Free Boundary Constant p-Mean Curvature Surfaces Intersecting the Pansu Sphere

In the paper, we introduce 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 . We introduce a stability criterion for free boundary CPMC surfaces. When M is the Pansu sphere S 1 in the Heisenberg group H 1 and N is the interior of M , we find examples of free boundary CPMC surfaces which are rotationally symmetric about the t -axis of H 1 . Besides the stable p-minimal disk { t = 0 } ∩ N , there exist Pansu spherical caps which are stable free boundary CPMC surfaces intersecting S 1 . We conclude the paper by two analogous problems as those for free boundary CMC surfaces intersecting with the Euclidean 2-sphere.