H-principles for regular Lagrangians

TL;DR

This paper establishes an h-principle for regular Lagrangians with Legendrian boundary in Weinstein domains of dimension six or higher, extending previous flexible domain results and providing new constructions of Lagrangian disks.

Contribution

It proves a comprehensive existence h-principle for regular Lagrangians in Weinstein domains and shows all such Lagrangians arise from this construction, also extending to Lagrangian caps.

Findings

Proved an existence h-principle for regular Lagrangians with Legendrian boundary.

All regular Lagrangians are obtained from the new construction.

Constructed infinitely many regular Lagrangian disks in the standard Weinstein ball.

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.