Minimal Surfaces: Boundary value problems

adjoint surface asymptotic lines boundary values bounded branch points Catalan's surface catenoid complete minimal surfaces conformal mapping convex defined denotes differential Dirichlet integral disk disk-type embedded minimal surfaces equations existence finite free boundary Gauss curvature Gauss map geodesic geometric given H-surface H½(B harmonic mappings holomorphic function implies infer intersects isoperimetric inequality Jordan curve Lemma lines of curvature manifold Math mean curvature Moreover normal obtain orthogonal P₁ parameter domain periodic minimal surfaces planar plane Plateau's problem proof Proposition proved real analytic regular representation formula result Riemann Riemannian Riemannian manifolds satisfies Schwarz's Section sequence simply connected sinh solution of Plateau's spherical image stationary minimal surface subset tangent Theorem topological Weierstrass whence X₁ yields zero