Tag Archives: Jordan curve theorem

Ricci Flow

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The departmental colloquium today was about certain aspects of the Ricci flow. Although the wikipedia page is quite intimidating, the geometric ideas behind it are quite intuitive. Several parts of the talk reminded me of the style of problems in our class. Since Aaron and I were the only students that went, I will try to explain some of it as well as I can (I really do not know a lot about this and perhaps someone will correct me if I make mistakes).

The Ricci flow is a generalization of the curve-shortening flow and mean curvature flow, which are themselves based on the heat equation. The heat equation tells us in good cases the heat will dissipate in finite time until it equal throughout. In simple cases, the other flows can be interpreted similarly, but their behavior is determined by the curvature of the geometric object. The curve-shortening flow shockingly enough shortens the lengths of curves in the plane and decreases the area at a constant rate so that it will go to zero in finite time. Also, it pushes out the curve where the curvature is negative and pushes in areas of positive curvature. That is, a loop will become more circular. The mean curvature flow is slightly more complicated but essentially acts the same on surfaces. The Ricci flow is a differential equation involving the curvature of a Riemannian manifold (smooth manifold with inner product on tangent spaces)  that basically generalizes these ideas. One key difference is that the Ricci flow runs on the manifold itself while the other two run on embedding or immersions in \( \mathbb{R}^2 \) and \( \mathbb{R}^3 \), respectively.

A large part of the talk and the part that reminded me of our class was discussing the singularities that arise under these flows for embedded loops, embedded spheres, and simply connected manifolds. As I mentioned above, the curve-shortening flow asymptotically sends loops to spheres and this happens to be the only singularity for embeddings of \( S^1 \), i.e., Jordan curves. Things are more interesting for surfaces. The mean curvature flow has three types of singularities for embedded spheres: the round sphere, a cylinder, and a sort of half cylinder (capped at one end). As I understand it, a big part in the Perelman proof is showing that these same types of singularities are the only ones that appear in the Ricci flow of a simply connected 3-manifold and that theses singularities are precisely the ones that can be removed via the connected sum without changing the topology of the manifold.

In our class, we proved the fundamental theorem of algebra by looking at what happens to the winding number as we passed through singularities (roots) and saw that the winding number changed. Similarly, a key part of the proof of the Jordan curve theorem is how the winding number changes passing through boundaries of connected components. This is the opposite of what happens with the Ricci flow on simply connected 3-manifolds: the topology does not change. I think this is quite interesting. Another idea that appears to be very close to these (sort of in between the two) is that the Sturm-Hurwitz theorem (analogue of fundamental theorem of algebra for trigonometric polynomials) can be proved using the heat equation. This proof was presented by Professor Tabachnikov in his MASS course last year, and if anyone is interested, I could show you my notes.

This is a lot of stuff at once, but hopefully you all find it interesting.

Exercise 12.1

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Suppose we have a compact \(K \subseteq \mathbb{C} \). We wish to show that the boundary of each path component of \(\mathbb{C} \setminus K \) is a subset of \(K\). Consider a path component \(P\), where \(\partial P \) is the collection of boundary points of \(P\). By definition, \( \forall q \in \partial P, q \notin P\). Thus \(q \in K \lor q\in L\), where \(L\) is a different path component of \(\mathbb{C} \setminus K\). Suppose \(q \in L\). We know that \( \mathbb{C} \setminus K\) is open, as its complement (\(K\)) is closed, as it is a compact subset of a metric space. Thus \(\exists \epsilon > 0 \ s.t. \ B\left(q, \epsilon\right) \subseteq \mathbb{C} \setminus K\). As \(q\) is a boundary point of \(P\), \( P \cap B\left(q, \epsilon\right) \neq \emptyset\). As \(B\left(q, \epsilon\right) \subseteq \mathbb{C} \setminus K\), there is a path between \(q\) and another point \(\hat q\), where \(\hat q \in B\left(q, \epsilon\right) \cap P\). This follows because \(B\left(q, \epsilon\right) \cap K \ = \emptyset\). As \(q\) and \(\hat q\) are path connected, they must be in the same path component by definition. This is a contradiction, so \(q \in K\). Thus, \(\forall q \in \partial P, q \in K\), so \(\partial P \subseteq K\).

Why is the Jordan curve theorem difficult?

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So, I decided to rearrange my course material and prove the Jordan curve theorem this week.

The JCT has a reputation for being hard to prove, and it is indeed a non-trivial result. But people don’t always understand why it is non-trivial.

For most curves that one thinks of, the theorem is elementary.  For example, I’ll review in class a very simple proof for polygonal curves, essentially that given in Courant and Hilbert’s classic What is Mathematics?  A similar proof works for differentiable or piecewise-differentiable curves.

Ross and Ross have an article called The Jordan Curve Theorem Is Nontrivial.  It contains some beautiful artwork – representational pen-and-ink drawings made with a single Jordan curve.  Nevertheless, the curves in these artworks are smooth; for them, JCT is elementary.

The real issue is the possibly exotic behavior of topological curves. For instance, the Julia sets of certain complex maps are topological circles, but unlikely to be smooth.  In the topological world, strange things can happen that do not happen in the smooth category.  A famous example is the Lakes of Wada: three connected open sets in the plane which share the same common boundary.  Of course, this boundary is not a Jordan curve.  But who is to say that it might not be!

It is this topological problem which is at the heart of the proof of the full version of JCT.