Explicit correspondences between gradient trees in \(\mathbb {R}\) and holomorphic disks in \(T^{*}\mathbb {R}\)

Abstract

Fukaya and Oh studied the correspondence between pseudoholomorphic disks in \(T^{*}M\) which are bounded by Lagrangian sections \(\{L_{i}^{\epsilon }\}\) and gradient trees in M which consist of gradient curves of \(\{f_{i}-f_{j}\}\). Here, \(L_{i}^{\epsilon }\) is defined by \(L_{i}^{\epsilon }=\) graph\((\epsilon df_{i})\). They constructed approximate pseudoholomorphic disks in the case \(\epsilon >0\) is sufficiently small. When \(M=\mathbb {R}\) and Lagrangian sections are affine, pseudoholomorphic disks \(w_{\epsilon }\) can be constructed explicitly. In this paper, we show that pseudoholomorphic disks \(w_{\epsilon }\) converges to the gradient tree in the limit \(\epsilon \rightarrow +0\) when the number of Lagrangian sections is three and four.