WP4: Geometric analysis

Nonlinear PDE find natural applications in curvature problems. While studying subriemannian minimal surfaces there is still a big gap between regularity results obtained with PDE methods and existence results, obtained with geometric analysis tools, which provide only BV solutions. We will fill the gap with a completely new approach, which complements the more geometric approach of the local group in Granada with the more analytic/PDE point of view of AS and PITTS/MIT/WPI groups. The first conjecture regarding the isoperimetric problem in the Heisenberg group is due to Pansu, who suggested that the minimizer could be a surface foliated by subriemannian geodesics. The problem is still open both in the riemannian and subriemannian setting. The last problem we will face involves conformal covariance and in particular the structure of conformally covariant elliptic operators.