Lagrangian submanifolds and moment convexity

Krötz, Bernhard; Otto, Michael

doi:10.1090/s0002-9947-05-03723-2

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…So, a coadjoint orbit in p carries two symplectic forms: the usual orbit symplectic form and the push forward of the dressing symplectic form by the map B b → log(b * b) ∈ p. Ginzburg and Weinstein [51] proved that there is a global diffeomorphism on p taking the Lie-Poisson tensor on p to the push forward of the Poisson Lie tensor on B to p by the map defined above. The symplectic proof of the Kostant Non-linear Convexity Theorem for all connected real semisimple Lie groups was given in Sleewaegen's 1999 Ph.D. thesis [134] (unpublished) and later by Krötz and Otto [88] each extending the Duistermaat Convexity Theorem in such a way that the symplectic proof outlined above works in full generality.…”

Section: "Non-linear" Symplectic Formulationsmentioning

confidence: 99%

Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

Raţiu¹,

Wacheux²,

Zung³

2017

Preprint

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This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently big then the local-global convexity principle breaks down, and the base spaces can be globally non-convex even for compact manifolds. As one of surprising examples, we construct a 2-dimensional "integral affine black hole", which is locally convex but for which a straight ray from the center can never escape.

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Section: "Non-linear" Symplectic Formulationsmentioning

confidence: 99%

Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

Raţiu¹,

Wacheux²,

Zung³

2017

Preprint

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“…However, they can be viewed as fixed point sets of an involution τ on the compact symplectic manifold M a as introduced in Chapter 3. We will define τ and show that it is compatible with the action of T in a way that the symplectic convexity theorem from [6] can be applied.…”

Section: The Complex Convexity Theorem For Non-complex Groupsmentioning

confidence: 99%

“…We then exhibit a Hamiltonian torus action on the compact symplectic manifold M a and show that that (1.1) becomes a consequence of the Atiyah-Guillemin-Sternberg convexity theorem. Finally, the general case of arbitrary G can be handled by descent to certain Lagrangian submanifolds Q a ⊆ M a by means of the recently discovered convexity theorem [6].…”

Section: Introductionmentioning

confidence: 99%

A refinement of the complex convexity theorem via symplectic techniques

Krötz

Otto

2005

Proc. Amer. Math. Soc.

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Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems

Raţiu¹,

Wacheux²,

Zung³

2023

Memoirs of the AMS

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This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

show abstract

Lagrangian submanifolds and moment convexity

Cited by 3 publications

References 12 publications

Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

Convexity of singular affine structures and toric-focus integrable Hamiltonian systems

A refinement of the complex convexity theorem via symplectic techniques

Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems

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