Geodesic Graphs with Constant Mean Curvature in Spheres

Abstract

We study the existence and unicity of graphs with constant mean curvature in the Euclidean sphere $\mathbb{S}^{n + 1} (a)$ of radius a. Given a compact domain Ω, with some conditions, contained in a totally geodesic sphere S of $\mathbb{S}^{n + 1} (a)$ and a real differentiable function χ on Ω, we define the graph of χ in $\mathbb{S}^{n + 1} (a)$ considering the ‘height’ χ(x) on the minimizing geodesic joining the point x of Ω to a fixed pole of $\mathbb{S}^{n + 1} (a)$. For a real number H such that |H| is bounded for a constant depending on the mean curvature of the boundary Γ of Ω and on a fixed number δ in (0,1), we prove that there exists a unique graph with constant mean curvature H and with boundary Γ, whenever the diameter of Ω is smaller than a constant depending on δ. If we have conditions on Γ, that is, let Γ′ be a graph over Γ of a function, if |H| is bounded for a constant depending only on the mean curvature of Γ and if the diameter of Ω is smaller than a constant depending on H and Γ, then we prove that there exists a unique graphs with mean curvature H and boundary Γ′. The existence of such a graphs is equivalent to assure the existence of the solution of a Dirichlet problem envolving a nonlinear elliptic operator.