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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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.

Article information

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

Dates
First available in Project Euclid: 31 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1414761911

Mathematical Reviews number (MathSciNet)
MR3273877

Zentralblatt MATH identifier
1328.53085

Subjects
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]

Citation

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. https://projecteuclid.org/euclid.ojm/1414761911


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