PhD in Mathematics

An introduction to Mean Curvature Flow

An introduction to Mean Curvature Flow

Academic year 2024/2025

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Program

The aim of this course is to give an introduction to the mean curvature flow and explain some applications, in particular in the case of two-convex hypersurfaces. We aim at finding a good balance between heuristic or intuitive arguments on the one hand (often illustrating the ideas and results with pictures and foregoing analytical rigor), and precise calculations and proofs on the other hand. We start with a discussion of the basics of mean curvature flow on hypersurfaces (explicit examples, evolution of geometric quantities, finite time singularities, parabolic rescaling, etc.). We then explain Huisken’s monotonicity formula and how it can be used to study singularity models of certain finite time singularities, before restricting to the two-convex case, where all local singularities are essentially modelled on shrinking cylinders. Next, still assuming two-convexity, we sketch the idea of mean curvature flow with surgery and explain a gluing construction to topologically undo the surgeries again. We then finish the course by discussing applications of this surgery and gluing approach towards the study of path-components of the spaces of two-convex embeddings of spheres and tori into Euclidean space.

• Buzano-Haslhofer-Hershkovits: The moduli space of two-convex embedded tori (article)
• Ecker: Regularity Theory for Mean Curvature Flow (book)
• Haslhofer-Kleiner: Mean curvature flow of mean convex hypersurfaces (article)
• Haslhofer-Kleiner: Mean curvature flow with surgery (article)
• Mantegazza: Lecture Notes on Mean Curvature Flow (book)

Notes

Hours: 20. Period: April - June 2025

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