February 2, 2007

Huisken on Uniformization, II

Posted by Urs Schreiber After some motivations in part I, here the second part of my transcript on G. Huisken’s talk on Uniformization via the Heat Equation.

So we are interested in one-parameter families $t \mapsto g(t)$ of metric on some manifold $M$, satisfying the Ricci flow equation $\frac{d}{dt} g_{ij} = -2 R_{ij} \,,$ where $R_{ij}$ are the components of the Ricci tensor obtained from $g$.

Here are some examples of such flows:

i)

Let the underlying manifold $M^n = S^n$ be the $n$-sphere and let it be equipped, at $t=0$ with the canonical metric $g_0 = g_\mathrm{canon}$ of that $n$-sphere. Then, under the above Ricci flow, this sphere simply contracts $g(t) = \psi(t) g(0) \,,$ where $\psi(t) = c(T-t)$ is simply a constant times the remaining time until complete collaps “to a point” (= $n$-sphere with everywhere vanishing metric).

ii)

Let the underlying manifild $M^n = S^2 \times S^1$ the the 2-sphere times the circle, equipped with the canonical metric $g_0 = g_\mathrm{canon} \,.$ Then, under the above Ricci flow, the 2-sphere factor shrinks as in the above example, while the circle factor remains invariant. This is of course due to the fact that the circle by itself has no intrinsic curvature at all.

iii)

there is a class of solutions that are called translating solutions. These are solutions where the flow is only by diffeomorphism.

In dimension $n=2$ an example of this is the cylinder which extends indefinitely in one direction and is glued to a cap in the other direction.

This is the Witten black hole (cigar) solution.

In dimension $n=3$ there is a similar solution, something that is asymptotic not to a cylinder but to a paraboloid.

Early results on Ricci flow:

Theorem (Hamilton):

i) In dimension $n=3$, after rescaling, every initial metric with positive Ricci-curvature converges to a metric of constant positive (sectional) curvature.

This proves the Poincaré cojecture for 3-manifolds that have positive Ricci-curvature.

ii) In dimension $n=2$, after rescaling, Ricci-flow always converges to a metric of constant Gauss curvature (due to Hamilton & Chow).

(two more points, which I cannot really reproduce from my notes)

In dimension $n=3$, main examples for structures of singularities are

i) The 3-sphere, which contracts to a point and the $S^2 \times \mathbb{R}$, which contracts to a line (this are the two examples from above).

ii) When two blobs of 3-manifolds are somehow connected, three things may happen:

Either one of these blobs is much smaller than the other, then it will simply shrink away and be absorbed by the other.

Or both of them are of comparable size. Then their connection will become thinner and thinner until it degenerate to a “neck” (an $S^2$ shrinks to a point).

iii) Or one of the blobs is right at the border between being much smaller and being comparable to the other. Then it will shrink to a point before being completely absorbed by the larger blob: a tip foms that locally looks like one of the “translating solutions” from above.

the hope was that these are the only three things that “can go wrong”, and that it may be possible to continue the flow after suitable removing the singularities of the above kind.

Hamilton’s startegy was:

i) given an arbitrary metric $g_0$ on a 3-dimensional $M^3$, let the metric flow until a singularity occurs

ii) prove that only the above canonical examples of singularities can occur and give a precise quantitative description of their asymptotic shape

iii) cut “necks” and replace cylinders by “caps”; cut “horns” and replace these by caps; reduce the curvature in this process.

iv) continue the smooth flow on the resulting pieces

v) prove that there are only finitely many such surgeries necessary.

There is one deep trouble: the interval between one and the next surgery is usually too short for sufficient smoothing to take place, and hence for recognizing cylinders (necks) as such. This is what makes the program so hard to carry through.

Another problem is to rule out doubly degenerate neckpinches. These are configurations asymptotic to a “translating 2-d solution” times a 1-dimensional interval. In this situation one encounters small volume in large balls of small curvature.

