The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at infinity would then corresponding to the unit $2$-sphere in the Poincare model).

There is a nice famous map, the embedding of $\mathbb{P}^1(\mathbb{C})$ as a rational normal curve of degree $d$ inside $\mathbb{P}^d(\mathbb{C})$. In homogeneous coordinates, it maps $v \in \mathbb{C}^2 \setminus \{ \mathbf{0} \}$ to $\operatorname{Sym}^d(v) \in \operatorname{Sym}^d(\mathbb{C}^2) \setminus \{\mathbf{0}\}$. Let us denote the induced embedding by $\nu_d$, so that $$ \nu_d: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^d(\mathbb{C}).$$

My question is this. Can one extend $\nu_d$, to some other map, say $\tilde{\nu}_d$, from $\overline{B(0,1)}$ (thought of as the closure of the Poincare open ball, so as to also include the sphere at infinity), to some real analytic manifold $M$ containing $\mathbb{P}^d$ as some kind of boundary "at infinity" in some sense, and such that one recovers $\nu_d$ when restricting to the sphere at infinity in the domain of $\tilde{\nu}_d$?

As such, I think the answer is most probably yes, since I did not impose many restrictions (and since my question is somewhat vague). However, I am interested in some natural construction, possibly involving what I like to call the hyperbolic $3$-Grassmannian, namely the manifold $$ \operatorname{SO}(3,d) \, / \, \operatorname{SO}(3) \times \operatorname{SO}(d)$$ (I suspect $M$ can be taken to be the previous manifold).

There may be an easy construction out there. I will think about this soon. I think it is an interesting question though.

The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at infinity would then corresponding to the unit $2$-sphere in the Poincare model).

There is a nice famous map, the embedding of $\mathbb{P}^1(\mathbb{C})$ as a rational normal curve of degree $d$ inside $\mathbb{P}^d(\mathbb{C})$. In homogeneous coordinates, it maps $v \in \mathbb{C}^2 \setminus \{ \mathbf{0} \}$ to $\operatorname{Sym}^d(v) \in \operatorname{Sym}^d(\mathbb{C}^2) \setminus \{\mathbf{0}\}$. Let us denote the induced embedding by $\nu_d$, so that $$ \nu_d: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^d(\mathbb{C}).$$

My question is this. Can one extend $\nu_d$, to some other map, say $\tilde{\nu}_d$, from $\overline{B(0,1)}$ (thought of as the closure of the Poincare open ball, so as to also include the sphere at infinity), to some real analytic manifold $M$ containing $\mathbb{P}^d$ as some kind of boundary "at infinity" in some sense, and such that one recovers $\nu_d$ when restricting to the sphere at infinity in the domain of $\tilde{\nu}_d$?

As such, I think the answer is most probably yes, since I did not impose many restrictions (and since my question is somewhat vague). However, I am interested in some natural construction, possibly involving some Grassmannian as $M$ (I think there may be a related construction in an article by Claude LeBrun).

I will think about this soon. I think it is an interesting question though.