Ahlfors-Weill extensions for harmonic mappings

Harmonic Mappings of Spheres

Introduction and statement of results. This announcement describes an elementary method of constructing harmonic maps in some cases not covered by the general existence theory. Recall that given smooth Riemannian manifolds N and M, with N compact, then the energy functional E:H(N,M) -> R is defined on a suitable manifold of maps H(iV, M ) and is given by E{ f ) = \ \n \df | . A map ƒ is said to...

Harmonic Mappings between Riemannian

Harmonic mappings between two Riemannian manifolds is an object of extensive study, due to their wide applications in mathematics, science and engineering. Proving the existence of such mappings is challenging because of the non-linear nature of the corresponding partial differential equations. This thesis is an exposition of a theorem by Eells and Sampson, which states that any given map from ...

The Schwarzian Derivative for Harmonic Mappings

The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to conformal mappings of polygons bounded by circular arcs. More recently, Nehari [5, 6, 7] and others have developed important criteria for global univalence in terms of the Schwarzian derivative, exploiting...

Coefficient inequality for perturbed harmonic mappings

Landau's theorem for planar harmonic mappings

Landau gave a lower estimate for the radius of a schlicht disk centered at the origin and contained in the image of the unit disk under a bounded holomorphic function f normalized by f(0) = f ′(0)− 1 = 1. Chen, Gauthier, and Hengartner established analogous versions for bounded harmonic functions. We improve upon their estimates.