Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?

1$\begingroup$ Could you say what is meant by a "deformation" and by a "trivial deformation", and also clarify whether you want an example of a closed manifold? $\endgroup$ – Igor Belegradek Mar 28 at 22:17

$\begingroup$ A charitable reading of your question is "Are there examples of compact Haken hyperbolic 3manifolds where the hyperbolic structure is locally rigid among all flat conformal structures." If this is indeed your question then such examples do exist and are due to Kevin Scannell. $\endgroup$ – Misha Mar 29 at 1:15

$\begingroup$ See Theorem 2 of this paper by @Misha link.springer.com/article/10.1007%2FBF01459788 $\endgroup$ – Ian Agol Mar 29 at 3:32

$\begingroup$ Thank you @Misha for making my question appropriate.I have edited now and used your sentence. $\endgroup$ – Gorapada Bera Mar 29 at 4:13
Let $M$ be a compact hyperbolic 3manifold and let $C(M)$ be the moduli space of flat conformal structures on $M$. The hyperbolic metric on $M$ defines a distinguished point $h\in C(M)$. One says that $h$ is locally rigid (in $C(M)$) if $h$ is an isolated point of $C(M)$. By Thurston's holonomy theorem, local rigidity of $h$ is equivalent to local rigidity of the holonomy representation $\rho_h$ of $h$ in $$ Rep(M):= Hom(\pi_1(M), O(4,1))/O(4,1).$$
Let $\Gamma< O(3,1)$ be the image of $\rho_h$ (it is a uniform torsionfree lattice in $O(3,1)$). The group $O(3,1)$ and, hence, $\Gamma$, acts naturally on the Lorentz space ${\mathbb R}^{3,1}$. A sufficient condition for local rigidity of $\rho_h$ in $Rep(M)$ is vanishing of the 1st cohomology group $$ H^1(M; {\mathcal F})\cong H^1(\Gamma; {\mathbb R}^{3,1}), $$ where ${\mathcal F}$ is a local system on $M$ determined by the action of $\Gamma$ on ${\mathbb R}^{3,1}$.
A representation $\rho_h$ is called infinitesimally rigid if this cohomology group vanishes.
Here is what is known:
Theorem. There are compact oriented hyperbolic 3manifolds $M$ for which $H^1(M; {\mathcal F})=0$. Hence, for such manifolds, $h$ is locally rigid in $C(M)$.
The earliest examples (or, rather, existence of examples) were obtained as Dehn surgery on hyperbolic 2bridge knots, [1]: Infinitely many Dehn surgeries yield infinitesimally rigid manifolds. These manifolds were nonHaken. Later, examples of infinitesimally rigid Haken hyperbolic manifolds (where incompressible surface is quasifuchsian) were constructed in [2]. In [5] it was proven that for every 2bridge knot all but finitely many Dehn surgeries yield infinitesimally rigid manifolds. In [6], the authors made a comprehensive numerical study of $H^1(M; {\mathcal F})$ for hyperbolic 3manifolds. They determined that out of the first 4500 2generator manifolds in the Hodgson–Weeks census, only 61 are not infinitesimally rigid.
Here is what is unknown:
Question 1. Are there compact hyperbolic nonHaken 3manifolds $M$ for which $h$ is not locally rigid in $C(M)$?
Question 2. Are there compact hyperbolic 3manifolds $M$ for which $H^1(M; {\mathcal F})\ne 0$ but $h$ is locally rigid in $C(M)$?
Given a nonzero cohomology class $c\in H^1(M; {\mathcal F})$, there is an infinite sequence of obstructions for "integrating" $c$ into a curve in $Rep(M)$ tangent to $c$. The first of these obstructions (the cup product) is known to vanish, [2], but higher obstructions (Massey products) are a mystery. There is a compelling evidence (coming from the study of projective deformations of hyperbolic structures, [7]) that Question 2 has positive answer.
Questions 1 and 2 are closely related since Scannell, [3], proved that for a nonHaken "Fibonacci" manifold $M$, $H^1(M; {\mathcal F})\ne 0$. We still do not know local rigidity in this case.
Question 3. Is there a compact hyperbolic $3$manifold $M$ for which $C(M)=\{h\}$, i.e. $h$ is globally rigid?
Most likely, Dehn surgeries on 2bridge knots have this property but it's hard to prove.
Question 4. What topological/geometric features of a hyperbolic 3manifold $M$ are responsible for vanishing/nonvanishing of $H^1(M; {\mathcal F})$, resp. local rigidity/flexibility of the hyperbolic structure?
It was realized early on (first, by Bill Thurston) that any closed embedded totallygeodesic hypersurface in $M$ yields nontrivial conformally flat deformations of the hyperbolic metric. On the other hand, as it was proven by Scannell, having merely an incompressible quasifuchsian surface is not enough. One can conjecture that if $M$ contains an embedded quasifuchsian surface which is "almost totally geodesic" (as in the work of Kahn and Markovi?), then $h$ is not locally rigid in $C(M)$. A weaker form of this conjecture is:
Conjecture. Let $M$ be a compact hyperbolic 3manifold. Then $M$ admits a sequence of finite coverings $M_i\to M$ such that ranks of $H^1(M_i; {\mathcal F})$ diverge to infinity.
This is known to be true for some classes of arithmetic manifolds (originally, for the simplest type, due to Millson, but also for other arithmetic types by Clozel and others), but is unknown for arbitrary arithmetic hyperbolic 3manifolds.
Question 5. What is the "right analogue" of Thurston's measured geodesic laminations which are responsible for nontrivial deformations of the hyperbolic structure $h$ in $C(M)$?
The evidence points towards some "unidentified laminar objects" which, as sets, are disjoint unions of complete geodesics and of totallygeodesic surfaces with geodesic boundary; they should be equipped with a kind of transverse metricmeasure structure. See [8] for some partial results in this direction.
[1] M. Kapovich, Deformations of representations of discrete subgroups of SO(3,1), Math. Ann. 299 (1994), 341–354.
[2] M. Kapovich and J. J. Millson, On the deformation theory of representations of fundamental groups of compact hyperbolic 3manifolds, Topology 35 (1996), 1085–1106.
[3] K. Scannell, Infinitesimal deformations of some SO(3,1) lattices. Pacific J. Math. 194 (2000), no. 2, 455–464.
[4] K. Scannell, Local rigidity of hyperbolic 3manifolds after Dehn surgery. Duke Math. J. 114 (2002), no. 1, 1–14.
[5] S. Francaviglia, J. Porti, Rigidity of representations in SO(4,1) for Dehn fillings on 2bridge knots. Pacific J. Math. 238 (2008), no. 2, 249–274.
[6] D. Cooper, D. Long, M. Thistlethwaite. Computing varieties of representations of hyperbolic 3manifolds into SL(4,?). Experiment. Math. 15 (2006), no. 3, 291–305.
[7] D. Cooper, D. Long, M. Thistlethwaite. Flexing closed hyperbolic manifolds. Geom. Topol. 11 (2007), 2413–2440.
[8] A. Bart, K. Scannell, A note on stamping. Geom. Dedicata 126 (2007), 283–291.

$\begingroup$ Again thank you so much @Misha for a very beautiful exposition of what I was looking for. I am very grateful. $\endgroup$ – Gorapada Bera Apr 1 at 5:34