The author uses Lagrangian Rabinowitz-Floer homology to address the question of whether or not a Lagrangian submanifold can be displaced from a hypersurface via a compactly supported Hamiltonian.
\smallpagebreak The main body of the paper discusses Lagrangian Rabinowitz-Floer homology in twisted cotangent bundles. However, most of the results in the paper are also discussed for the following special case: Suppose that \((X_0,\lambda_0)\) is a Liouville domain, i.e., \(X_0\) is a compact manifold with boundary \(\Sigma := \partial X_0\), and \(d\lambda_0\) is a symplectic form on \(X_0\) such that \(\eta := \lambda_0|_\Sigma\) is a positive contact form on \(\Sigma\). Define \(X := X_0 \cup_\Sigma (\Sigma \times [1,\infty))\) and extend \(\lambda_0\) to a 1-form on \(X\) by setting \(\lambda := r \eta\) on \(\Sigma \times \{r \geq 1\}\). The pair \((X,\lambda)\) is called the completion of the Liouville domain \((X_0,\lambda_0)\).
\smallpagebreak Let \(L \subset X\) be an exact Lagrangian submanifold that is transverse to \(\Sigma\) such that \(K := \Sigma \cap L\) is a closed Legendrian submanifold of \((\Sigma, \eta)\). Let \(\theta^t: \Sigma \rightarrow \Sigma\) denote the Reeb flow of \(\eta\). The author asks the following questions.
\smallpagebreak Question 1.1: Is it possible to displace \(\Sigma\) from \(L\) via a compactly supported Hamiltonian diffeomorphism?
\smallpagebreak Question 1.2: Must there exist a Reeb chord with endpoints in \(K\)? That is, a point \(p \in K\) such that \(\theta^\tau(p) \in K\) for some \(\tau\neq 0\)?
\smallpagebreak The first main result in the paper says that a positive answer to the first question implies a positive answer to the second question.
\smallpagebreak Theorem 1.3: Suppose \((X,\lambda)\) is a completion of a Liouville domain as above, and suppose that \(L \subset X\) is an exact Lagrangian submanifold transverse to \(\Sigma\) with the property that \(K := \Sigma \cap L\) is a closed Legendrian submanifold of \((\Sigma, \eta := \lambda|_\Sigma)\). If one can displace \(\Sigma\) from \(L\) via a compactly supported Hamiltonian diffeomorphism, then there exists a Reeb chord of \(\eta\) with endpoints in \(K\).
\smallpagebreak Now let \(\psi:X \rightarrow X\) be a compactly supported Hamiltonian diffeomorphism. A point \(p \in K\) is called a relative leaf-wise intersection point of \(\psi\) if there exists a \(\tau \in {\mathbb R}\) such that \(\psi(\theta^\tau(p)) \in L\). When the answer to Question 1.1 is ‘no’, the author asks the following questions:
\smallpagebreak Question 1.5: Suppose that it is not possible to displace \(\Sigma\) from \(L\) via a compactly supported Hamiltonian diffeomorphism. Is it true that every \(\psi\) has a relative leaf-wise intersection point?
\smallpagebreak Question 1.6: When is it true that for a generic \(\psi\) there are always infinitely many relative leaf-wise intersection points?
The author addresses the above questions by studying the Lagrangian Rabinowitz-Floer homology on a twisted cotangent bundle \((T^\ast M,\omega)\), where \(\omega = d\lambda_{\mathrm{can}} + \pi^\ast \sigma\) is the canonical symplectic form twisted by the pullback of a 2-form \(\sigma\) on \(M\) via the footprint map \(\pi:T^\ast M \rightarrow M\), with a Tonelli Hamiltonian \(H:T^\ast M \rightarrow {\mathbb R}\). In this setting, the author considers a conormal bundle \(N^\ast S\), where \(\sigma|_S = 0\), in which case \(N^\ast S\) is Lagrangian submanifold of \((M,\omega)\) and there is defined a Mañé critical value \(c(H,\sigma,S) \in {\mathbb R} \cup \{\infty\}\). The author makes the following definition.
