H-principles for regular Lagrangians

Abstract

We prove an existence h-principle for regular Lagrangians with Legendrian boundary in arbitrary Weinstein domains of dimension at least six; this extends a previous result of Eliashberg, Ganatra, and the author for Lagrangians in flexible domains. Furthermore, we show that all regular Lagrangians come from our construction and describe some related decomposition results. We also prove a regular version of Eliashberg and Murphy's h-principle for Lagrangian caps with loose negative end. As an application, we give a new construction of infinitely many regular Lagrangian disks in the standard Weinstein ball.

Simplifying Weinstein Morse functions

“…W \W f lex is an exact symplectic cobordism, perhaps without a compatible Weinstein Morse function. The decomposition in Theorem 1.5 has several applications, which are explored in [25,24]; for example, it is used to prove an existence h-principle for regular Lagrangians with boundary in arbitrary Weinstein domains and construct 'maximal' Weinstein domains.…”

Maximal contact and symplectic structures

“…The proof is similar to the proof of Theorem 4.1, which uses the existence of smoothly trivial Weinstein cobordisms from overtwisted contact structures to arbitrary contact structures [14]. Here we will need the following Legendrian analog, proven in [44]: for any Legendrian Λ n−1 ⊂ (Y 2n−1 , ξ), n ≥ 3, there is a smoothly trivial regular Lagrangian cobordism L n in the trivial Weinstein cobordism (Y, ξ) × [0, 1] such that ∂ + L n = Λ n−1 and ∂ − L n = Λ n−1 loose . Here Λ n−1 loose ⊂ (Y 2n−1 , ξ) is the (unique) loose Legendrian in the same formal class as Λ n−1 .…”

“…Since Λ i are formally Legendrian isotopic, Λ i,loose are Legendrian isotopic by the h-principle for loose Legendrians [52]; we will fix one representative Λ loose of these Legendrians. Then by [44] There is a Lagrangian cobordism Λ loose ×[0, k] ⊂ X 2n . Its positive boundary Λ loose ×{k} ⊂ ∂ + X 2n is not loose since the attaching spheres of X 2n are symplectically linked with Λ loose .…”