New lower bounds on the number of intersections of monotone Lagrangian submanifolds

Abstract

Considering a closed monotone Lagrangian submanifold L , L, we give, under some hypotheses, a lower bound on the intersection number of L L with its image by a generic Hamiltonian isotopy. First for monotone Lagrangian submanifolds L L which are K ( π , 1 ) \mathbf {K}(\pi ,1) and, in particular, for monotone Lagrangian submanifolds with negative sectional curvature this bound is 1+ β 1 ( L ) . \beta _{1}(L). In more general cases the lower bound is weaker. We generalise some results previously obtained by L. Buhovsky in [J. Topol. Anal. 2 (2010), pp. 57–75] and P. Biran and O. Cornea in [Geom. Topol. 13 (2009), pp. 2881–2989].