标题： 梯度树在$\mathbb{R}$中与全纯盘在$T^{*}\mathbb{R}$中的显式对应关系 显示英文标题

标题： Explicit correspondences between gradient trees in $\mathbb{R}$ and holomorphic disks in $T^{*}\mathbb{R}$

摘要： 福家和奥研究了在$T^{*}M$中由拉格朗日截面$\{L_{i}^{\epsilon}\}$限定的伪全纯盘与由$\{f_{i}-f_{j}\}$的梯度曲线组成的$M$中的梯度树之间的对应关系。 这里，$L_{i}^{\epsilon}$由$L_{i}^{\epsilon}=$图像$(\epsilon df_{i})$定义。 他们构造了在$\epsilon>0$足够小的情况下近似伪全纯盘。当$M=\mathbb{R}$和拉格朗日截面是仿射的时候，可以显式构造伪全纯盘$w_{\epsilon}$。在本文中，我们证明当拉格朗日截面的数量为三和四时，伪全纯盘$w_{\epsilon}$在极限$\epsilon\to+0$下收敛到梯度树。 显示英文摘要

摘要： 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.