Prescribing the Maslov form of Lagrangian immersions
type: Item
instance of: scholarly article
title: Prescribing the Maslov form of Lagrangian immersions
zbMATH Open document ID: 1034.53068
DOI: 10.1023/A:1016238817734
author: Knut Smoczyk
published in: Geometriae Dedicata
publication date: 2002-08-21T00:00:00Z
review text: Let \(({\mathbb R}^{2n},J,\omega)\) be the Euclidean space equipped with the standard symplectic structure \(\omega\) and complex structure \(J\). An immersion \(F:L\to{\mathbb R}^{2n}\), where \(L\) is an \(n\)-dimensional manifold, is called Lagrangian, if \(F^*\omega=0\). Equivalently \(L\) is Lagrangian, if \(J\) gives an isomorphism between the tangent and the normal bundle of \(L\). It follows that normal variations of a Lagrangian immersion can be described in terms of \(1\)-forms \(\theta\in\Omega^1(L,{\mathbb R})\). If \(\theta\) is closed, then the variation preserves the Lagrangian condition. In case of an exact \(1\)-form \(\theta=d\alpha\), the corresponding flow is called Hamiltonian. \smallskip The author studies the quasilinear parabolic system \[ (*)\quad d F/dt=-g^{ij}(H_ i-m_ i)\nu_ j, \] where \(H=H_ idx^ i\) denotes the mean curvature form of the evolving immersions and \(m=m_ idx^ i\in\Omega^1(L,{\mathbb R})\) is a fixed, time independent \(1\)-form on \(L\) such that for the initial immersion \([H]=[m]\). \(g^{ij}\) denotes the inverse of the induced metric on \(L\) and \(\nu_j\) is a local frame in the normal bundle. It then follows that \((*)\) is a Hamiltonian flow for all \(t\) for which a smooth solution of \((*)\) exists and that the evolving submanifolds stay Lagrangian.\smallskip The author shows that the flow exists for all \(t\in[0,\infty)\) and that the Lagrangian immersions converge to a Lagrangian submanifold with prescribed Maslov form \(M=m/\pi\) provided the second fundamental form of the evolving Lagrangian hypersurfaces is uniformly bounded and the induced metrics are uniformly equivalent for all \(t\). In the second part of the paper, the author derives sharp results for the \(1\)-dimensional situation of evolving starshaped curves. In particular, he obtains the following results: Theorem 1.3: Assume that \(R : S^ 1\to{\mathbb R}\) is a smooth positive function and \(\gamma_ 0\) the initial curve given by the immersion \(F:S…
reviewed by: Knut Smoczyk
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 1786838
zbMATH Keywords: Lagrangian
zbMATH Keywords: Lagrangian mean curvature flow
zbMATH Keywords: Maslov form
zbMATH Keywords: prescribed curvature
zbMATH Keywords: starshaped curves
MaRDI profile type: Publication
full work available at URL: a:1016238817734
OpenAlex ID: W1526657481
description: scientific article; zbMATH DE number 1786838