Manifold variational problems

Given a q-manifold, a valuation X like Euler characteristic or volume and an integer , we can look at the variational problem to maximize or minimize X(H) among all submanifolds of X. A submanifold is given by a function . This is a finite problem so that we can in principle look at all possible cases. The problem also works for manifolds with boundary. In the case of manifolds with boundary, we can also look at the constrained variational problem where the function is fixed on the boundary. This then produces a solution to a cobordism problem.

These type of problems are very beautiful, not only because they remind of some of the most beautiful problems in the continuum, like the Plateau problem, which asks for minimal surfaces given some boundary constraint. These are soap bubble problems. It also reminds about the Dirichlet problem, especially if we look at , the number of ‘points’ of the submanifold as this is some sort of Dirichlet energy.

An other beautiful gem is to look at the case k=q, where we look for point packing problems. In the discrete these are coloring problems. It is already highly non-trivial in the case of a 2-sphere G, where the problem of coloring with 4 colors is equivalent to the 4-color problem! If you can color a 2-sphere with 4 colors, then you can color any planar graph with 4 colors as Whitney already knew. Heawood looked at the problem when we can color with 3 colors. This is topologically interesting and requires the 2-sphere to be Eulerian. It is actually equivalent to being Eulerian. Every Barycentrically refined 2-sphere is Eulerian and can be colored with 3 colors. The coloring is the dimension of the point, when it was the original graph.

But this is just the tip of the iceberg. An interesting problem is the case q=4, k=2, where we want to find a 2-manifold in a 4-manifold with minimal Euler characteristic. This means to find the sub-manifold with maximal genus. When maximizing the Euler characteristic, we have a sphere packing problem.

If this is not exciting, I do not know what is. We have a discrete finite problem which relates to some of the most iconic problems we know in mathematics: sphere packing, map coloring, minimal surfaces or Dirichlet problems. Very obvious is also the physics connection, for obvious reasons. The world sheet of a 2-manifold can be seen as a string trajectory in space time.

Here is the talk (August 1 is the Swiss national holiday). As a dual citizen, I of course celebrate both birth days July 4th and August 1). I have on both holidays had 360 degree videos in my feed. Here was the 4th of July movie featuring the american flag. The Swiss flag had been given to me once by a calculus student (his name was Miles) about 10 years ago. The intro sequence of the following movie was taken last week, when the sky was more clear. This summer, my favorite route (done dozens of times now) is to run through Sommerville, pass the highschool in Sommerville, and follow the Green line. One can run on a nice path until the new park between the Science Museum and the bunker Hill Memorial bridge and then take return back along the river. The shortest version is about 15 k. I sometimes extend to the Bunker Hill monument and Constitution getting a bit into the American Revolution vibe. Both Switzerland as well as the USA were born in a rebellion spirit. Switzerland has Wilhelm Tell , the US has the Paul Revere.

In any case, this topic looks like a rich field. There are many, many interesting questions. I see for example that given a boundary value problem, the number of solutions which are maxima is small, sometimes even unique. This is a bit unexpected, given that Euler characteristic is a quantized quantity.