We discuss recent progress in the study of the space of harmonic maps from the 2sphere to the unit n-sphere in Euclidean (n+1)-space. We consider the structure of this space as an algebraic variety, the existence of non-manifold points in this space, and the relation between this question and the integrability of Jacobi fields along harmonic maps. One of the main tools used is that of the twist...

We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual ∂-problems or loop group factorizations. We interpret ...

We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections and avoiding the usual ∂-problems or loop group factorizations. We interpret ...

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 ...

In this paper, we extend a Hardy-Littlewood type theorem to the exponentially p-harmonic Bergman space on real unit ball B in Rn. As an application, characterize spaces terms of Lipschitz conditions. Furthermore, some derivative-free characterizations for n-harmonic Qk are established.

We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H. Conversely, any nowhere antiholomorphic harmonic map into H is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of compl...

A complete discrete set of spherical single-particle wave functions for studies of weakly-bound many-body systems is proposed. The new basis is obtained by means of a local-scale point transformation of the spherical harmonic os-cillator wave functions. Unlike the harmonic oscillator states, the new wave functions decay exponentially at large distances. Using the new basis, characteristics of w...

Just as the harmonic map equation is a geometric analogue of the classical Laplace equation for harmonic functions, so the classical linear evolution PDEs, the heat, wave, and Schrödinger equations, have geometric “map” analogues: the harmonic map heat-flow, wave map, and Schrödinger map equations. These equations are nonlinear when the target space geometry is nontrivial. Quite remarkably, the...