\smallpagebreak Definition 1.10: Consider a closed connected hypersurface \(\Sigma \subset T^\ast M\) and a closed connected submanifold \(S\) such that \(\sigma|_S=0\), with \(\Sigma \cap N^\ast S \neq \emptyset\) and \(\Sigma \pitchfork N^\ast S\). The pair \((\Sigma,S)\) is called a Mañé supercritical pair if there exists a Tonelli Hamiltonian \(H:T^\ast M \rightarrow {\mathbb R}\) with \(c(H,\sigma,S)< 0\), and such that \(\Sigma\) is the regular level set \(H^{-1}(0)\).
\smallpagebreak The two main theorems in the paper can now be stated.
\smallpagebreak Theorem 1.11: Suppose \((X,\omega)\) is a geometrically bounded symplectically aspherical symplectic manifold with \(c_1(TX) = 0\). Let \(\Sigma \subset X\) denote a closed connected \(\pi_1\)-injective hypersurface that encloses a compact connected component of \(X \backslash \Sigma\), and let \(L \subset X\) denote a \(\pi_1\)-injective Lagrangian submanifold transverse to \(\Sigma\) with \(\Sigma \cap L \neq \emptyset\). Let \(\widetilde{X} \rightarrow X\) denote the universal cover of \(X\). Assume there exists a primitive \(\lambda\) of the lifted symplectic form \(\widetilde{\omega}\) such that:
1. \(\sup_{\tilde{\Sigma}} |\lambda| < \infty\) and \(\inf_{\tilde{\Sigma}} \lambda(R)>0\), where \(R\) is a vector field generating ker \(\omega|_\Sigma\) pulled back to \(\widetilde{X}\).
2. \(\lambda|_{\widetilde{L}} = d\) (bounded function).
If there exists a compactly supported Hamiltonian diffeomorphism \(\psi:X \rightarrow X\) with no relative leaf-wise intersection points (e.g., if one can displace \(\Sigma\) from \(L\)), then there exists a characteristic chord of \(\Sigma\) with endpoints in \(\Sigma \cap L\).
\smallpagebreak Theorem 1.13: Let \((M^n,g)\) denote a closed connected orientable Riemannian manifold of dimension \(n \geq 2\), and let \(\sigma \in \Omega^2(M)\) denote a weakly exact 2-form whose lift to the universal cover admits a bounded primitive. Equip \(T^\ast M\) with the twisted symplectic form \(\omega := d \widetilde{\lambda}_{\mathrm{can}} + \pi^\ast \sigma\). Let \(S^d \subseteq M\) denote a closed connected submanifold such that \(\sigma|_S = 0\), and let \(\Sigma \subset T^\ast M\) denote a hypersurface such that \((\Sigma, N^\ast S)\) form a Mañé supercritical pair (c.f. Definition 1.10). Assume that one of the following two conditions hold:
1. \(d < n/2\), or \(d = n/2\) and \(n \geq 4\),
2. The double coset space \(\pi_1(S)\backslash \pi_1(M) / \pi_1(S)\) is non-trivial.
Then it is not possible to displace \(\Sigma\) from \(N^\ast S\), and the answer to Question 1.5 is ‘yes’. Moreover, if dim \(H_\ast(P(M,S);{\mathbb Z}_2) = \infty\) and the pair \((\Sigma, N^\ast S)\) is non-degenerate (this condition is satisfied generically, see Section 2.1), then a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.
Note: \(P(M,S)\) denotes the set of all smooth paths \(q:[0,1] \rightarrow M\) with \(q(0)\) and \(q(1)\) both lying in \(S\).
The main theorems are proved by studying the Rabinowitz-Floer homology of two different functionals \({\mathcal A}\) and \({\mathcal A}_\psi\), which yield the same homology. The first functional is a Morse-Bott functional, and the author uses cascades to define its homology. The critical points of the second functional detect relative leaf-wise intersection points of \(\psi\). Thus, if \(\psi\) has no relative leaf-wise intersection points, then the Rabinowitz-Floer homology is zero. The author computes the Rabinowitz-Floer homology using a short exact sequence that relates the Lagrangian Rabinowitz-Floer homology to the Morse complex of a free time action functional.