Quantum Lunch: Schrödinger Operators in Some Curvature Problems

Speaker: Niels Martin Møller

Schrödinger Operators in Some Curvature Problems

I will explain how some of the "finer" properties of Schrödinger operators that one can study have become directly important in the modern theory of minimal surfaces and curvature flows.

Why, for example do we need to know (parts of) the spectrum of the quantum harmonic oscillator on the disk and the sphere, and accurate estimates of its Green's function on a cone, in order to understand singularity formation for the mean curvature flow?

And what is the connection between the behavior of the eigenvalues of the Laplacian on a round sphere minus small(er and smaller) disks and the "moduli space" of complete, embedded minimal surfaces? This can be tricky business: Some imagined minimal surfaces can't possibly exist - and the ones that do, we would like to be able to construct as directly as possible, starting from our "fine" understanding of the Schrödinger operators vs. the geometry.

Time permitting:
What does the comparison of quantum double wells to single wells have to do with the question of variational stability of (pieces of) the catenoid minimal surface?