A geometric inequality in the Heisenberg group and its applications to stable solutions of semilinear problems

In the Heisenberg group framework, we obtain a geometric inequality for stable solutions of Δℍ u = f (u) in a domain Ω⊆ℍ. More precisely, if we denote the horizontal intrinsic Hessian by Hu, the mean curvature of a level set by h, its imaginary curvature by p, the intrinsic normal by ν and the unit tangent by υ, we have that Mathematic expression for any Mathematic expression. Stable solutions in the entire ℍ satisfying a suitably weighted energy growth and such that Mathematic expression are then shown to have level sets with vanishing mean curvature.