Illinois Journal of Mathematics

On isometric Lagrangian immersions

John Douglas Moore and Jean-Marie Morvan

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Abstract

This article uses Cartan-Kähler theory to show that a small neighborhood of a point in any surface with a Riemannian metric possesses an isometric Lagrangian immersion into the complex plane (or by the same argument, into any Kähler surface). In fact, such immersions depend on two functions of a single variable. On the other hand, explicit examples are given of Riemannian three-manifolds which admit no local isometric Lagrangian immersions into complex three-space. It is expected that isometric Lagrangian immersions of higher-dimensional Riemannian manifolds will almost always be unique. However, there is a plentiful supply of flat Lagrangian submanifolds of any complex $n$-space; we show that locally these depend on $\frac{1}{2}n(n+1)$ functions of a single variable.

Article information

Source
Illinois J. Math., Volume 45, Number 3 (2001), 833-849.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138154

Digital Object Identifier
doi:10.1215/ijm/1258138154

Mathematical Reviews number (MathSciNet)
MR1879238

Zentralblatt MATH identifier
0988.53008

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53D12: Lagrangian submanifolds; Maslov index

Citation

Moore, John Douglas; Morvan, Jean-Marie. On isometric Lagrangian immersions. Illinois J. Math. 45 (2001), no. 3, 833--849. doi:10.1215/ijm/1258138154. https://projecteuclid.org/euclid.ijm/1258138154


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