adjustment of diffeomorphism in such a way that the behaviour of the singularities is under better control. More on that in part II. -urs)

Posted at February 2, 2007 12:16 PM UTC

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6 Comments & 4 Trackbacks

Read the post Huisken on Uniformization, II
Weblog: The n-Category Café
Excerpt: Something about Ricci flow and the proof of the Poincare conjecture.
Tracked: February 2, 2007 3:07 PM

Re: Huisken on Uniformization, I

Hamilton’s co-author in the theorem is Michael Gage from University of Rochester.

Posted by: Deane on February 8, 2007 5:23 PM | Permalink | Reply to this

Re: Huisken on Uniformization, I

Hamilton’s co-author in the theorem is Michael Gage from University of Rochester.

Thanks a lot! I’ll include that in the above entry.

Posted by: urs on February 8, 2007 5:30 PM | Permalink | Reply to this
Read the post Report-Back on BMC
Weblog: The n-Category Café
Excerpt: Bruce Bartlett reports from the British Mathematics Colloquium 2007
Tracked: April 22, 2007 8:19 PM
Read the post Report from "Workshop on Higher Gauge Theory"
Weblog: The n-Category Café
Excerpt: Report-back on a little symposium titled "Higher Gauge Theory" (but concerned just with abelian gerbes) that took place at the AEI in Golm.
Tracked: May 9, 2007 11:52 AM
Read the post The G and the B
Weblog: The n-Category Café
Excerpt: How to get the bundle governing Generalized Complex Geometry from abstract nonsense and arrow-theoretic differential theory.
Tracked: August 25, 2007 9:02 PM

Re: Huisken on Uniformization, I

A comment in mathoverflow relates Ricci flows to renormalization. Because the later seems to be a universal concept with connections to arithmetics (e.g. in arxiv articles by Connes,Marcolli), I’d like to know more about that. Do you know where one could read about such a connection between Ricci flow and renormalization?

Posted by: Thomas on November 15, 2009 9:29 PM | Permalink | Reply to this

Re: Huisken on Uniformization, I

Perhaps the commentator of my question meaned some other “renormalization” than that in QFT?

Posted by: Thomas on November 15, 2009 9:45 PM | Permalink | Reply to this

Re: Huisken on Uniformization, I

Yes, Ricci flow is the special case of renormalization group flow for the Polyakov sigma-model 2-d QFT – usually interpreted as describing the wolrdvolume theory of the bosonic string propagating in a background gravitational field, encoded by the Riemannian metric. The background Riemannian metric is the collection of coupling constants for the worldvolume theory, and these “flow” under renormalization of the worldsheet theory. This is the Ricci flow.

This is mentioned a bit more explicitly in the followuo entry Huisken on renormalization II. In particular, there it is mentioned that Perelman’s method involves considering the more general situation where the string sees not only a gravitationa background field, but also the Kalb-Ramand and notably the dilaton background fields. These three fields are precisely the three massless bosonic background fields of the bosonic string.

And notice that this Ricci-flow problem for the bosonic string is effectively the source where much of the interest in string theory origininated in: the worldvolume theory of the string is required to be conformal, which means that it sits at a fixed point of the renormalization group/Ricci flow. Which in the presence of only the gravitational background field says hence that the background metric has to be Ricci flat. Which happens to be precisely the Einstein-equation for gravity in the absence of other fields. If the other fields are taken into account, the equation for the fixed point of the Ricci flow changes accordingly to the equation of gravity coupled to other fields.

So the Ricci flow equation is one of the indications that strings may know about gravity.

Posted by: Urs Schreiber on November 16, 2009 9:29 AM | Permalink | Reply to this

Re: Huisken on Uniformization, I

Many thanks! An other expert’s answer as copy here. Amazing how things are interconnected, Manin’s castle seems to live in a very high dim space, as so many different parts of it are within a short distance :)

Posted by: Thomas on November 16, 2009 10:51 AM | Permalink | Reply to this

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