Geodesics on the space of Lagrangian submanifolds in cotangent bundles

Milinković, Darko

doi:10.1090/s0002-9939-00-05851-2

Cited by 11 publications

(6 citation statements)

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“…It remains to prove the claim. Since f is autonomous, the path of Lagrangians it generates is a geodesic with respect to Hofer's distance for small times, see Milinković [17,Theorem 8]. That is, for s ∈ (0, 1) small enough…”

Section: Preliminariesmentioning

confidence: 99%

Coisotropic rigidity and C0-symplectic geometry

Humilière¹,

Leclercq²,

Seyfaddini³

2015

Duke Math. J.

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Abstract. We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C 0 -dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C 0 -Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.

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Section: Preliminariesmentioning

confidence: 99%

Coisotropic rigidity and C0-symplectic geometry

Humilière¹,

Leclercq²,

Seyfaddini³

2015

Duke Math. J.

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“…The argument for the proof of Proposition 3.1 suggests that in the setting where L and L ′ are Hamiltonian isotopic and exact (the same would hold in the weakly exact case: ω, µ| π 2 (M,L) = 0), assuming that the cobordism is monotone, one can replace w(L, L ′ ) in the statement of the Proposition by d S (L, L ′ ), the spectral distance between L and L ′ (introduced in [33], see also [21] for additional references). For a fixed Lagrangian L recall from the work of Milinkovic [24] that, if L ′ is sufficiently C 1 -close to L, then d S (L, L ′ ) = d H (L, L ′ ). Therefore, we expect that, at least under this additional proximity assumption, d * (L, L ′ ) = d H (L, L ′ ) for all * ≥ m.…”

Section: Additional Commentsmentioning

confidence: 99%

Lagrangian cobordism and metric invariants

Cornea¹,

Shelukhin²

2015

Preprint

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“…Remark. In [7,8] Milinković studied geodesics in the space L(T * X, ω, O X ) of Lagrangian submanifolds which are Hamiltonian isotopic to the zero section O X of the cotangent bundle T * X of a smooth closed manifold X. In his terminology, the above condition is said to be strongly quasi-autonomous (see [8,Definition 10]).…”

Section: A Variational Definition Of Geodesicsmentioning

confidence: 99%

“…By contrast there are few results on Hofer's geodesics on the space of Lagrangian submanifolds which are Hamiltonian isotopic to one. Milinković [7,8] studied the case of Hamiltonian isotopy class of the zero section of the cotangent bundle of a compact manifold. Akveld-Salamon [1] proved length minimizing properties of some Lagrangian loops of RP n in CP n .…”

Section: Introductionmentioning

confidence: 99%