Singular solutions of the Yamabe problem in the Heisenberg group and their bifurcation

We prove the existence of a homogeneous singular solution of the critical equation $$-\mathrm{\Delta }u=u^{\frac{Q+2}{Q-2}}$$ on the Heisenberg group $H^n$, where $Q$ is the $\mathit{\text{homogeneous}}\mathit{\text{dimension}}$. In order to do this, we introduce a suitable concept of normal curvature for hypersurfaces. Furthermore we study the bifurcation of non-homogeneous solutions from the homogeneous one.