Infinitely many exotic monotone Lagrangian tori in \(\mathbb{C}\mathbb{P}^{2}\) (Q2813667)

By using some techniques of symplectic field theory, in this paper the authors characterize the convex hull of all relative homotopy classes in \(\pi_2(\mathbb{CP}^2; T(a^2,b^2,c^2))\), which can be represented by holomorphic discs with Maslov index 2. For any Markov triple \((a,b,c)\) there exists a monotone Lagrangian torus \(T(a^2,b^2,c^2)\) in \(\mathbb{CP}^2\), associated to each projective space \(\mathbb{CP}(a^2,b^2,c^2)\). This constitutes an example of a monotone Lagrangian torus, which is Lagrangian isotopic but not Hamiltonian isotopic, in a compact symplectic manifold. More exactly, the authors show here that the monotone Lagrangian tori corresponding to two distinct Markov triples are not Hamiltonian isotopic. Next they prove that the boundary of a neighbourhood of an orbifold point in the projective space \(\mathbb{CP}(a^2,b^2,c^2)\) is contactomorphic to a lens space of the form \(L(k^2,kl-1)\). Another result is that for any Markov triple \((a,b,c)\), there exists a monotone Lagrangian torus \(T(a^2,b^2,c^2)\), that is the ``barycentric fiber'' described in a base diagram of an almost toric fibration. The authors prove that an almost toric fibration having \(T(a^2,b^2,c^2)\) as its central fiber can be obtained from the moment polytope of the standard torus action on \(\mathbb{CP}^2\), thus solving a claim from \textit{W. Wu} [Compos. Math. 151, No. 7, 1372--1394 (2015; Zbl 1325.53105)].NEWLINENEWLINEThen, the authors describe the neck-stretching technique from the symplectic field theory, used here for lens spaces (seen as contact manifolds). They consider a symplectic manifold \((M,\omega)\), a hypersurface \(V\) of contact type, and the Liouville vector field \(X\) transversal to \(V\). The manifold \(M\) is assumed to be divided by \(V\) into two components, \(M_+,\;M_-\), chosen such that \(X\) points inwards along \(M_+\) and outwards along \(M_-\).NEWLINENEWLINEA compactness theorem is given here for a symplectic manifold \(M\), with an adjusted almost complex structure \(J\), a Lagrangian submanifold \(L\subset M_+\), and a sequence of stable \(J^n\)-holomorphic discs \(u_n:(\mathbb{D},\partial \mathbb{D})\rightarrow (M^n,L)\), in the same relative homotopy class, where \(M^n=M_-\cup V\times [-n,n]\cup M_+ \). This theorem, which is an adapted version of Theorem 1.6.3 of [\textit{Y. Eliashberg} et al., in: GAFA 2000. Visions in mathematics -- Towards 2000. Proceedings of a meeting 1999. Part II. Basel: Birkhäuser. 560--673 (2000; Zbl 0989.81114)], shows that the limit of a subsequence of \(u_n\) is a stable curve of height \(k\geq 1\) (see also Theorem 10.6 of [\textit{F. Bourgeois} et al., Geom. Topol. 7, 799--888 (2003; Zbl 1131.53312)].