Quasiregular mapping by branched circle packing

We prove that the quasiregular mappings given by the (normalized) principal solution of the linear Beltrami equation (1) and the principal solution of the quasilinear Beltrami equation are inverse to each other. This basic fact is deduced from the Liouville for generalized analytic functions. It essentially simplifies the known proofs of the measurable Riemann mapping theorem and its holomorphic dependence on parameters.

We use Andreev-Thurston’s theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.

O. Introduction 1. Notation 2. Some exterior algebra 3. Differential forms in Lnm (i2, A n) 4. Differential systems for quasiregular mappings 5. Liouville Theorem in even dimensions 6. Hodge theory in LP(R n) 7. The Beltrami equation in even dimensions 8. The Beurling-Ahlfors operator 9. Regularity theorems for quasiregular mappings 10. The Caccioppoli type estimate 11. Removability theorems for quasiregular mappings 12. Some examples Appendix: The 4-dimensional case

We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. The respective results for the Hausdorff and upper box-counting dimension follow immediately from their quasiconformal counterparts by employing elementary properties of these dimension notions (e.g. countable stability and Lipschitz stability). However, the Assouad dimension and spectrum do not share such properties. We obtain upper bounds on the Assouad dimension and spectrum of images of compact sets under holomorphic and planar quasiregular maps by studying their behavior around their critical points. As an application, the invariance of porosity of compact subsets of the plane under quasiregular maps is established.

PROOF. Assuming (a) and given a proper mapf: X--Y, let Z denote the topological sum of X and Y and define g: Z-+ Y by g X =f and gi Y=iy. The g is clearly surjective and proper, hence closed, and since X is closed in Z, f is closed. Hence (a) =(b). Trivially (b) =*(c), while (c)=>(d) is immediate from the Lemma. Since (b)=*(a) is trivial it remains to deduce (b) from (d). Let f: X-Y be a proper map and F a closed subset of X. We must show f(F) is closed in Y, and since we are assuming (d), it suffices to show that if K is a compact subset of Y then Knf(F) is compact. But Knf(F) =f(f-1(K)CF).

We show that if E is a closed convex set in {mathbb {C}}^n(n>1) contained in a closed halfspace H such that Ecap bH is nonempty and bounded, then the concave domain Omega = {mathbb {C}}^n{setminus } E contains images of proper holomorphic maps f:Xrightarrow {mathbb {C}}^n from any Stein manifold X of dimension <n, with approximation of a given map on closed compact subsets of X. If in addition 2dim X+1le n then f can be chosen an embedding, and if 2dim X=n, then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.

The properties of the extremal sets of extremal quasiconformal mappings are discussed. It is proved that if an extremal Beltrami coefficient μ(z) is not uniquely extremal, then there exists an extremal Beltrami coefficient ?(z) in its equivalent class and a compact subset E ⊂ △ with positive measure such that the essential upper bound of ?(z) on E is less than the norm of [μ].

For an isotropic elasticity problem in stresses in three-dimensional space minus the origin, we study solutions that have the singularity 1/r2 and, after the multiplication by r2, polynomially depend on the direction cosines. In this polynomial class, for the equilibrium equation we write out the general solution that is a statically admissible (in the sense of Castigliano) solution of the Kelvin problem. We show that if one or several Beltrami equations are not satisfied, then the classical Kelvin solution becomes nonunique. A method for constructing nonunique solutions of this kind is given. The equivalence of various statements of the elasticity problem in stresses is discussed. For the problem on the action of a lumped force at the vertex of an arbitrary conical elastic body, we write out the exact solution in stresses for the case of an incompressible material. The solution for a compressible material is represented in the form of series in a parameter characterizing the deviation of the Poisson ratio from 1/2. We obtain iterative chains of problems in stresses and conditions for the finiteness of these chains. We also analyze the realizability of a linear-fractional dependence of the solution in stresses on the Poisson ratio.

A property of the Bergman projection associated to a bounded circular domain containing the origin in ${{\mathbf {C}}^N}$ is proved: Functions which extend to be holomorphic in large neighborhoods of the origin are characterized as Bergman projections of smooth functions with small support near the origin. For certain circular domains $D$, it is also shown that functions which extend holomorphically to a neighborhood of $\overline D$ are precisely the Bergman projections of smooth functions whose supports are compact subsets of $D$. Two applications to proper holomorphic mappings are given.

Rotationally symmetric biharmonic maps between models

We examine the transmission problem in a two-dimensional domain, which consists of two different homogeneous media. We use boundary integral equation methods on the Maxwell equations governing the two media and we study the behaviour of the solution as the two different wave numbers tend to zero. We prove that as the boundary data of the general transmission problem converge uniformly to the boundary data of the corresponding electrostatic transmission problem, the general solution converges uniformly to the electrostatic one, provided we consider compact subsets of the domains.

In this paper we consider a system of first order nonlinear Beltrami equations with singular point in the angular unbounded region of the complex plane. This system of equations is used in the theory of surfaces of positive infinitesimal curvature with a density point and for the construction of isometric nodal coordinates on surfaces of positive curvature with a density point. In this paper, for this system of equations we obtain sufficient condition for the solution of the Dirichlet type problem in the space of continuous functions. For this purpose, we will use the general solution of the system of corresponding linear elliptic differential equations with singular point. The proof of existence of continuous solutions of the Dirichlet problem is based on the Schauder fixed point principle.