That’s where Perelman came in. His idea was to extend the Ricci flow equation by adding on another degree of freedom encoded in a scalar function $f$ on $M$ and consider the combined flow defined by \begin{aligned} & \frac{d}{d t} g_{ij} = -2 R_{ij} + 2 D_i D_j f \\ & \frac{d}{d t} f = -\Delta f + R \end{aligned} \,. One can show that this is, upt to diffeomorphism, equivalent to the original pure Ricci flow. Moreover, this is still a gradient flow, but now for the quantity given by $E(g,f) = \int_M ( R(g) + (\nabla f)^2 ) e^{-f}\; d\mu \,,$ in the class where $dm = e^{-f}d\mu$ is fixed.

Perelman calls this quantity an entropy. But in fact, this is the Lagrangian density for Einstein-Dilaton gravity, where the function $f$ is what is called the dilaton field.

(G. Huisken remarked that he wondered whether Perelman first had this idea and then found that it is the same structure as appearing in string theory, or whether he was inspired by the string theoretic description in the first place.)

The above system of equations is well-known to describe the renormalization-group flow of a string propagating on $M$, and coupled to the “gravitational background field” encoded in the metric $g$ and the “dilaton background field” encoded in $f$.

Of some relevance is the fact that in the equation for the dilaton, the Laplace operator appears with the opposite sign. This “backward heat equation” is what allowed Perelman to prescribe not initial but final data and use this to rule out the occurence of certain problematic collaps singularities.

In non-technical manner, Perelman’s statements can be described as follows:

the a-priori estimate for the smooth flow and the reduction of entropy and volume can be used to prove:

given $g_0$, there is a precise quantitative algorithm that realizes Hamilton’s strategy with finitely many surgeries.

On all Riemannian 3-manifolds $(M^3,g_0)$ that are simply connected, $\pi_1(M^3) = 0$, the algorithm stops after finite time, when $\mathrm{vol}(g(t)) \to 0$.

This implies the truth of the Poincaré conjecture.

For $t \to \infty$ one obtains geometrization according to Thurston: at the end of the flow only $S^3$s remain, so $M^3$ must have been a connected sum of $S^3$s in the first place, hence itself an $S^3$.

Finally, Gerhard Huisken ended by mentioning a result by himself, which is an analog of this for mean curvature flow (i.e. for embedded 3-manifolds):

Theorem (Huisken, 2006): Given any $(M^3,g) \hookrightarrow \mathbb{R}^4$ with $R \gt 0$, there is an algorithm with surgery of the above kind that ends in finite time.

Corollary: Any such 3-hypersurface in $\mathbb{R}^4$ is diffeomorphic to $S^3$ or a finite connected sum of $S^2 \times S^1$. It is the boundary of a handle body.

That’s the end of my transcript.

Posted at February 2, 2007 1:36 PM UTC

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Where is the Kalb-Ramond field?

it seems unlikely that Perelman’s modified Ricci flow is just by coincidence the same as the RG flow of a string in gravitational and dilaton background.

Now, the gravitational field and the dilaton field are just two of the three massless fields that are present in this context.

In addition, there is also the (“Kalb-Ramond” curvature) 3-form field $H \in \Omega^3(M) \,,$ which would add an additional term of the form $\int_M H \wedge \star H$ to Perelman’s “entropy” functional, and accordingly lead to a more general gradient flow.

What happens with Hamilton’s strategy when we include this?

Would this affect all these collaps singularities? Maybe enhance them? Or – maybe reduce them?

We know that with the standard metric, the 3-sphere defines a conformal fixed point (except for the anomaly) of the renormalization group flow when the above 3-form $H$ is the parallelizing torsion 3-form of $S^3$ (and the dilaton is constant). This corresponds to the $\mathrm{SU}(2)$ Wess-Zumino-Witten model.

Shouldn’t that gradient flow in the case where we include the KR 3-form tend to want to converge to this WZW model?

Wouldn’t this seem to say that the 3-form contribution prevents the 3-sphere from collapsing under the RG flow?

Probably not, since that would seem too goo to be true (given all the trouble that people went through with the Poincaré conjecture.)

But, given that apparently the role of the dilaton was only recently appreciated in this business, is it clear that nothing would be gained from also including the KR field?

It would seem unnatural not to consider this degree of freedom.

Posted by: urs on February 2, 2007 3:50 PM | Permalink | Reply to this

String theory and Poincare conjecture

This is brilliant! I had no idea that Perelman’s proof can be interpreted in terms of standard quantum field theory and renormalization group ideas. I was sure that the ‘Ricci flow’ must have an elegant physical interpretation, but I didn’t know it was so concrete.

