I am reading about the deformation theory of hyperbolic manifolds (say in dimension $3$ or higher). I am somewhat familiar with the deformation theory of compact complex manifolds, so I am trying to use the intuition I have about those to guide me in understanding hyperbolic structures. While I was reading about Mostow Rigidity, I realized that I am really lacking some very basic understanding of what happens on the "model case" of the hyperbolic space itself.
If I understood correctly, a basic fact in the theory is that hyperbolic space $\mathbb{H}^n$ is rigid, meaning that if we have a family of complete hyperbolic metrics $\{h_t\mid t\in(-\varepsilon,\varepsilon)\}$ on the disk $\mathbb{D}^n$ then in fact there is a smooth isotopy $\phi_t$ such that $h_t=\phi_t^*h_0$. My understanding for why this should be true is that we all know that a complete, simply connected Riemannian manifold of constant sectional curvature $-1$ is isometric to $\mathbb{H}^n$. A subtler point is the smooth dependence on the parameter $t$. So, first question: is this correct, and is there a canonical reference for the dependence of the isometries on $t$?
My second question regards the infinitesimal rigidity of $\mathbb{H}^n$, which can be stated as follows: let $\dot{g}$ be a first order deformation of the hyperbolic structure $h$ of $\mathbb{H}^n$. Is there a vector field $X$ such that $\dot{g}=\mathcal{L}_Xh$? Again, my guess is that it should be easy to say "yes", but I need some help to prove it (or to find a reference).
To be more clear, by first-order deformation I mean a symmetric $2$-tensor $\dot{g}$ such that for every $x\in\mathbb{D}^n$ and every plane $\sigma$ in $T_x\mathbb{D}^n$, the curvature of $h+\varepsilon\dot{g}$ at $\sigma$ satisfies $K(h+\varepsilon \dot{g},\sigma)=-1+O(\varepsilon^2)$. This corresponds to a second-order system of PDEs for $\dot{g}$, which does not seem particularly easy to attack.
If we knew somehow that the deformation theory of $\mathbb{H}^n$ is unobstructed then the two questions would be equivalent and I would be reasonably happy. But again, I have not been able to find much discussion of the obstruction theory... Let's say that this is a third question.
Thanks in advance for the help!
Your first question doesn't really have an answer because you have not specified any conditions on the family $\{h_t \mid t \in (-\epsilon,+\epsilon)\}$. So, for instance, I could pick two different complete hyperbolic metrics $\mu_1,\mu_2$ on $\mathbb D^n$ and set $h_t = \mu_1$ if $t$ is rational and $h_t = \mu_2$ if $t$ is irrational. Nope, there is no smooth isotopy at all.
Regarding your second question, I do not know the answer to the problem that you pose. However, what I can tell you is that the kind of deformation you are asking about in your second question is not what is studied in the deformation theory of hyperbolic manifolds.
That theory is solely concerned with complete hyperbolic manifolds $M$ that have nontrivial fundamental group $\pi_1 M$ (whic rules out $M= \mathbb H^n$ entirely). By using the theory of universal covering spaces one obtains a universal covering map $\mathbb H^n \mapsto M$ and a deck transformation action of $\pi_1 M$ on $\mathbb H^n$ given by a representation $h : \pi_1 M \to \text{Isom}(\mathbb H^n) = SO(n,1)$. What deformation theory of hyperbolic manifolds is concerned with is deforming the representation $h$, in the space of all representations $\pi_1 M \mapsto SO(n,1)$.