Search results for "Conformal mapping"

$n$-harmonic coordinates and the regularity of conformal mappings

2014

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \infty$ on any Riemannian manifold.

Improved Hölder regularity for strongly elliptic PDEs

2019

We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of H\"older regularity, higher than what is given by the classical exponent $1/K$.

Planar Quasiconformal Mappings; Deformations and Interactions

1998

The theory of quasiconformal mappings divides traditionally into two branches, the mappings in the plane and the case of higher dimensions. Basically, this is not due to the history of the topic but rather since planar quasiconformal mappings admit flexible methods (so far) not available in space. In this expository paper we wish to describe some recent trends and activities in quasiconformal theory peculiar to the plane. It is obvious, though, that not all topics can be covered no matter which point of view is taken; many important advances and connections must necessarily be bypassed. Therefore we concentrate on a specific theme, a property that singles out the difference between mappings…

Boundary regularity and the uniform convergence of quasiconformal mappings

1979

Uniform continuity of quasiconformal mappings and conformal deformations

2008

We prove that quasiconformal maps onto domains satisfying a suitable growth condition on the quasihyperbolic metric are uniformly continuous even when both domains are equipped with internal metric. The improvement over previous results is that the internal metric can be used also in the image domain. We also extend this result for conformal deformations of the euclidean metric on the unit ball of R n \mathbb {R}^n .

Mappings of finite distortion from generalized manifolds

2014

We give a definition for mappings of finite distortion from a generalized manifold with controlled geometry to a Euclidean space. We prove that the basic properties of mappings of finite distortion are valid in this context. In particular, we show that under the same assumptions as in the Euclidean case, mappings of finite distortion are open and discrete.

A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

2017

We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between subdomains of $\mathbb{H}_1$ are proven, including distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in $\mathbb{H}_1$. The theorems are discussed for…

Nonexistence of Quasiconformal Maps Between Certain Metric Measure Spaces

2013

We provide new conditions that ensure that two metric measure spaces are not quasiconformally equivalent. As an application, we deduce that there exists no quasiconformal map between the sub-Riemannian Heisenberg and roto-translation groups.

Dimension gap under conformal mappings

2012

Abstract We give an estimate for the Hausdorff gauge dimension of the boundary of a simply connected planar domain under p -integrability of the hyperbolic metric, p > 1 . This estimate does not degenerate when p tends to one; for p = 1 the boundary can even have positive area. The same phenomenon is extended to general planar domains in terms of the quasihyperbolic metric. We also give an example which shows that our estimates are essentially sharp.

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

2011

Abstract In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces B ˙ p , q s and Triebel–Lizorkin spaces F ˙ p , q s for all s ∈ ( 0 , 1 ) and p , q ∈ ( n / ( n + s ) , ∞ ] , both in R n and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve F ˙ n / s , q s on R n for all s ∈ ( 0 , 1 ) and q ∈ ( n / ( n + s ) , ∞ ] . A metric measure space version of the above morphism property is also established.