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Self-similar solutions to the mean curvature flows on Riemannian cone manifolds and special Lagrangians on toric Calabi-Yau cones. (English) Zbl 1328.53085

Summary: 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.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
55N91 Equivariant homology and cohomology in algebraic topology
53C38 Calibrations and calibrated geometries
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References:

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