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

*(English)*Zbl 1048.53058Subtle interrelations between optical hypersurfaces (OH) and Lagrangian submanifolds (LS) are discussed: certain LS lying on a given OH cannot be deformed into the domain bounded by that OH (the boundary rigidity) and even if this rigidity fails, the intersection of the deformed LS and the OH contains a dynamically significant set related to the Aubry-Mather theory (a non-removable intersection).

In more detail, let \({\mathcal L}\) be the class of all LS of \(T^*M\) which are Lagrangian isotopic to the zero section, let \(\omega= d\lambda\) be the canonical symplectic form on \(M\). Given \(\Lambda\in{\mathcal L}\), we obtain the cohomology class \(a_\Lambda= [\lambda]\in H^1(M,\mathbb{R})\). On the other hand, a smooth, closed and fibrewise convex hypersurface \(\Sigma\subset TM\) is called optical and we denote by \(\sigma\) the characteristic one-dimensional foliation on \(\Sigma\).

Theorem 1.1: Let \(\Lambda\in{\mathcal L}\) be a Lagrangian submanifold lying in an optical hypersurface \(\Sigma\). Let the flow of the characteristic foliation \(\sigma\) on \(\Sigma\) preserve an absolutely continuous measure. If \(K\in{\mathcal L}\) is a Lagrangian submanifold lying in the domain bounded by \(\Sigma\) and such that \(a_K= a_\Lambda\), then \(K=\Lambda\).

We cannot state more results since they employ rather advanced concepts from the actual theory of dynamical systems and symplectic geometry: the shape \(sh(U)\) of a subset \(U\subset T^*M\), the graph selector of a LS, the Mañés critical value of \(c(L)\) of a Lagrangian function, the Mather set \({\mathcal M}\subset M\), the Aubry set \(A\subset M\), the chain recurrence of points \(x\in M\) (to name a few).

The article provides an instructive insight into the recent achievement in symplectic geometry.

In more detail, let \({\mathcal L}\) be the class of all LS of \(T^*M\) which are Lagrangian isotopic to the zero section, let \(\omega= d\lambda\) be the canonical symplectic form on \(M\). Given \(\Lambda\in{\mathcal L}\), we obtain the cohomology class \(a_\Lambda= [\lambda]\in H^1(M,\mathbb{R})\). On the other hand, a smooth, closed and fibrewise convex hypersurface \(\Sigma\subset TM\) is called optical and we denote by \(\sigma\) the characteristic one-dimensional foliation on \(\Sigma\).

Theorem 1.1: Let \(\Lambda\in{\mathcal L}\) be a Lagrangian submanifold lying in an optical hypersurface \(\Sigma\). Let the flow of the characteristic foliation \(\sigma\) on \(\Sigma\) preserve an absolutely continuous measure. If \(K\in{\mathcal L}\) is a Lagrangian submanifold lying in the domain bounded by \(\Sigma\) and such that \(a_K= a_\Lambda\), then \(K=\Lambda\).

We cannot state more results since they employ rather advanced concepts from the actual theory of dynamical systems and symplectic geometry: the shape \(sh(U)\) of a subset \(U\subset T^*M\), the graph selector of a LS, the Mañés critical value of \(c(L)\) of a Lagrangian function, the Mather set \({\mathcal M}\subset M\), the Aubry set \(A\subset M\), the chain recurrence of points \(x\in M\) (to name a few).

The article provides an instructive insight into the recent achievement in symplectic geometry.

Reviewer: Jan Chrastina (Brno)