# 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}
}
• Published 2002
• Mathematics
• Moscow Mathematical Journal
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

#### Figures from this paper

Exact Lagrangian submanifolds, Lagrangian spectral invariants and Aubry–Mather theory
• Mathematics
• Mathematical Proceedings of the Cambridge Philosophical Society
• 2017
Abstract We construct graph selectors for compact exact Lagrangians in the cotangent bundle of an orientable, closed manifold. The construction combines Lagrangian spectral invariants, developed byExpand
Rigid fibers of integrable systems on cotangent bundles
(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing theExpand
$\mathcal{C}^0$-rigidity of Lagrangian submanifolds and punctured holomorphic discs in the cotangent bundle
• Mathematics
• 2017
Our main result is the $\mathcal{C}^0$-rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic discs in cotangentExpand
Spectral invariants towards a Non-convex Aubry-Mather theory
Aubry-Mather is traditionally concerned with Tonelli Hamiltonian (convex and super-linear). In \cite{Vi,MVZ}, Mather's $\alpha$ function is recovered from the homogenization of symplectic capacities.Expand
Rigid fibers of spinning tops
• Mathematics
• 2019
(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing theExpand
On the stability of Mañé critical hypersurfaces
• Mathematics
• 2010
We construct examples of Tonelli Hamiltonians on $${\mathbb{T}^n}$$ (for any n ≥ 2) such that the hypersurfaces corresponding to the Mañé critical value are stable (i.e. geodesible). We also provideExpand
Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization
• Mathematics
• 2011
For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T^*N, and a family of functions on the space of smooth functions with compactExpand
Punctured holomorphic curves and Lagrangian embeddings
• Mathematics
• 2014
We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. ApplicationsExpand
Graph selectors and viscosity solutions on Lagrangian manifolds
Let be a Lagrangian submanifold of for some closed manifold X. Let be a generating function for which is quadratic at infinity, and let W(x) be the corresponding graph selector for in the sense ofExpand
Viscosity Solutions on Lagrangian Manifolds and Connections with Tunnelling Operators
We consider a geometrical approach to constructing viscosity solutions to Hamilton-Jacobi-Bellman equations. This takes the Lagrangian manifold M on which the characteristic curves for the CauchyExpand

#### References

SHOWING 1-10 OF 95 REFERENCES
Lagrangian embeddings and critical point theory
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 theExpand
Lagrangian flows: The dynamics of globally minimizing orbits
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 resultsExpand
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 forExpand
Mather Sets for Twist Maps and Geodesics on Tori
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 interestExpand
On minimizing measures of the action of autonomous Lagrangians
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 modExpand
Lois de conservation et géométrie symplectique
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 correspondingExpand
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 valueExpand
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 valueExpand
First steps in symplectic topology
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 LagrangianExpand
New invariants of open symplectic and contact manifolds
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. FloerExpand