I have some points to raise:

1. The geometry of renormalization is cool! I raised this point last year, when I talked about Segal’s seminal talk on ‘A mathematician looks at quantum field theory’ given at the Kavli Institute in 1999.

Indeed, on page 2 of his first talk, he precisely makes the point about how the gradient flow of the action functional (the one which computes the area) has a lot of interesting geometry. For instance, in 2d the flow can form “bubbles” while in 3d it can “cavitate”. He explicitly makes the point about how this is related to renormalization.

I always knew he was saying something very deep there… and indeed it was true, as Perelman’s proof of the Poincare [Ed : how do you get an accent?] conjecture shows! I’d be very interested to see what others have to say about the geometry of renormalization, as Segal presented it there.

2. Can you explain how these kinds of ideas interact with the ideas behind the Nambu Goto action in string theory versus the Polyakov action? What I’m getting at is that this idea - of using an auxiliary dilaton field as a convenient technical tool - has been used for ages in quantum field theory (at least, in the primitive way I understood it!). I mean, consider statistical field theory, etc, etc.

3. Could you expand more upon the difference between the intrinsic and extrinsic (i.e. manifolds embedded in $\mathbb{R}^n$) Ricci flows? I mean, we know from Nash’s embedding theorem that any Riemannian manifold can be isometrically embedded in $\mathbb{R}^n$. Wouldn’t it be technically simpler to perform the whole thing inside $\mathbb{R}^n$? Because, when its living in $\mathbb{R}^n$, it really *does* shrink or expand as the flow proceeds, whereas in the intrinsic version, the abstract set of the manifold stays the same - only the metric changes. The former is a bit more intuitive, to me at least.

4. The idea of adding in the Kalb-Ramond field is great. I’m convinced it would do some really interesting stuff.

Posted by: Bruce Bartlett on February 2, 2007 5:31 PM | Permalink | Reply to this
Posted by: urs on February 2, 2007 5:44 PM | Permalink | Reply to this

Re: how do you get an accent?

I have a more complete list of such entities (not just alphabetic ones) here, where you can also test how the funny ones appear in your browser. (I also have a list of numerical code entities here, but it’s an extremely large list that may crash your browser.)

Posted by: Toby Bartels on February 2, 2007 10:14 PM | Permalink | Reply to this

Re: String theory and Poincare conjecture

Wouldn’t it be technically simpler to perform the whole thing inside $\mathbb{R}^n$?

I believe G. Huisken briefly made a comment on that. But, alas, I forget what it was…

Posted by: urs on February 2, 2007 5:45 PM | Permalink | Reply to this

Re: String theory and Poincare conjecture

Just to quickly add that to appreciate Segal’s comments on the geometry of renormalization - page 2 of his talk - you need to listen to the audio - there’s a lot more information there :-).

On the topic of audio, it should be emphasized that all the talks at Fields from the Thematic Program on Geometric Applications of Homotopy Theory, such as eg. Andre Joyal’s course on quasicategories, or the talks from the workshop, all have audio to go with the slides. ‘Not being there’ is no excuse for not attending the talks! :-)

Posted by: Bruce Bartlett on February 2, 2007 6:02 PM | Permalink | Reply to this

Re: String theory and Poincare conjecture

Another excuse for not listening to the Fields Institute talks is not wanting to use the obnoxious, ad-ridden ‘Real Player’ software, which seems to be the only option!

I had an old version of Real Player until my computer died in Shanghai. The new version has so many more ads and nasty stuff built in that I’ve avoided getting it.

I guess I’ll need to try taming the wild RealPlayer.

Or maybe I’ll try some alternative software.

Anyway, thanks for noticing that the talks are available now. I’ll add links to my page.

From some anonymous people in the know:

Real’s advertising company at the time was working in parallel to us to conduct consumer research on the Real brand. What they found was shocking. Real, at that time, had almost complete brand awareness among people online. Their brand was the only brand other than Microsoft, Netscape, and Yahoo! to attain this level of recognition. But unlike these other companies, Real was universally disdained by customers. Almost every customer mentioned the deceptive download process and the way in which RealPlayer took over their system, installing software they didn’t want and didn’t need.

