Compact graphs over a sphere of constant second order mean curvature

Constant mean curvature surfaces with boundary on a sphere

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must be spherical. Our results apply in many scenarios in physics where in absence...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Constant Mean Curvature Graphs in a Strip of R

where H is a given nonzero number and φ is a smooth function on ∂Ω. The graph of a solution u ∈ C2(Ω) ∩ C0(Ω) is a surface with constant mean curvature H spanning the space curve given by the graph of φ. The orientation of this graph is given by N = (∇u,−1)/√1 + |∇u|2, that is, N points downward. From the physical viewpoint, a soap film in equilibrium between two regions of different gas pressu...

A note on radial graphs with constant mean curvature

Let be a smooth domain on the unit sphere S whose closure is contained in an open hemisphere and denote by H the mean curvature of ∂ as a submanifold of with respect to the inward unit normal. It is proved that for each real number H that satisfies inf H > −H ≥ 0, there exists a unique radial graph on bounded by ∂ with constant mean curvature H . The orientation on the graph is based on the nor...

A height estimate for constant mean curvature graphs and uniqueness

In this paper, we give a height estimate for constant mean curvature graphs. Using this result we prove two results of uniqueness for the Dirichlet problem associated to the constant mean curvature equation on unbounded domains. 2000 Mathematics Subject Classification. 53A10. Introduction The surfaces with constant mean curvature are the mathematical modelling of soap films. These surfaces appe...