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

Hidemasa Suzuki

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 $\{L_i^ε=\operatorname{graph}(εdf_i)\}$ 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 $w_ε$ is a polygon in $\mathbb{C}\simeq T^*\mathbb{R}$, we can describe $w_ε$ by a Schwarz-Christoffel map. In \cite{S25}, we proved that pseudoholomorphic disks $w_ε$ converge to the gradient tree in the limit $ε\to+0$ when the image of $w_ε$ is a generic convex quadrilateral. In this paper, we show such a convergence for any convex quadrilaterals by studying the non-generic case.

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

Abstract

For given smooth functions on , Fukaya and Oh showed that the moduli space of pseudoholomorphic disks in which are bounded by Lagrangian sections is diffeomorphic to the moduli space of gradient trees in which consist of gradient curves of . When the image of the pseudoholomorphic disk is a polygon in , we can describe by a Schwarz-Christoffel map. In \cite{S25}, we proved that pseudoholomorphic disks converge to the gradient tree in the limit when the image of is a generic convex quadrilateral. In this paper, we show such a convergence for any convex quadrilaterals by studying the non-generic case.
Paper Structure (13 sections, 27 theorems, 120 equations, 14 figures, 1 table)

This paper contains 13 sections, 27 theorems, 120 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Gradient trees and pseudoholomorphic disks.
  4. Schwarz-Christoffel maps, hypergeometric function and their properties.
  5. Interior angles in convex polygons.
  6. Conformal Modulus of Quadrilaterals.
  7. Gradient trees and holomorphic disks in the case $M=\mathbb{R},k=4$
  8. Main results and their proofs.
  9. Main results.
  10. Power series representations of holomorphic disks.
  11. The behavior of $z_{4,\epsilon}$ and trees.
  12. Proof of Theorem \ref{['k4corr1']}.
  13. Proof of Theorem \ref{['k4corr2']}.

Key Result

Theorem 2.1

Let $\pi$ be the cotangent bundle $\pi:T^{*}M\rightarrow M$. We set $p_{i}=\pi(x_{i}^{\epsilon})$. Let $J=J_{g}$ be the canonical almost complex structure on $T^{*}M$ associated to the metric $g$ on $M$. For each genericThe term "generic" in this sentence refers to the transversality of the unstable

Figures (14)

  • Figure 1: The ribbon tree ($k=6$) (Reproduced from S25).
  • Figure 2: The quadrilateral (Reproduced from S25).
  • Figure 3: The situation of Prop.\ref{['mpocm1']} and Prop.\ref{['mpocm2']}.
  • Figure 4: Candidates of the tree $T$ of the ribbon tree $(T,i)\in Gr_{4}$ (Reproduced from S25).
  • Figure 5: The transformation $\phi_{0,\delta}$ (Reproduced from S25).
  • ...and 9 more figures

Theorems & Definitions (40)

  • Definition 2.1: FO97
  • Definition 2.2: FO97
  • Definition 2.3: FO97
  • Definition 2.4: FO97
  • Theorem 2.1: FO97
  • Theorem 2.2: DT02,Ahl-ca,Neh75 etc.
  • Definition 2.5: Erd50,Mim22 etc.
  • Proposition 2.1: Erd50,Mim22 etc.
  • Definition 2.6: Ols64,Mim22 etc.
  • Lemma 2.1: Ols64,Mim22 etc.
  • ...and 30 more