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

Akito Futaki, Kota Hattori+1 more|arXiv (Cornell University)|Dec 27, 2011

Geometric Analysis and Curvature Flows13 references7 citations

TL;DR

This paper extends the theory of self-similar solutions to the mean curvature flow from Euclidean space to Riemannian cone manifolds, introducing a generalized definition of self-similarity using the position vector's normal component. It proves that under type I_c singularities—where the flow collapses to the cone apex with controlled radial decay—parabolic rescaling yields a limit that is a self-similar solution, generalizing Huisken's classical result to singular 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.

Motivation & Objective

• To generalize the concept of self-similar solutions in mean curvature flow from Euclidean space to Riemannian cone manifolds.
• To define and analyze type I_c singularities, where the flow collapses to the cone apex with controlled radial decay.
• To extend Huisken's rescaling technique and monotonicity formula to Riemannian cone manifolds.
• To construct special Lagrangian submanifolds on toric Calabi-Yau cones via self-similar solutions.
• To relate the infinitesimal deformation space of special Lagrangian cones to the spectrum of the Laplacian on the link manifold.

Proposed method

• Define the Riemannian cone manifold $C(N) = N \times \mathbb{R}^+$ with metric $\bar{g} = dr^2 + r^2 g$.
• Introduce the position vector $\vec{F} = r \frac{\partial}{\partial r}$ and define self-similar solutions via $H = \lambda \vec{F}^\perp$.
• Introduce type I_c singularities via three conditions: Type I decay of second fundamental form, radial collapse to apex, and $K_1(T-t) \leq \min_M r^2(F_t) \leq K_2(T-t)$.
• Apply parabolic rescaling $F^\lambda(p,s) = (\pi_N(F(p,T + s/\lambda^2)), \lambda r(F(p,T + s/\lambda^2)))$ to analyze singular limits.
• Use monotonicity and rescaling techniques to show subsequential convergence of rescaled flows to self-similar solutions.
• Relate special Lagrangian cones to harmonic 1-forms and eigenfunctions of the Laplacian on the link via $\beta = r\varphi\,dr + r^2\gamma$ with $\gamma = \frac{1}{2}d\varphi$ and $\Delta_\Sigma \varphi = 2n\varphi$.

Experimental results

Research questions

• RQ1 Can the classical theory of self-similar solutions in mean curvature flow be extended beyond Euclidean space to Riemannian cone manifolds?
• RQ2 What conditions ensure that parabolic rescaling of a mean curvature flow on a cone manifold yields a self-similar limit?
• RQ3 How do type I_c singularities—characterized by radial collapse to the apex—generalize Huisken's original result on singularity models?
• RQ4 What is the structure of the infinitesimal deformation space of special Lagrangian cones over Sasaki-Einstein manifolds?
• RQ5 How is the space of special Lagrangian cones on toric Calabi-Yau cones related to the spectrum of the Laplacian on the link?

Key findings

• The paper establishes that for a mean curvature flow on a Riemannian cone manifold with a type I_c singularity, there exists a subsequence of parabolic rescalings that converges to a self-similar solution.
• The limit solution satisfies $H = \lambda \vec{F}^\perp$, generalizing the self-similar condition from Euclidean space to cones.
• The infinitesimal deformation space of special Lagrangian cones is isomorphic to $\mathrm{Ker}(\Delta_\Sigma - 2n)$, where $\Sigma$ is the link of the cone.
• Special Lagrangian cones arise precisely when the Lagrangian angle is constant and the associated 1-form $\beta$ satisfies $d\beta = d*\beta = 0$, leading to $\Delta_\Sigma \varphi = 2n\varphi$.
• The Reeb vector field $\xi$ does not generate special Lagrangian deformations, as $\hat{\omega}(\xi) = -r\,dr$ fails to satisfy $d*\beta = 0$, confirming that such deformations are not special Lagrangian.
• The construction of self-similar solutions on Riemannian cone manifolds is achieved via the spectral condition $\Delta_\Sigma \varphi = 2n\varphi$, linking geometry to analysis on the link.