Convex functions and harmonic maps

Certain Convex Harmonic Functions

Convex Hull Properties of Harmonic Maps

In 1975, Yau [Y] proved, by way of a gradient estimate, that a complete manifold M with non-negative Ricci curvature must satisfy the strong Liouville property for harmonic functions. The strong Liouville property (Liouville property) asserts that any positive (bounded) harmonic function defined on M must be identically constant. In 1980, Cheng [C] generalized the gradient estimate to harmonic ...

Harmonic functions in non-locally convex spaces

Quasiconformal Harmonic Functions between Convex Domains

We generalize Martio’s paper [14]. Indeed the problem studied in this paper is under which conditions on a homeomorphism f between the unit circle S1 := {z : |z| = 1} and a fix convex Jordan curve γ the harmonic extension of f is a quasiconformal mapping. In addition, we give some results for some classes of harmonic diffeomorphisms. Further, we give some results concerning harmonic quasiconfor...

Superrigidity, Generalized Harmonic Maps and Uniformly Convex Spaces

We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs re...