Products and connected sums of spheres as monotone Lagrangian submanifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

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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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