The n-Category Café

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.