Osaka Journal of Mathematics

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

Akito Futaki, Kota Hattori, and Hikaru Yamamoto

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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.

Article information

Osaka J. Math., Volume 51, Number 4 (2014), 1053-1081.

First available in Project Euclid: 31 October 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 55N91: Equivariant homology and cohomology [See also 19L47]


Futaki, Akito; Hattori, Kota; Yamamoto, Hikaru. Self-similar solutions to the mean curvature flows on Riemannian cone manifolds and special Lagrangians on toric Calabi--Yau cones. Osaka J. Math. 51 (2014), no. 4, 1053--1081.

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