Self-similar solutions to the mean curvature flows on Riemannian cone manifolds and special Lagrangians on toric Calabi--Yau cones

Abstract

The self-similar solutions to the mean curvature flow have been defined and studied on the Euclidean space. In this paper we propose a general treatment of the self-similar solutions to the mean curvature flow on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend results on special Lagrangian submanifolds on $\mathbb{C}^{n}$ to the toric Calabi--Yau cones over Sasaki--Einstein manifolds.