My company was asked to recommend changes to the download process. We created a new process that allowed users to download the player in 3 steps, a reduction of 10 steps from the previous process.

What we found in the process of this project was very concerning. Real as a strategy had for years intentionally obscured the free download link. Even when users found the link, the download process would try and trick (there is no other word for it) the user into downloading the pay version of the software. Real would even test multiple versions of their design to see which ones were more effective at this.

Real would resort to even more disgusting (and probably illegal) tricks. One page in the process would show the user some very legitimate choices above the “fold” (the bottom of the area of a web page that can be shown in a window without forcing the user to scroll). However, beneath the fold, Real had options for additional plug-ins with dubious value such as sound enhancers and web accelerators, that were selected BY DEFAULT. If a user did not scroll down (and the design cleverly did not signal to the user that they had any reason to do so) they would not see that there were choosing to purchase around $50 in additional software. Real told us that this page alone was responsible for driving their average order up by$25. They even told us that most people didn’t even know they were buying the additional software. When we told them that we found these tactics user un-friendly, unsustainable, and bad for the brand, they agreed. But they also told us that they were “hooked on! it like heroin” and didn’t want to change anything for fear of loosing revenue. Despite the fact that Real was knowingly misleading customers, and that it was driving down the value of the company and the brand, and we had designed a solution to the problem, Real refused to implement it.

Posted by: John Baez on February 2, 2007 7:59 PM | Permalink | Reply to this

Re: String theory and Poincare conjecture

Can you explain how these kinds of ideas interact with the ideas behind the Nambu Goto action in string theory versus the Polyakov action?

The fact that the string’s RG flow is of the form discussed above applies directly only to the Polyakov action.

Even though canonical quantization of the NG action is equivalent to that of the Polyakov action, I am not sure if the RG flow argument goes through for the NG version. I have never seen it done in that context. (Maybe somebody else reading this here can comment on that.)

I don’t know if it helps, but: the NG action is better thought of as describing not the “fundamental string” but the $D$-string”, i.e. the 1-dimensional D-brane (which sweeps out a 2-dimensional world-volume).

The natural generalization of the NG action including various background fields is the Dirac-Born-Infeld-like action for D-branes.

While string theory started out with playing around with the NG action, the more modern perspective is that a “fundamental string” is a (super-)conformal 2-dimensional QFT of fixed central charge (namely 15).

The Polyakov action defines certain such CFTs, namely those that arise as sigma-models.

Posted by: urs on February 6, 2007 12:57 PM | Permalink | Reply to this

Re: Where is the Kalb-Ramond field?

I just came across this old thread while googling something else. Yes, it is possible to include the NS-NS B-field and study RG flows in string theory with B-field using Ricci flow-inspired techniques. My collaborators and I do this in Nucl.Phys. B739 (2006) 441-458; arXiv:hep-th/0510239. For example, one can show that this RG flow is monotonic on compact target spaces, by constructing a Perelman-like monotonic functional. We found that this functional equals the central charge at RG fixed points, making it the closest thing to a C function.

Posted by: Suneeta on January 13, 2009 7:18 PM | Permalink | Reply to this

Re: Where is the Kalb-Ramond field?

Interesting, thanks. It’s good that these blog posts are around for a while…

I remember back then when I talked to him, Huisken was interested in learning more about the string theory side of this story and how it might generalize to inclusion of the $B$-field. Maybe you could drop him a note (unless of course this is obsolete by now).

So, Perelman used the dilaton field to get better control on some technicalities in the geometrization program, as recalled in the above entry. Is it at all conceivable that introducing on top of that the $B$-field, as you do, provides even more useful control over the singularities under the Ricci flow?

Posted by: Urs Schreiber on January 13, 2009 7:29 PM | Permalink | Reply to this

Re: Huisken on Uniformization, II

It is definitely *not* easier to do the Ricci flow by embedding the manifold into Euclidean space, because you need to use a large dimension, and the embedding is highly non-unique. This means that the relationship between the extrinsic and intrinsic geometry is quite messy and complicated. So you end up trying to study the intrinsic geometry (which is what really matters) indirectly in a much more complicated way.

Posted by: Deane on February 8, 2007 5:29 PM | Permalink | Reply to this
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