Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ for $q_0,q_1 \in M$.

It is claimed that by using $L^{\infty}$ estimates one can show that the moduli spaces $\mathcal{M}(x^-,x^+)$ are pre-compact in $C^{\infty}_{loc}(\mathbb{R}\times [0,1],T^*M)$. To do this I belive one can just adapt the argument that is done in floer theory for the periodic case and use the $L^{\infty}$ estimates to be able to use the Arzela-Ascoli theorem.

However I am having some difficulty in proving that we have uniform gradient bounds, which is a key part of the argument , and the authors claim that one can obtian such a result by using a bubbling-off argument to produce a $J$-holomorphic disk. I have tried to replicate the argument done in the periodic case and in the end I am able to find a J-Holomorphic map $u:\mathbb{R}\times \bar {\mathbb{R}} \rightarrow T^*M$ such that $\int_{\mathbb{R}\times \bar {\mathbb{R}}}u^*\omega <\infty$ and this integral is non-zero, and $u(s,\infty)\subset T_{q_0^*M}, u(s,-\infty)\subset T_{q_1}^*M$. However I am not sure one is able to use this map to produce a $J$-holomorphic disk with boundary in either $T_{q_0}^*M$ or $T_{q_1}^*M$.

Any enlightment is appreciated, thanks in advance .

Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ for $q_0,q_1 \in M$.

It is claimed that by using $L^{\infty}$ estimates one can show that the moduli spaces $\mathcal{M}(x^-,x^+)$ are pre-compact in $C^{\infty}_{loc}(\mathbb{R}\times [0,1],T^*M)$. To do this I belive one can just adapt the argument that is done in floer theory for the periodic case and use the $L^{\infty}$ estimates to be able to use the Arzela-Ascoli theorem.

However I am having some difficulty in proving that we have uniform gradient bounds in the space $\mathcal{M}(x^{-},x^{+})$. I have tried to follow the argument for the periodic case but I was not able to obtain in this case a holomorphic disk with boundary in either $T_{q_0}^*M$ or $T_{q_1}^*M.$

Any enlightment is appreciated, thanks in advance .

Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ for $q_0,q_1 \in M$.

It is claimed that by using $L^{\infty}$ estimates one can show that the moduli spaces $\mathcal{M}(x^-,x^+)$ are pre-compact in $C^{\infty}_{loc}(\mathbb{R}\times [0,1],T^*M)$. To do this I belive one can just adapt the argument that is done in floer theory for the periodic case and use the $L^{\infty}$ estimates to be able to use the Arzela-Ascoli theorem.

However I am having some difficulty in proving that we have uniform gradient bounds, which is a key part of the argument . I have tried to replicate the argument done in the periodic case and in the end I am able to find a J-Holomorphic map $u:\mathbb{R}\times \bar {\mathbb{R}} \rightarrow T^*M$ such that $\int_{\mathbb{R}\times \bar {\mathbb{R}}}u^*\omega <\infty$ and this integral is non-zero, and $u(s,\infty)\subset T_{q_0^*M}, u(s,-\infty)\subset T_{q_1}^*M$. However I am not sure one is able to use this map to produce a $J$-holomorphic disk with boundary in either $T_{q_0}^*M$ or $T_{q_1}^*M$.

Any enlightment is appreciated, thanks in advance .

Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ for $q_0,q_1 \in M$.

It is claimed that by using $L^{\infty}$ estimates one can show that the moduli spaces $\mathcal{M}(x^-,x^+)$ are pre-compact in $C^{\infty}_{loc}(\mathbb{R}\times [0,1],T^*M)$. To do this I belive one can just adapt the argument that is done in floer theory for the periodic case and use the $L^{\infty}$ estimates to be able to use the Arzela-Ascoli theorem.

However I am having some difficulty in proving that we have uniform gradient bounds, which is a key part of the argument , and the authors claim that one can obtian such a result by using a bubbling-off argument to produce a $J$-holomorphic disk. I have tried to replicate the argument done in the periodic case and in the end I am able to find a J-Holomorphic map $u:\mathbb{R}\times \bar {\mathbb{R}} \rightarrow T^*M$ such that $\int_{\mathbb{R}\times \bar {\mathbb{R}}}u^*\omega <\infty$ and this integral is non-zero, and $u(s,\infty)\subset T_{q_0^*M}, u(s,-\infty)\subset T_{q_1}^*M$. However I am not sure one is able to use this map to produce a $J$-holomorphic disk with boundary in either $T_{q_0}^*M$ or $T_{q_1}^*M$.

Any enlightment is appreciated, thanks in advance .