Spherical CR Dehn Surgery

Consider a three dimensional cusped spherical $\mathrm{C}\mathrm{R}$ manifold $M$ and suppose that the holonomy representation of ${\pi }_1(M)$ can be deformed in such a way that the peripheral holonomy is generated by a non-parabolic element. We prove that, in this case, there is a spherical $\mathrm{C}\mathrm{R}$ structure on some Dehn surgeries of $M$. The result is very similar to R. Schwartz’s spherical $\mathrm{C}\mathrm{R}$ Dehn surgery theorem, but has weaker hypotheses and does not give the unifomizability of the structure. We apply our theorem in the case of the Deraux-Falbel structure on the Figure Eight knot complement and obtain spherical $\mathrm{C}\mathrm{R}$ structures on all Dehn surgeries of slope $-3+r$ for $r\in {\mathbb{Q}}^{+}$ small enough.