In a continuation of our previous work [20], we outline a theory which should lead to the construction of a universal pre-building and versal building with a φ-harmonic map from a Riemann surface, in the case of two-dimensional buildings for the group SL3. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for S...

(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be ...

We generalize the results of Song-Zelditch on geodesics in spaces of Kähler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kähler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C topology by harmonic maps into the spaces of Bergman metrics. In pa...

Harmonic morphism is a smooth map between Riemannian manifolds which pulls back germs of harmonic functions to germs of harmonic functions. It may be charactrized as harmonic maps which are horizontally weakly conformal [5,9]. One task of studying harmonic morphism is constructing concrete examples; Another one is classification of all harmonic morphisms between all special manifolds (in partic...

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 ...

We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability compensation in one space dimension are related harmonic with free boundary. consider new these boundary is actually motivated rather unusual for maps. construct then weak solutions prove their partial regularity time via Ginzburg-Landau approximation. Th...