Fall 2025
Math 597: Minimal submanifolds and Willmore surfaces
MWF 1:25-2:15pm, McAllister 106
Office: 221 McAllister Building
Office Hours: Th 10-11am, or by appointment
Lecture Notes. I am typing these up as we go along. In particular, I am not investing much time into proofreading or referencing. If you find any errors, please let me know. Also, if you find these valuable and want extra detail somewhere, please let me know and may try to incorporate your request.
The rough plan of the course is to cover the following topics:
Introduction to Riemannian and conformal geometry of manifolds and their submanifolds.
First and second variation of the area functional.
Some nonexistence results.
The stable Bernstein theorem.
Classification of minimal surfaces in the three-sphere of low index.
The Willmore functional for surfaces immersed in conformal manifolds.
The Li-Yau conformal volume and a lower bound for the Willmore functional in round spheres.
The Willmore Conjecture in Euclidean 3-space.