We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set (i.e., the set where the (p-)area integrand vanishes), we formulate some extension theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the x y-plane) in the Heisenberg group H 1 . In H 1 , identified with the Euclidean space R 3 , the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, C 2 smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S 3 . This fact continues to hold when S 3 is replaced by a general pseudohermitian 3-manifold.

We consider a C 1 smooth surface with prescribed p (or H )-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed p-mean curvature H ∈ C 0 , we show that any characteristic curve is C 2 smooth and its (line) curvature equals −H in the nonsingular domain. By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are C 2 smooth. We also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. These results can be applied to more general situations. Mathematics Subject Classification (2000)Primary 35L80; Secondary 35J70 · 32V20 · 53A10 · 49Q10

For n ≥ 2 we define a notion of umbilicity for hypersurfaces in the Heisenberg group Hn. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant p(or horizontal)-mean curvature in Hn up to Heisenberg translations.

Abstract. We study immersed, connected, umbilic hypersurfaces in the Heisenberg group Hn with n ≥ 2. We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.