On the graph theorem for Lagrangian minimizing tori

Mario Jorge Dias Carneiro; Rafael O. Ruggiero

doi:10.3934/dcds.2018260

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2018, Volume 38, Issue 12: 6029-6045. Doi: 10.3934/dcds.2018260

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On the graph theorem for Lagrangian minimizing tori

Mario Jorge Dias Carneiro^1, and
Rafael O. Ruggiero^2, ,

1.
Dep. de Matemática - ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, MG, 31270-901, Brazil
2.
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ, 22453-900, Brazil

* Corresponding author: Rafael O. Ruggiero
* Corresponding author: Rafael O. Ruggiero

Received: August 31, 2017

Revised: March 31, 2018

Published: November 2018

The research project is partially supported by CNPq, FAPERJ (Cientistas do nosso estado), Pronex de Geometria, Pronex de Sistemas Dinómicos (Brazil), CNRS, unité FR2291 FRUMAM.

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Abstract

We study the graph property for Lagrangian minimizing submanifolds of the geodesic flow of a Riemannian metric in the torus $ (T^{n},g) $, $ n>2 $. It is well known that the transitivity of the geodesic flow in a minimizing Lagrangian submanifold implies the graph property. We replace the transitivity by three kind of assumptions: (1) $ r $-density of the set of recurrent orbits for some $ r>0 $ depending on $ g $, (2) $ r $-density of the limit set, (3) every point is nonwandering. Then we show that a Lagrangian, minimizing torus satisfying one of such assumptions is a graph.

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Mathematics Subject Classification: Primary: 53D25; Secondary: 53D12, 37J50.

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