Geometry of Lagrangian Submanifolds Related to Isoparametric Hypersurfaces

2014 ◽  
pp. 117-127
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Lagrangian Submanifolds ◽  
Isoparametric Hypersurfaces

scholarly journals Geometry of G2 orbits and isoparametric hypersurfaces

2011 ◽  
Vol 203 ◽  
pp. 175-189
Author(s):  
Reiko Miyaoka
Keyword(s):  
Lie Algebra ◽  
Lagrangian Submanifolds ◽  
Point Of View ◽  
Parameter Family ◽  
Isoparametric Hypersurfaces ◽  
Singular Orbits

AbstractWe characterize the adjointG2orbits in the Lie algebragofG2as fibered spaces overS6with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case (g,m) = (6,2) with case (3,2). The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations ofS6andG2/SO(4). From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.

scholarly journals Lagrangian geometry of the Gauss images of isoparametric hypersurfaces in spheres

2019 ◽  
Vol 6 (1) ◽  
pp. 265-278
Author(s):  
Reiko Miyaoka ◽  
Yoshihiro Ohnita
Keyword(s):  
Lagrangian Submanifolds ◽  
Joint Work ◽  
Standard Sphere ◽  
Isoparametric Hypersurfaces ◽  
Survey Article ◽  
Complex Hyperquadric ◽  
Hypersurfaces In Spheres ◽  
Lagrangian Geometry

AbstractThe Gauss images of isoparametric hypersufaces of the standard sphere Sn+1 provide a rich class of compact minimal Lagrangian submanifolds embedded in the complex hyperquadric Qn(ℂ). This is a survey article based on our joint work [17] to study the Hamiltonian non-displaceability and related properties of such Lagrangian submanifolds.

scholarly journals Geometry of G2 orbits and isoparametric hypersurfaces

2011 ◽  
Vol 203 ◽  
pp. 175-189 ◽  
Author(s):  
Reiko Miyaoka
Keyword(s):  
Lie Algebra ◽  
Lagrangian Submanifolds ◽  
Point Of View ◽  
Parameter Family ◽  
Isoparametric Hypersurfaces ◽  
Singular Orbits

AbstractWe characterize the adjoint G2 orbits in the Lie algebra g of G2 as fibered spaces over S6 with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case (g,m) = (6,2) with case (3,2). The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of S6 and G2/SO(4). From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.

scholarly journals Linear Boundary Port Hamiltonian Systems defined on Lagrangian submanifolds

2020 ◽  
Vol 53 (2) ◽  
pp. 7734-7739
Author(s):  
Bernhard Maschke ◽  
Arjan van der Schaft
Keyword(s):  
Hamiltonian Systems ◽  
Lagrangian Submanifolds ◽  
Linear Boundary

On Lagrangian Catenoids

2007 ◽  
Vol 50 (3) ◽  
pp. 321-333 ◽  
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.

Minimal Lagrangian submanifolds in ℂP 3 and the sinh-Gordon equation

2001 ◽  
Vol 40 (1-4) ◽  
pp. 130-143 ◽  
Author(s):  
Ildefonso Castro ◽  
Luc Vrancken
Keyword(s):  
Gordon Equation ◽  
Lagrangian Submanifolds
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