Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

Abstract

In the unit cone ${\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} < {z}^{2}, {z} > {0}\}$ we establish a geometric maximum principle for H-surfaces, where its mean curvature ${H = H(x, y, z)}$ is optimally bounded. Consequently, these surfaces cannot touch the conical boundary ${\partial \mathcal{C}}$ at interior points and have to approach ${\partial \mathcal{C}}$ transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains ${\Omega}$ which are strictly convex in the following sense: On its whole boundary ${\partial \mathcal{C}(\Omega)}$ their associate cone ${\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}$ admits rotated unit cones ${O \circ \mathcal{C}}$ as solids of support, where ${O \in {\mathbb R}^{3\times3}}$ represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains ${\Omega}$ bounding a given Jordan-contour ${\Gamma \subset \mathcal{C} \backslash \{0\}}$ with one-toone central projection.