More on explicit correspondence between gradient trees in $\mathbb{R}$ and holomorphic convex quadrilaterals in $T^{*}\mathbb{R}$

TL;DR

This paper extends previous results by proving that pseudoholomorphic disks in the cotangent bundle of rica converge to gradient trees for all convex quadrilaterals, including non-generic cases, thus broadening the understanding of their correspondence.

Contribution

It generalizes the convergence result of pseudoholomorphic disks to gradient trees from generic to all convex quadrilaterals, including non-generic configurations.

Findings

Convergence of pseudoholomorphic disks to gradient trees for all convex quadrilaterals.

Extension of previous generic case results to non-generic quadrilaterals.

Use of Schwarz-Christoffel maps to describe disk images.

Abstract

For given smooth functions $(f_1,\dots,f_n)$ on $M$, Fukaya and Oh showed that the moduli space of pseudoholomorphic disks in $T^*M$ which are bounded by Lagrangian sections $\{L_i^\epsilon=\operatorname{graph}(\epsilon df_i)\}$ is diffeomorphic to the moduli space of gradient trees in $M$ which consist of gradient curves of $\{f_i-f_j\}$. When the image of the pseudoholomorphic disk $w_\epsilon$ is a polygon in $\mathbb{C}\simeq T^*\mathbb{R}$, we can describe $w_\epsilon$ by a Schwarz-Christoffel map. In \cite{S25}, we proved that pseudoholomorphic disks $w_\epsilon$ converge to the gradient tree in the limit $\epsilon\to+0$ when the image of $w_\epsilon$ is a generic convex quadrilateral. In this paper, we show such a convergence for any convex quadrilaterals by studying the non-generic case.