Question about and good reference for Kahn and Markovic result As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary:
Let $M$ be a compact hyperbolic $3$-manifold. Then for any $x\in M$ and for any direction $v\in T_xM$ one may find an immersed surface $S$ in $M$ which is: arbitrarily $\varepsilon$-quasi-Fuchsian, passes arbitrarily closes to $x$ and with normal arbitrarily close to $v$.
First question: do I understand correctly?
Second question: where I can find good reference for such kind of results? Do there exist surveys which are more accessible than the original paper to a broad audience?
 A: I think that a good reference could be the following:
Proposition ([1, Proposition 4.1 (or 5.1 in the arXiv version)]):
Let $M$ be a closed hyperbolic 3-manifold, and regard $\pi_1(M)$ as acting on $\mathbb{H}^3 \simeq \widetilde{M}^3$. For each great circle $C \subset \partial \mathbb{H}^3$ there is a sequence of immersions $(f_i \colon F_i \to M)_i$, where each $F_i$ is a surface and $f_*(\pi_1(F_i))$ is a quasi-Fuchsian subgroup in $\pi_1(M)$, such that $\partial \widetilde{f}_i (\widetilde{F}_i) \subset \mathbb{H}^3$ pointwise converges to $C$.
The "pointwise convergence" can be misleading; the proof of Corollary 4.2 (that you can find right after in the paper) clarifies this notion.
The authors say that this is implied by [2, Theorem 1.1], which should be the same as [3, Theorem 1.1]. To me, this implication is not crystal clear; I think that an expert will be able to clarify this.
Once you have the Proposition, your result should follow easily: take a point and a tangent vector in $\mathbb{H}^3$ that map to your point and vector in $M$. Consider the plane perpendicular to this vector. The sequence of surfaces given by the Proposition should satisfy your requests (to be 100% sure, I should ask which notion of $\varepsilon$-quasi-Fuchsian we are using).
[1]: Bergeron, Nicolas, and Daniel T. Wise. "A boundary criterion for cubulation." American Journal of Mathematics 134.3 (2012): 843-859.
[2]: J. Kahn and V. Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, preprint, 2009.
[3]: Kahn, Jeremy, and Vladimir Markovic. "Immersing almost geodesic surfaces in a closed hyperbolic three manifold." Annals of Mathematics (2012): 1127-1190.
