论文标题
Brieskorn的单调Lagrangians的家庭 - Pham Hypersurfaces
Families of monotone Lagrangians in Brieskorn--Pham hypersurfaces
论文作者
论文摘要
我们介绍了受单一考虑的启发的技术，用于在某些仿射性高空（主要是Brieskorn-pham型）中构建紧凑的单调Lagrangians。我们专注于尺寸2和3，尽管结构概括为更高的构建。这些技术在控制Lagrangian的同源性类别，Maslov类和单调性常数以及一系列可能的差异类型方面具有显着的纬度。它们也足够明确，可以适合计算伪形态曲线不变的。 应用程序包括单调拉格朗日$ s^1 \ timesσ_g$ in $ \ mathbb {c}^3 $的无限族，由任何属于任何属$ g \ geq 2 $;而且，对于固定的软不变式，Brieskorn-Pham Hypersurfaces的一系列无限的Lagrangian家族。这些通常与哈密顿的同位素相同。在特定情况下，我们还建立了明确定义的Maslov零态圆形annuli计数，该计数区分了拉格朗日人以紧凑支持的同构。除此之外，这些赋予了精确单调拉格朗日摩托的家族，这些家族既不与几何突变或紧凑型的符号呈现形态相关。
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 $S^1 \times Σ_g$ in $\mathbb{C}^3$, distinguished by soft invariants for any genus $g \geq 2$; 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.
