Self-shrinkers for mean curvature flow
Mean curvature flow can be defined as the gradient flow of area which means that it deforms a given surface such that its area decreases the fastest. Surfaces which shrink homothetically under mean curvature flow without changing their shape are called self-shrinkers. They are of great importance for the general theory of mean curvature flow because they model local singularities developing along the flow.
In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Min-max techniques allow a rigorous existence proof for these surfaces. Conjecturally, they have precisely one asymptotically conical end. For large genus this can be confirmed via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity.
References
- R. Buzano, H. Nguyen and M. B. Schulz, Noncompact self-shrinkers for mean curvature flow with arbitrary genus, J. reine angew. Math. (2024).
