Special Metric Invariants

Sosov, E. N.

doi:10.1134/s1995080218020269

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Variations On the Hausdorff Metric1

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2023

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“…Following a result of Sosov [Sos01], the Hausdorff metric between 𝐿 and 𝐿 ′ is given by the infinimum over all 𝛿 𝐻 -continuous paths of closed subsets from 𝐿 to 𝐿 ′ . In fact, this infimum is even realized by a geodesic.…”

Section: Variations On the Hausdorff Metricmentioning

confidence: 99%

Convergence and Riemannian bounds on Lagrangian submanifolds

Jean-Philippe

2023

Int. J. Math.

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We consider collections of Lagrangian submanifolds of a given symplectic manifold which respect uniform bounds of curvature type coming from an auxiliary Riemannian metric. We prove that, for a large class of metrics on these collections, convergence to an embedded Lagrangian submanifold implies convergence to it in the Hausdorff metric. This class of metrics includes well-known metrics such as the Lagrangian Hofer metric, the spectral metric and the shadow metrics introduced by Biran et al. [Lagrangian shadows and triangulated categories, Astérisque 426 (2021) 1–128]. The proof relies on a version of the monotonicity lemma, applied on a carefully-chosen metric ball.

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Section: Variations On the Hausdorff Metricmentioning

confidence: 99%

Convergence and Riemannian bounds on Lagrangian submanifolds

Jean-Philippe

2023

Int. J. Math.

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Special Metric Invariants

Cited by 1 publication

References 3 publications

Convergence and Riemannian bounds on Lagrangian submanifolds

Convergence and Riemannian bounds on Lagrangian submanifolds

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