Geometric aspects of PDEs on sub-Riemannan manifolds

We consider sone analytic problems arising in sub-Riemannian geometry. First, we construct singular solutions to the CR Yamabe equation in the Heisenberg group in two regimes: nearly cylindrical solutions via bifurcation theory and nearly locally spherical solutions via a refined implicit function argument. We then consider the Einstein-Hilbert action in the CR setting. We characterize stationary points as pseudo-Einstein structures, then showing the role of embeddability in determining the sign of the second variation at standard spheres.