Boundary Rigidity for Lagrangian Submanifolds, Non-Removable Intersections, and Aubry—Mather Theory

AbstractWe construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants, developed by Oh, and results, by Abouzaid, about the Fukaya category of a cotangent bundle. We also introduce the notion of Lipschitz-exact Lagrangians and prove that these admit an appropriate generalisation of graph selector. We then, following Bernard–Oliveira dos Santos, use these results to give a new characterisation of the Aubry and Mañé sets of a Tonelli Hamiltonian and to generalise a result of Arnaud on Lagrangians invariant under the flow of such Hamiltonians.

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On equivalence of two constructions of invariants of Lagrangian submanifolds

Pacific Journal of Mathematics ◽

10.2140/pjm.2000.195.371 ◽

2000 ◽

Vol 195 (2) ◽

pp. 371-415 ◽

Cited By ~ 5

Author(s):

Darko Milinković

Keyword(s):

Lagrangian Submanifolds

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Linear Boundary Port Hamiltonian Systems defined on Lagrangian submanifolds

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2020.12.1526 ◽

2020 ◽

Vol 53 (2) ◽

pp. 7734-7739

Author(s):

Bernhard Maschke ◽

Arjan van der Schaft

Keyword(s):

Hamiltonian Systems ◽

Lagrangian Submanifolds ◽

Linear Boundary

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On Lagrangian Catenoids

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-031-4 ◽

2007 ◽

Vol 50 (3) ◽

pp. 321-333 ◽

Cited By ~ 1

Author(s):

David E. Blair

Keyword(s):

General Problem ◽

Lagrangian Submanifolds ◽

Minimal Submanifold ◽

Conformally Flat ◽

Totally Geodesic ◽

Schouten Tensor ◽

Multiplicity One

AbstractRecently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

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