Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and each fiber is a relatively compact strictly pseuddoconvex (complex) domain with, maybe, different complex structures but all of them diffeomorphic.

Let $Y_0$ be the central fiber. And let $f:Y_0 \to D$ be an holomorphic map where $D$ is the (open) unit disk in $\mathbb{C}$.

What are the obstructions to extend $f$ to a small neighborhood of $Y_0$?. That is, to find a family of holomorphic maps $f_t:Y_t \to D$ for $t \in M$ near $0$.

There is a theorem in On Deformations of Holomorphic Maps. III. by Horikawa that solves this problem for families of compact complex manifolds. The sufficient conditions are that the map $H^1(D, \Gamma D) \to H^1(Y_0, f^{\ast} \Gamma D)$ is surjective and the analogous map for $H^2$ is surjective. (Where $\Gamma D$ denotes the sheaf of germs of vector fields on $D$. I am not sure that this theorem also applies in this situation.

I would appreciate any related problems or some insight in the current state of the art of these kind of questions.

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and each fiber is a relatively compact strictly pseuddoconvex (complex) domain with, maybe, different complex structures but all of them diffeomorphic.

Let $Y_0$ be the central fiber. And let $f:Y_0 \to D$ be an holomorphic map where $D$ is the (open) unit disk in $\mathbb{C}$.

What are the obstructions to extend $f$ to a small neighborhood of $Y_0$?. That is, to find a family of holomorphic maps $f_t:Y_t \to D$ for $t \in M$ near $0$.

There is a theorem in On Deformations of Holomorphic Maps. III. by Horikawa that solves this problem for families of compact complex manifolds. The sufficient conditions are that the map $H^1(D, \Gamma D) \to H^1(Y_0, f^{\ast} \Gamma D)$ is surjective and the analogous map for $H^2$ is injective. (Where $\Gamma D$ denotes the sheaf of germs of vector fields on $D$. I am not sure that this theorem also applies in this situation.

I would appreciate any related problems or some insight in the current state of the art of these kind of questions.

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and each fiber is a relatively compact strictly pseuddoconvex (complex) domain with, maybe, different complex structures but all of them diffeomorphic.

Let $Y_0$ be the central fiber. And let $f:Y_0 \to D$ be an holomorphic map where $D$ is the (open) unit disk in $\mathbb{C}$.

What are the obstructions to extend $f$ to a small neighborhood of $Y_0$?. That is, to find a family of holomorphic maps $f_t:Y_t \to D$ for $t \in M$ near $0$.

There is a theorem in On Deformations of Holomorphic Maps. III. by Horikawa that solves this problem for families of compact complex manifolds. The sufficient conditions are that the map $H^1(Y, \Gamma Y) \to H^1(X, f^{\ast} \Gamma Y)$ is surjective and the analogous map for $H^2$ is surjective. (Where $\Gamma Y$ denotes the sheaf of germs of vector fields on $Y$. I am not sure that this theorem also applies in this situation.

I would appreciate any related problems or some insight in the current state of the art of these kind of questions.

Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and each fiber is a relatively compact strictly pseuddoconvex (complex) domain with, maybe, different complex structures but all of them diffeomorphic.

Let $Y_0$ be the central fiber. And let $f:Y_0 \to D$ be an holomorphic map where $D$ is the (open) unit disk in $\mathbb{C}$.

What are the obstructions to extend $f$ to a small neighborhood of $Y_0$?. That is, to find a family of holomorphic maps $f_t:Y_t \to D$ for $t \in M$ near $0$.

There is a theorem in On Deformations of Holomorphic Maps. III. by Horikawa that solves this problem for families of compact complex manifolds. The sufficient conditions are that the map $H^1(D, \Gamma D) \to H^1(Y_0, f^{\ast} \Gamma D)$ is surjective and the analogous map for $H^2$ is surjective. (Where $\Gamma D$ denotes the sheaf of germs of vector fields on $D$. I am not sure that this theorem also applies in this situation.

I would appreciate any related problems or some insight in the current state of the art of these kind of questions.