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 flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows 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 the results on special Lagrangian submanifolds on ${\mathbb{C}}^n$ to the toric Calabi-Yau cones over Sasaki-Einstein manifolds.

References in corpus (3)

• Uniqueness and examples of compact toric Sasaki-Einstein metrics
• A Compactness Theorem for the Second Fundamental Form
• Special Lagrangians and Lagrangian self-similar solutions in cones over toric Sasaki manifolds

Cited by in corpus (6)

• Ricci-mean curvature flows in gradient shrinking Ricci solitons
• On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking Ricci solitons
• Special Lagrangians and Lagrangian self-similar solutions in cones over toric Sasaki manifolds
• Some topics on Ricci solitons and self-similar solutions to mean curvature flow
• Deformations of special Legendrian submanifolds in Sasaki-Einstein manifolds
• Self-similar solutions of curvature flows in warped products