These notes will contain material you should know from the final lectures of the course, which will be drawn from sources other than the text. The Riemann mapping theorem is a key theoretical result in complex analysis. The basic question is to determine the existence of conformal maps between two general domains D,D′/-. The simplest formulation is for two simplyconnected domains. We can also s...

A. Both bi-harmonic map and f -harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f -bi-harmonic maps as the critical points of the f -bi-energy functional 1 2 ∫ M f |τ(φ)| dvg. This class of maps generalizes both concepts of harmonic maps and biharmonic maps. We first derive the f -biharmonic map ...

The objective of this research paper is to describe the behaviour boundary derivative conformal maps from polygon domain onto unit disk through construct some an interesting cases, and its inverse maps. Moreover, we study existence finiteness integrability over infinite sector W.

Let Σ be a compact Riemann surface. Any sequence fn : Σ —> M of harmonic maps with bounded energy has a "bubble tree limit" consisting of a harmonic map /o : Σ -> M and a tree of bubbles fk : S 2 -> M. We give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the fn converge pointwise. We then give explicit counterexam...

Harmonic morphisms are smooth maps between Riemannian manifolds which preserve Laplace's equation. They are characterised as harmonic maps which are horizontally weakly conformal 14, 20]. R.L. Bryant 7] proved that there are precisely two types of harmonic morphisms with one-dimensional bres which can be deened on a constant curvature space of dimension at least four. Here we prove that, on an ...

A “fat slit” is a compact domain in the upper half plane bounded by a curve with endpoints on the real axis and a segment of the real axis between them. We consider conformal maps of the upper half plane to the exterior of a fat slit parameterized by harmonic moments of the latter and show that they obey an infinite set of Lax equations for the dispersionless KP hierarchy. Deformation of a fat ...

Surface harmonic map between genus-0 surfaces plays an important role in applied mathematics and engineering, with applications in medical imaging and computer graphics. Previous work [1] introduces a variational approach for computing surface harmonic maps. It obtains global conformal parameterizations of genus-0 surfaces through minimizing the harmonic energy. Two weaknesses of this approach ...

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply co...

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply co...