Discrete geometry and isotropic surfaces

Jauberteau, F.; Rollin, Yann; Tapie, Samuel

doi:10.24033/msmf.469

Cited by 4 publications

(14 citation statements)

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“…Indeed, such an approach was successful in the analogous situation of polyhedral Lagrangian tori (cf. [5] and [8]). At this stage, we can merely show that the modified moment maps flow is well behavec in a neighborhood of regular points of the moduli space: Theorem 3.9.2.…”

Section: Polyhedral Symplectic Geometrymentioning

confidence: 99%

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Polyhedral symplectic maps and hyperKähler geometry

Rollin

2021

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We provide an interpretation of the space of symplectic diffeomorphism of a 4-dimensional real torus T 4 , as the vanishing locus of a hyperKähler moment map. This construction leads to a notion of modified hyperKähler moment map flow, whose regular fixed points coincide with the symplectic diffeomorphisms of T 4 .Given a Euclidean triangulation of T 4 , we introduce a notion of polyhedral maps and polyhedral symplectic maps of the torus. We show that all the construction carried out in the smooth setting have analogues in the polyhedral setting. In this context, the hyperKähler modified moment map flow is an ordinary differential equation, which is always convergent. Under a non-degeneracy assumption of the limit, this provides an effective construction of polyhedral symplectic maps of the torus.This paper lays out the ground work for some experimental symplectic geometry. In an upcoming work with François Jauberteau [3], we write a computer program that runs a numerical version of the flow and generates non trivial examples of symplectic polyhedral maps of T 4 .

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Section: Polyhedral Symplectic Geometrymentioning

confidence: 99%

“…From a broader scope, this paper is part of an ongoing research project, concerned with symplectic polyhedra piecewise linear symplectic maps, and their relations with smooth symplectic geometry (cf. [5,8,4,3]). Constructions of piecewise linear symplectic maps are scarce.…”

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confidence: 99%

Polyhedral symplectic maps and hyperKähler geometry

Rollin

2021

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“…Although we shall not prove anything about short time existence of the moment map flow in this work, we provide at least a heursitic evidence. In the next section, we compute the variation of the moment map and show that the variation of µ(f ), when f is deformed in the direction of the complexified action JY h , is expressed as a Laplacian of h. However, the systematic study of the moment map flow in the smooth setting is not our purpose here, and we shall return to this question in a sequel to this paper [6].…”

Section: 2mentioning

confidence: 99%

“…This could be relied upon to define a discrete analogue of the gauge group action G = Ham(Σ, σ). This idea will be explored in a sequel to this work [6].…”

Section: 33mentioning

confidence: 99%

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Discrete geometry and isotropic surfaces

Jauberteau¹,

Rollin²,

Tapie³

2018

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We consider smooth isotropic immersions from the 2-dimensional torus into R 2n , for n ≥ 2. When n = 2 the image of such map is an immersed Lagrangian torus of R 4 . We prove that such isotropic immersions can be approximated by arbitrarily C 0 -close piecewise linear isotropic maps. If n ≥ 3 the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well.The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flow in finite dimension, whose limits provide, from an experimental perspective, many examples of piecewise linear Lagrangian tori in R 4 . The DMMF program, which is freely available, is based on the Euler method and shows the evolution equation of discrete surfaces in real time, as a movie.

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Polyhedral approximation by Lagrangian and isotropic tori

Rollin

2022

Journal of Symplectic Geometry

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Discrete geometry and isotropic surfaces

Cited by 4 publications

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Polyhedral symplectic maps and hyperKähler geometry

Polyhedral symplectic maps and hyperKähler geometry

Discrete geometry and isotropic surfaces

Polyhedral approximation by Lagrangian and isotropic tori

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