Archive for the ‘Matt’ Category

My Favorite Thing About Soapy Surfaces, I

July 3, 2007

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 $\mathbb{E}^3$. Any smooth surface in $\mathbb{E}^3$ 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 $f : D \rightarrow \mathbb{E}^3$ with $D \subset \mathbb{C}$.

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 $\mathbb{E}^3$ the curvature vector $\kappa_p$ is exactly the force felt by $p$ due to the tugging of its neighboring points.