Infinitely many planar fillings and symplectic Milnor fibers

Abstract

We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find new examples of infinitely many exact Lagrangian spheres and tori 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality. We also provide a generalization of a criterion for when the symplectic homology of a Weinstein 4-manifold is non-vanishing given an explicit Weinstein handlebody diagram.

On topologically distinct infinite families of exact Lagrangian fillings

“…exact Lagrangian cobordisms with empty negative ends) for some given Legendrian submanifolds has become one of the central questions in contact and symplectic topology. First it has been positively answered by Casals-Gao [5], and then later it has been investigated using different methods in the works of An-Bae-Lee [1,2], Casals-Zaslow [7], Gao-Shen-Weng [16,17], Casals-Ng [6], Capovilla-Searle [4] and the author [19].…”