I’m Matt, also still a grad student at Cornell. I wanted to start of with a post about a problem which has motivated (sometimes very indirectly) much of my interest in PDEs, geometry and category theory (!)
Let’s talk about elastic membranes sitting in Euclidean 3-space . Any smooth surface in inherits a complex structure which is given by the cross-product with the unit normal, so we might as well assume that we are always working with a conformal parametrization with .
To understand minimal surfaces geometrically, it is helpful to first think about curves. Imagine holding a rubber band along a curve and pinning it at its endpoints. If we let go, each point will get pulled on by its neighbors and move off some distance in the normal direction. The force felt by each point is proportional to the curvature. In other words, for curves in the curvature vector is exactly the force felt by due to the tugging of its neighboring points.
Now what if we had a whole surface made of elastic instead of just a curve? Each normal plane at intersects the surface in some curve — the curvature of this curve tells us how much is being tugged on by its neighbors in this particular direction. So to find out the total influence of neighboring points on , we just need to add up all these curvatures. It isn’t too hard to convince yourself that the total force felt by is the mean curvature vector . So for the surface to be sitting still like a nice soap film and not wiggling all over the place, the Laplacian had better be zero everywhere; this property characterizes minimal surfaces.
We can say all that in a nice invariant way by utilizing the operator (precomposition by a 90 degree rotation): the Laplacian is given (more-or-less) by the operator , so defines a minimal surface iff . Can we interpret this condition more geometrically? You bet!
Let’s start with a minimal surface and a closed, contractible curve . cuts out a disc-shaped subset . Now meditate on this: how does tug on ? You should be able to convince yourself that the force felt by is where is the unit tangent vector field which is perpendicular to . So in order for to stay put, it had better be true that for all contractible loops . In other words, is a closed form ().
Now if is closed and is closed, then certainly is as well: we could replace with the unit tangent vector at angle from and all calculations in the previous paragraph would still be true. So we could start with a conformally parametrized minimal surface , replace with and then integrate to get a new surface . Everything that applied to the old surface applies to these new ones as well — in particular, they are all minimal!
This is weird: minimal surfaces don’t come alone, but in families! What is even stranger is that each surface in a family has the same Gaussian curvature () and the same mean curvature function () as the others. For a generic surface in , and are enough to determine the surface up to Euclidean motion, so there is something truly exceptional going on here.
Well, that was an awful lot for a first post and I didn’t even get to the really neat and weird stuff yet. I think I’ll break here — there are ten thousand other incredible properties of minimal surfaces that I could go in to, but the internet might run out of space first. But before you leave, check out the family associated to the catenoid up close and personal: http://www.foundalis.com/mat/helicoid.htm