Daren Cheng, Department of Pure Mathematics, University of Waterloo
"Stable minimal surfaces in R4"
It is well-known that complex submanifolds of Kahler manifolds are stable critical points of the volume, and it is natural to ask when the converse is true. In this talk I present an important such result, due to Micallef, in the setting of surfaces in R4. The main theorem implies that a complete, oriented, immersed, stable minimal surface in R4 with quadratic area growth must be holomorphic with respect to some orthogonal complex structure on R4. The main tool is a complexified version of the second variation formula of the area, which I'll briefly review at the beginning of the talk along with some other necessary background. I'll then describe the proof of the above statement, which roughly goes as follows: First we average the second variation of area along suitable deformations. Then, using the stability assumption and a result due to Fischer-Colbrie and Schoen, we play the previous computation against the quadratic area growth to show that the surface is "infinitesimally holomorphic", from which the desired conclusion follows without too much work.
Reference: Micallef, "Stable minimal surfaces in Euclidean space", JDG 1984.
Zoom meeting: https://zoom.us/j/93359434940?pwd=NEtnaC9IOFFlZkZ3NkFYNjFLd1FSQT09