Geometry of generating functions and Lagrangian spectral invariants

Abstract

Partially motivated by the study of topological Hamiltonian dynamics, we prove various $C^0$-aspects of the Lagrangian spectral invariants and the basic phase functions $f_H$, that is, a natural graph selector constructed by Lagrangian Floer homology of $H$ (relative to the zero section $o_N$). In particular, we prove that $${\gamma }^{lag}({\varphi }_H^1(o_N)):={\rho }^{lag}(H;1)-{\rho }^{lag}(H;[pt{]}^{\#})\to 0$$ as ${\varphi }_H^1\to id$, \emph{provided} $H$'s satisfy $\supp X_H \subset D^R(T^*N) \setminus o_B$ for some $R>0$ and a closed subset $B\subset N$ with nonempty interior. We also study the relationship between $f_H$ and ${\rho }^{lag}(H;1)$ and prove a structure theorem of the micro-support of the singular locus $\Sing(σ_H)$ of the function $f_H$. Based on this structure theorem and a classification theorem of generic Lagrangian singularity in $\dim N=2$ obtained by Arnold's school, we define the notion of cliff-wall surgery when $\dim N=2$: the surgery replaces a multi-valued Lagrangian graph ${\varphi }_H^1(o_N)$ by a piecewise-smooth Lagrangian cycle that is canonically constructed out of the single valued branch $Σ_H: = \Graph df_H \subset ϕ_H^1(o_N)$ defined on an open dense subset of $N \setminus \Sing(σ_H)$ of codimension 1.

38 pages, 2 figures; This paper is largely a reorganization of Part I and section 11, 12 of the withdrawn article arXiv:1111.5992v5; v2) presentation improved, introduction re-written, old sections 2, 3 removed, Example 6.1 added and some imprecise remark corrected

References in corpus (3)

• On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows
• Quasi-morphisms on cotangent bundles and symplectic homogenization
• Localization of Floer homology of engulfable topological Hamiltonian loop

Cited by in corpus (2)

• Localization of Floer homology of engulfable topological Hamiltonian loop
• Continuous Hamiltonian dynamics and area-preserving homeomorphism group of $D^2$