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 $\epsilon\to+0$ when the number of Lagrangian sections is three and four.
Categories: math.SG math.CA math.CV