Special Lagrangians, stable bundles and mean curvature flow

By Richard P. Thomas, Shing-Tung Yau. 2001

187 citations recorded by the source corpus.

We make a conjecture about mean curvature flow of Lagrangian submanifolds of Calabi-Yau manifolds, expanding on \cite{Th}. We give new results about the stability condition, and propose a Jordan-Holder-type decomposition of (special) Lagrangians. The main results are the uniqueness of special Lagrangians in hamiltonian deformation classes of Lagrangians, under mild conditions, and a proof of the conjecture in some cases with symmetry: mean curvature flow converging to Shapere-Vafa's examples of SLags.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/54a69c1ddf36c1e1dabfaed0e1b3d7e28d6853b9

How researchers see this paper

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In Applied Mathematics

1. Knut Smoczyk
2. Jingyi Chen
3. N. Kapouleas
4. I. Smith
5. Jiayu Li
6. M. Haskins
7. M. Li
8. D. Joyce

In Geometry and Topology

1. Knut Smoczyk
2. Jingyi Chen
3. N. Kapouleas
4. I. Smith
5. Jiayu Li
6. M. Haskins
7. M. Li
8. M. Abouzaid

Citations in the corpus

• Flat surfaces and stability structures (2014, unknown): “Investigations into this direction were started by Thomas [48] and Thomas–Yau [49] with the aim of finding an algebraic notion of stability describing special Lagrangian submanifolds and their mean curvature flow.”
• Self-shrinkers of the mean curvature flow in arbitrary codimension (2005, background): “While until recently people were mostly interested in mean curvature flow of hypersurfaces, this non-linear evolution equation is getting more and more attention in higher codimension, see for instance [AS96], [Anc00], [BN99], [CLT02], [TW04], [SW02], [ TY02 ].”
• Immersed Lagrangian Floer Theory (2008, unknown): “Define ik : p k copies q (B ⊗ Λ0nov) × · · · × (B ⊗ Λ 0 nov) → A⊗ Λ 0 nov by ik = { ı̂+ ∑ T i1,T , k = 1,∑ T ik,T , k = 0, 2, 3, 4, . . . , where the sums are over all planar rooted trees T with k leaves.”
• Special Lagrangian cones with higher genus links (2005, background): “is a natural extension of special Lagrangian geometry to any Calabi-Yau manifold M by replacing with the natural parallel holomorphic (n,0)-form on M. Special Lagrangian submanifolds appear to play an important role in a number of interesting geometric properties of Calabi-Yau manifolds, e.g. Mirror”
• Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, result): “Among the first results in this direction are the results on mean curvature flow of space curves by Altschuler and Grayson [2, 3], measure-theoretic approaches to higher codimension mean curvature flows by Ambrosio and Soner [4], existence and convergence results for the Lagrangian mean curvature fl”
• Self-similar solutions and translating solitons for Lagrangian mean curvature flow (2008, unknown): “Research by the first author (unpublished) indicates that an important condition in the improved Thomas–Yau conjecture described above is that the Lagrangians should have unobstructed Lagrangian Floer homology, in the sense of Fukaya, Oh, Ohta and Ono [7].”
• Quiver algebras as Fukaya categories (2013, background): “4], this decomposition should reflect the Harder-Narasimham filtration of Lγ in the stability condition associated to φ, and (especially in light of [50]) there are obvious potential connections to mean curvature flow.”
• Singularity of mean curvature flow of Lagrangian submanifolds (2003, background): “where � is the Laplacian of the induced metric on �t ([Sm1-3], also see [ TY ] for a derivation of (4)).”