From Morse Trees to $J$-Holomorphic Discs -- Rigid Y-Graphs

Abstract

The correspondence between Morse flow trees and $J$-holomorphic discs was established by Fukaya--Oh and Ekholm. We revisit this correspondence and present an alternative approach, designed to generalize naturally to the equivariant setting and to certain Morse graph configurations. The central ingredient is a gluing construction that produces $J$-holomorphic discs from Morse flow trees. A well-known difficulty is that this gluing is of Morse--Bott type, equivalently, in an appropriate Fredholm framework, pieces to be glued together are obstructed. We resolve this via the obstruction bundle gluing technique of Hutchings--Taubes. Given a rigid, transversely cut-out Y-shaped Morse flow tree, we show that for every sufficiently small $ε> 0$ there exists at least one corresponding $J$-holomorphic discs in the cotangent bundle, with boundaries inside corresponding Lagrangian submanifolds of height $ε$. This is the first paper in a series; subsequent work will extend the result to all ribbon trees and to moduli spaces of all dimensions and establish the injectivity and surjectivity of the correspondence.