Preface

These notes are based on a course on minimal hypersurfaces given at the Beijing Institute of Mathematical Sciences and Applications (BIMSA). The goal is to give a concise introduction to several central ideas in the theory of minimal hypersurfaces, with an emphasis on stability, curvature estimates, regularity, and applications to scalar curvature and mathematical general relativity.

The material begins with basic Riemannian geometry and the first and second variation formulas. It then discusses geometric measure theory, regularity and compactness theorems for stable minimal hypersurfaces, stable Bernstein-type problems, and recent applications of minimal hypersurface methods to positive scalar curvature, the positive mass theorem, and related rigidity questions. The exposition is intended to be self-contained enough for graduate students in geometry, while still keeping the main analytic and geometric mechanisms visible.

Comments, corrections, and suggestions are welcome and may be sent to [email protected].