MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.52 Yamamoto Lagrangian shrinkers in Calabi–Yau cones26AC43F0ACEF

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. Yamamoto — 2012-03-18 (first arXiv submission).
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For every integer $g\ge 1$, Yamamoto constructs a real six-dimensional toric Calabi–Yau cone $C(S)=M_ g$ and a compact Lagrangian shrinking immersion with parameter manifold $\Sigma _ g\times S^1$, where $\Sigma _ g$ is a closed orientable surface of genus $g$. The submanifold has real dimension three and codimension three. This is not a genus-$g$ surface in Euclidean three-space, nor a construction in the flat space $\mathbb C^3$. The construction uses the real form, moment-map level sets and torus action of the chosen toric cone, followed by the radial/phase ODE in Theorem 7.1. For the cone metric $dr^2+r^2g_ S$, write $D_ a(s,r)=(s,ar)$ and $R=r\partial _ r$. After normalizing $\mathbf H=-R^\perp /2$, the flow is $D_{\sqrt{-t}}(L)$ for $t<0$. Compactness of the parameter manifold gives completeness and properness of the immersion; a general embedding assertion is not recorded. The cone vertex is outside the smooth ambient manifold and is approached only at the singular endpoint. The figure is a radial and topology diagram, not a Euclidean isometric rendering of the Calabi–Yau metric.

Sources: [Yamamoto2012].

Yamamoto Lagrangian shrinkers in Calabi--Yau cones. Radial cone coordinate and topology symbol for Σ_g × S¹. Exact radial time scaling is applied to a schematic, not to an explicit Euclidean parametrization.
Source-informed schematic. Radial cone coordinate and topology symbol for Σ_g × S¹. Exact radial time scaling is applied to a schematic, not to an explicit Euclidean parametrization. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Threefold in the six-dimensional cone M_g · Riemannian ambient manifold · Not recorded
Ambient space
Riemannian ambient manifold
Topology
Σ_g × S¹, g ≥ 1
Embeddedness
Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness
Compact
Surface genus
Not a smooth surface; surface genus is inapplicable.
Topological ends
No ends (compact parameter manifold).
Entropy
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area
Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

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References on this page

  1. [Yamamoto2012] Hikaru Yamamoto. Special Lagrangians and Lagrangian self-similar solutions in cones over toric Sasaki manifolds. arXiv preprint, 2012. URL: https://arxiv.org/abs/1203.3934, arXiv:1203.3934.All uses in this project ↩ 1
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