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I believe this question to be common knowledge for people working in the area. I have posted it on Mathstackexchange but nobody replied.

Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex structure on $M$. My naive question is: when does $\gamma$ bound a pseudoholomorphic curve? More precisely,

when there does exists a Riemann surface with boundary $\Sigma$, and a J-holomorphic map $u:\Sigma\to (M,J)$ such that $u|_{\partial \Sigma} \equiv \gamma$ ?

I do not know much about pseudoholomorhic curves, I see that some people requires some conditions on the boundary like, contact type or totally real, if you want you can consider these assumptions or deal with these cases.

I believe this question to be common knowledge for people working in the area. I have posted it on Mathstackexchange but nobody replied.

Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex structure on $M$. My naive question is: when does $\gamma$ bound a pseudoholomorphic curve? More precisely,

when there does exists a Riemann surface with boundary $\Sigma$, and a J-holomorphic map $u:\Sigma\to (M,J)$ such that $u|_{\partial \Sigma} \equiv \gamma$ ?

I do not know much about pseudoholomorhic curves, I see that some people requires some conditions on the boundary like, contact type or totally real, if you want you can consider these assumptions or deal with these cases.

I have posted it on Mathstackexchange but nobody replied.

Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex structure on $M$. My naive question is: when does $\gamma$ bound a pseudoholomorphic curve? More precisely,

when there does exists a Riemann surface with boundary $\Sigma$, and a J-holomorphic map $u:\Sigma\to (M,J)$ such that $u|_{\partial \Sigma} \equiv \gamma$ ?

I do not know much about pseudoholomorhic curves, I see that some people requires some conditions on the boundary like, contact type or totally real, if you want you can consider these assumptions or deal with these cases.

Source Link
Overflowian
Overflowian
  • 2.5k
  • 1
  • 12
  • 25

Which curves are boundary of pseudoholomorphic curves?

I believe this question to be common knowledge for people working in the area. I have posted it on Mathstackexchange but nobody replied.

Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex structure on $M$. My naive question is: when does $\gamma$ bound a pseudoholomorphic curve? More precisely,

when there does exists a Riemann surface with boundary $\Sigma$, and a J-holomorphic map $u:\Sigma\to (M,J)$ such that $u|_{\partial \Sigma} \equiv \gamma$ ?

I do not know much about pseudoholomorhic curves, I see that some people requires some conditions on the boundary like, contact type or totally real, if you want you can consider these assumptions or deal with these cases.