Geometry of generating functions and Lagrangian spectral invariants

الأصل · EN

Partially motivated by the study of topological Hamiltonian dynamics, we prove various C⁰-aspects of the Lagrangian spectral invariants and the basic phase functions fₕ, that is, a natural graph selector constructed by Lagrangian Floer homology of H (relative to the zero section oₙ). In particular, we prove that γlag(ϕₕ¹(oₙ)): = ρlag(H;1) - ρlag(H;[pt]#) → 0 as ϕₕ¹ → id, provided H's satisfy Xₕ ⊂ Dʳ(T*N) oB for some R > 0 and a closed subset B ⊂ N with nonempty interior. We also study the relationship between fₕ and ρlag(H;1) and prove a structure theorem of the micro-support of the singular locus (σₕ) of the function fₕ. Based on this structure theorem and a classification theorem of generic Lagrangian singularity in N = 2 obtained by Arnold's school, we define the notion of cliff-wall surgery when N = 2: the surgery replaces a multi-valued Lagrangian graph ϕₕ¹(oₙ) by a piecewise-smooth Lagrangian cycle that is canonically constructed out of the single valued branch Σₕ: = dfₕ ⊂ ϕₕ¹(oₙ) defined on an open dense subset of N (σₕ) of codimension 1.