Free boundary constant p-mean curvature surfaces intersecting the Pansu sphere

The Pansu conjecture in ℍ1 is an interesting and difficult problem. The present paper introduces the notion of free boundary constant p-mean curvature (CPMC) surface in a 3-dimensional pseudohermitian manifold N with boundary M. It arises as the critical point of p-area among surfaces which divides N into two subsets of preassigned volumes and whose boundary is free to move in M. The authors introduce a stability criterion for free boundary CPMC surfaces. When M is the Pansu sphere S1 in the Heisenberg group ℍ1 and N is the interior of M, we find examples of free boundary CPMC surfaces which are rotationally symmetric about the t-axis of ℍ1. Besides the stable p-minimal disk {t=0}∩N, there exist Pansu spherical caps which are stable free boundary CPMC surfaces intersecting S1. The authors conclude the paper by discussing two problems analogous to known problems for free boundary CMC surfaces intersecting with the Euclidean 2-sphere.