Recently, I have been trying to a construction of a gluing map regarding the Lagrangian Floer Homology of two fibers in the cotangent bundle $T^*M$ of a manifold , in order to prove that the map $\Theta$ defining the isomorphism , in section $3$ of https://arxiv.org/pdf/math/0408280.pdf. is a chain map.
Now, from all the places I have seen/studied Floer Homology that actually do the gluing construction the idea seems to be standard. Let's take two maps $(u,v)$ that we want to glue. We due a pre-gluing $u\#_{\rho}v$and then we use the Newton-Picard method to a find a section $\gamma_{\rho}$ such that $\exp_{u_{\rho}v}\gamma_{p}$ satisfies the Floer equation. This exponential map is with respect to some metric, for example in Classic Floer Homology for the periodic orbit problem I don't think we need to worry about the metric we use, however if we are doing Lagrangian Floer Homology we need to pick a family of metrics $g_t$ such that $L_0$ is totally geodesic with respect to $g_0$ and $L_1$ is totally geodesic with respect to $g_1$ and then take the exponential with respect to this family of metrics $g_t$. More generally , if we have different boundary conditions what I think happens is that we wanna use a metric such that the boundary conditions will be satisfied.
My problem is the following : I am doing the gluing of a map $u:\mathbb{R}^{+}\times [0,1]\rightarrow T^*M$ and $v:\mathbb{R}\times [0,1]\rightarrow T^*M$ where $\pi(u(\cdot,0))\in W^{u}(q)$ and I am able to find a section $\gamma_{\rho}$ such that $\exp_{u\#_{\rho}v}(\gamma_{p})$ satisfies the Floer equation using the Newton-Picard method. However, I want the boundary condition $\pi(\exp_{u\#_{\rho}v}(\gamma_{p}))(0,\cdot)\in W^{u}(q)$ and so far I haven't been able to find a metric that will make this happen. Therefore I was wondering, if there other approaches to the gluing construction that use other alternatives to the exponential map so that I would try and fix this problem some other way. Any insight is appreciated, thanks in advance.
