Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\phi\colon S^1\tilde\to \partial X$ with the standard unit circle $S^1$ which is the boundary of the standard unit disk $D$. Consider the closed topological surface $Y:=X\cup_\phi D$ obtained from $X$ by gluing the disk $D$ along $\phi$.

Q1. Is it true that $Y$ has a unique complex structure such that the interiors of $X$ and of $D$ are complex subspaces of $Y$ with their original complex structures?

Q2. If the answer to Q1 is 'yes', is it true that if one chooses another orientation preserving diffeomorphism $\phi'\colon S^1\tilde\to \partial X$ then the resulting Riemann surface $Y'$ is isomorphic (as complex manifold) to $Y$?

Q3. Let $X_0\subset X$ be another Riemann surface which is obtained from $X$ by a small deformation of the boundary of $X$ (thus topologically $X_0$ is a deformation retract of $X$). Let $Y_0$ be obtained from $X_0$ as in Q1. Are $Y$ and $Y_0$ isomorphic?

Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\phi\colon S^1\tilde\to \partial X$ with the standard unit circle $S^1$ which is the boundary of the standard unit disk $D$. Consider the closed topological surface $Y:=X\cup_\phi D$ obtained from $X$ by gluing the disk $D$ along $\phi$.

Q1. Is it true that $Y$ has a unique complex structure such that the interiors of $X$ and of $D$ are complex subspaces of $Y$ with their original complex structures?

Q2. If the answer to Q1 is 'yes', is it true that if one chooses another orientation preserving diffeomorphism $\phi'\colon S^1\tilde\to \partial X$ then the new Riemann surface $Y'$ (as in Q1) is isomorphic (as a complex manifold) to $Y$?

Q3. Let $X_0\subset X$ be another Riemann surface which is obtained from $X$ by a small deformation of the boundary of $X$ (thus topologically $X_0$ is a deformation retract of $X$). Let $Y_0$ be obtained from $X_0$ as in Q1. Are $Y$ and $Y_0$ isomorphic?

Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\phi\colon S^1\tilde\to \partial X$ with the standard unit circle $S^1$ which is the boundary of the standard unit disk $D$. Consider the closed topological surface $Y:=X\cup_\phi D$ obtained from $X$ by gluing the disk $D$ along $\phi$.

Q1. Is it true that $Y$ has a unique complex structure such that the interiors of $X$ and of $D$ are complex subspaces of $Y$ with their original complex structures?

Q2. If the answer to Q1 is 'yes', is it true that if one chooses another orientation preserving isomorphism $\phi'\colon S^1\tilde\to \partial X$ then the resulting Riemann surface $Y'$ is isomorphic (as complex manifold) to $Y$?

Q3. Let $X_0\subset X$ be another Riemann surface which is obtained from $X$ by a small deformation of the boundary of $X$ (thus topologically $X_0$ is a deformation retract of $X$). Let $Y_0$ be obtained from $X_0$ as in Q1. Are $Y$ and $Y_0$ isomorphic?

Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\phi\colon S^1\tilde\to \partial X$ with the standard unit circle $S^1$ which is the boundary of the standard unit disk $D$. Consider the closed topological surface $Y:=X\cup_\phi D$ obtained from $X$ by gluing the disk $D$ along $\phi$.

Q1. Is it true that $Y$ has a unique complex structure such that the interiors of $X$ and of $D$ are complex subspaces of $Y$ with their original complex structures?

Q2. If the answer to Q1 is 'yes', is it true that if one chooses another orientation preserving diffeomorphism $\phi'\colon S^1\tilde\to \partial X$ then the resulting Riemann surface $Y'$ is isomorphic (as complex manifold) to $Y$?

Q3. Let $X_0\subset X$ be another Riemann surface which is obtained from $X$ by a small deformation of the boundary of $X$ (thus topologically $X_0$ is a deformation retract of $X$). Let $Y_0$ be obtained from $X_0$ as in Q1. Are $Y$ and $Y_0$ isomorphic?