The n-Category Café

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 $$\frac{d}{d t} u = \Delta u$$ for a 1-parameter family of functions $$u : (x,t) \mapsto u(t,x)$$ on the real line.

It is noteworthy that this equation describes the gradient flow of the Dirichlet-Integral $$E(u) := \frac{1}{2}\int_\mathbb{R} |u'|^2 \; d x \,,$$ i.e. the fastest way to descrease this functional in the $L^2$-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 $n$-dimensions) $$u(x,t) = \frac{1}{(2\pi t)^{n/2}} \exp \left( - \frac{|x|}{4t} \right)$$ are self-similar in that for any $\lambda \gt 0$ we have $$u(x,t) = u(\lambda x, \lambda^2 t) \,.$$

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

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

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 $$\int u \mathrm{log}u \; d x$$ but now with respect to the Wasserstein metric (this applies to $u \gt 0$).

There is an estimate, called the Li-Yau Harnack estimate, which says that $$\Delta u \geq |D \mathrm{log}u|^2 - \frac{n}{2t} \,.$$ This holds as stated for $\mathbb{R}^n$ 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 $$\Gamma(\cdot,0) : S^1 \to \mathbb{R}^2$$ in the real plane and let $t \mapsto \Gamma(\cdot,t)$ be a 1-parameter family of such curves satisfying the equation $$\frac{d}{dt} \Gamma(p,t) = \vec \kappa(p,t) = \kappa(p,t)\vec \nu(p,t) \,,$$ where $\vec\kappa$ is the second arc-length derivative of $\Gamma$ $$\vec \kappa(p,t) = \kappa(p,t) \vec \nu(p,t) = \frac{d}{d s_t^2} \Gamma(p,t) \,,$$ $\vec \nu(p,t)$ is the unit normal vector to the curve at $(p,t)$ and $\kappa(p,t)$ 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 $L^2$-norm), now simply of the length of the curve $$E(\Gamma) = \int_{S^1} d s \,.$$

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 $R_0$ centered at $x_0$ in $\mathbb{R}^2$, $$\Gamma_0(S^1) = S^1_{R_0}(x_0) \,.$$ Then under the above flow it will remain circular and centered at $x_0$ $$\Gamma_t(S^1) = S^1_{R_t}(x_0)$$ but shrink in radius according to $$R_t = \sqrt{R_0^2 - 2t} \,.$$ This goes on until $$t = T := R_0^2/2$$ 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 $\Gamma_0(S^1)$ is an embedded curve then $\Gamma_t(S^1)$ 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 $$|\vec \kappa| \to \infty$$ can become singular for $t \to T$. 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 $n$-dimensional something embedded in $\mathbb{R}^{n+1}$ $$F_0 : M^n \to \mathbb{R}^{n+1}$$ we define a flow by the quasi-linear parabolic system $$\frac{d}{dt}F(p,t) = \vec H(p,t) = (\lambda_1 + \cdots + \lambda_n)(p,t) \cdot \vec\nu(p.t) := \Delta_t F(p,t) \,.$$ Here the $\lambda_i$ are the principal extrinsic curvatures, i.e. the $n$ eigenvalues of the the second fundamental form of the hypersurface.

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

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

The intrisic analog of this is Hamilton’s Ricci-Flow (from 1982). This is a a manifold with a family of metrics $t \mapsto g(t)$ that flow according to the equation $$\frac{d}{dt}g_{ij} = -r R_{ij}(g) \,,$$ where $R_{ij}$ 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 $$R_{ij} = \Delta g_{ij} + \cdots \,.$$ 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)