Families of monotone Lagrangians in Brieskorn–Pham hypersurfaces

Abstract

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn–Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian $$S^1 \times \Sigma _g$$ S 1 × Σ g in $${\mathbb {C}}^3$$ C 3 , distinguished by soft invariants for any genus $$g \ge 2$$ g ≥ 2 ; and, for fixed soft invariants, a range of infinite families of Lagrangians in Brieskorn–Pham hypersurfaces. These are generally distinct up to Hamiltonian isotopy. In specific cases, we also set up well-defined counts of Maslov zero holomorphic annuli, which distinguish the Lagrangians up to compactly supported symplectomorphisms. Inter alia, these give families of exact monotone Lagrangian tori which are related neither by geometric mutation nor by compactly supported symplectomorphisms.

Homological Lagrangian monodromy for some monotone tori

“…If L were not a critical point then D L ψ would have an eigenvector not contained in ker D L W . Then (12) would require the corresponding eigenvalue to be 1, contradicting our assumption.…”

Families of relatively exact Lagrangians, free loop spaces and generalised homology

“…Question 1.1 has been studied in few other places: by Hu, Lalonde and Leclercq in [HLL11], by Mangolte and Welschinger in [MW12], by Ono in [Ono15], by Keating in [Kea21], and by Augustynowicz, Smith and Wornbard in [ASW22]. Out of these, only Hu, Lalonde and Leclercq focus on the relatively exact setting, which is where we will focus.…”