# Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory

@article{Paternain2002BoundaryRF, title={Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory}, author={Gabriel P. Paternain and Leonid Polterovich and Karl Friedrich Siburg}, journal={Moscow Mathematical Journal}, year={2002}, volume={3}, pages={593-619} }

We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface.
We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain.
Moreover, we study the "non-removable intersection set" between the Lagrangian and the hypersurface, and show that it contains… Expand

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#### References

SHOWING 1-10 OF 94 REFERENCES

Lagrangian embeddings and critical point theory

- Mathematics
- 1985

Abstract We derive a lower bound for the number of intersection points of an exact Lagrangian embedding of a compact manifold into its cotangent bundle with the zero section. To do this the… Expand

Lagrangian flows: The dynamics of globally minimizing orbits

- Mathematics
- 1997

The objective of this note is to present some results, to be proved in a forthcoming paper, about certain special solutions of the Euler-Lagrange equations on closed manifolds. Our main results… Expand

Lagrangian flows: The dynamics of globally minimizing orbits-II

- Mathematics
- 1997

Define the critical levelc(L) of a convex superlinear LagragianL as the infimum of thek ∈ ℝsuch that the LagragianL+k has minimizers with fixed endpoints and free time interval. We provide proofs for… Expand

Mather Sets for Twist Maps and Geodesics on Tori

- Physics
- 1988

The title refers to a theory which is based on independent research in three different fields—differential geometry, dynamical systems and solid state physics—and which has attracted growing interest… Expand

On minimizing measures of the action of autonomous Lagrangians

- Mathematics
- 1995

We apply J Mather's theory (1991) on minimizing measures to the case of positive definite autonomous Lagrangians L:TM to R. We show that the minimal action function beta (h)=min( integral Ld mu mod… Expand

Lois de conservation et géométrie symplectique

- Mathematics
- 1991

Using recent work in symplectic geometry, we construct a reasonably unique weak solution of a Hamilton-Jacobi type equation under fairly general hypotheses. The generic structure of the corresponding… Expand

Anosov magnetic flows, critical values and topological entropy

- Mathematics
- 2002

We study the magnetic flow determined by a smooth Riemannian metric g and a closed 2-form Ω on a closed manifold M. If the lift of Ω to the universal cover is exact, we can define a critical value… Expand

Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values

- Mathematics
- 1998

Abstract. Let
$\Bbb L$ be a convex superlinear Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical value… Expand

First steps in symplectic topology

- Mathematics
- 1986

CONTENTSIntroduction § 1. Is there such a thing as symplectic topology? § 2. Generalizations of the geometric theorem of Poincare § 3. Hyperbolic Morse theory § 4. Intersections of Lagrangian… Expand

New invariants of open symplectic and contact manifolds

- Mathematics
- 1991

The first specifically symplectic invariant of open symplectic manifolds, the width, was defined by M. Gromov in [Gr]. Gromov's width was generalized by H. Hofer, I. Ekeland, E. Zehnder, and A. Floer… Expand