Unwrapped continuation invariance in Lagrangian Floer theory: energy and C⁰ estimates

2009/10/06 by OH, Yong-Geun
#53D40 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.0910.1131

Abstract

We consider pairs of Lagrangian submanifolds (L₀,L), (L₁, L) belonging to the class of Lagrangian submanifolds with conic ends on Weinstein manifolds. The main purpose of the present paper is to define a canonical chain map h_\CL: CF(L₀,L) → CF(L₁,L) of Lagrangian Floer complex inducing an isomorphism in homology, under the Hamiltonian isotopy \CL=\L_s\_{0 ≤ s≤ 1} generated by conic Hamiltonian functions such that the intersections L ∩ L_s do not escape to infinity. The main ingredients of the proof is an a priori bound for general isotopy of the energy quadratic at infinity and a C⁰-bound for the C¹-small isotopy \CL = \L_s\, for the associated pseudo-holomorphic map equations with moving Lagrangian boundary induced by a conic Hamiltonian isotopy. For the Lagrangian submanifolds with asymptotically conic ends, we construct a natural homomorphism h_\LL: HF(L₀,L) → HF(L₁,L) for which the corresponding chain map may not necessarily exist. This provides a more conventional construction of the chain isomorphism which replaces the sophisticated method using the Lagrangian cobordism via the machinery of \citekasturi-oh1,kasturi-oh2 whose details were only outlined in \citeoh:gokova.

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