Geometric properties of harmonic mappings in stable Riemannian domains

H. Kneser (Jahresber Dt Math Vereinigung 35:123-124, 1926) showed by an ingenious method that plane harmonic mappings on the unit disc B, which attribute the circumference partial derivative B in a topological way to a convex curve Gamma, necessarily yield a diffeomorphism of B onto the interior G of the contour Gamma and a homeomorphism between their closures. E. Heinz has generalized this method to solutions of nonlinear elliptic systems [see Chap.13, Sect.6 of Sauvigny (Partial differential equations. 1. Foundations and integral representations; 2. Functional analytic methods; with consideration of lectures by E. Heinz. Springer, London, 2012], however, this reasoning is restricted to the local situation and requires Lipschitz conditions for certain linear combinations of their coefficient functions. These Lewy-Heinz-systems comprise the equations for harmonic mappings with respect to a Riemannian metric and were utilized by Jost (J Reine Angew Math 342:141-153, 1981) to prove univalency for harmonic mappings between Riemannian surfaces. A global result is achieved by reconstruction of the solution for the Dirichlet problem, since this problem is uniquely determined by the uniqueness result of Jager and Kaul (Manuscr Math 28:269-291, 1979). Here we shall adapt the original method of H. Kneser for harmonic mappings with respect to Riemannian metrics in order to receive harmonic diffeomorphisms from B onto stable Riemannian domains Omega. We construct a global nonlinear auxiliary function associated with an embedding into a field of geodesics. In the special case of planar harmonic mappings under semi-free boundary conditions, this procedure already appears in Proposition 3 of Hildebrandt and Sauvigny (J Reine Angew Math 422:69-89, 1991). By our present method to show univalency and to obtain a diffeomorphism between the domains, we can dispense of the uniqueness for the associate Dirichlet problem. The crucial idea consists of the notion stable Riemannian domains Omega, which possess a family of non-intersecting geodesic rays emanating from each boundary point and furnish a simple covering of the whole domain. Furthermore, we establish a convex hull property for harmonic mappings within Omega. On the basis of investigations by Hildebrandt et al. (Acta Math 138:1-16, 1977), we construct harmonic embeddings within the hemisphere by direct variational methods.