Search results for "Conformal map"

Extremal length and Hölder continuity of conformal mappings

1986

Reciprocal lower bound on modulus of curve families in metric surfaces

2019

We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\zeta_1, \zeta_2, \zeta_3, \zeta_4$ and let $\Gamma(\zeta_i, \zeta_j; Q)$ denote the family of curves in $Q$ connecting $\zeta_i$ and $\zeta_j$; then $\text{mod} \Gamma(\zeta_1, \zeta_3; Q) \text{mod} \Gamma(\zeta_2, \zeta_4; Q) \geq 1/\kappa$ for $\kappa = 2000^2\cdot (4/\pi)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconfor…

Hardy spaces and quasiconformal maps in the Heisenberg group

2023

We define Hardy spaces $H^p$, $00$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential maximal function of $f$. As an application, we characterize Carleson measures on $B$ via integral inequalities for quasiconformal mappings on $B$ and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from $\mathbb{R}^n$ to $\mathbb{H}^1$. A crucial difference between the proofs in $\mathbb{R}^n$ and $\mathbb{…

$n$-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 ($r &gt; 1$) 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 $1 &lt; p &lt; \infty$ on any Riemannian manifold.

Conformality and $Q$-harmonicity in sub-Riemannian manifolds

2016

We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.

Matter dependence of the four-loop QCD cusp anomalous dimension: from small angles to all angles

2019

We compute the fermionic contributions to the cusp anomalous dimension in QCD at four loops as an expansion for small cusp angle. As a byproduct we also obtain the respective terms of the four-loop HQET wave function anomalous dimension. Our new results at small angles provide stringent tests of a recent conjecture for the exact angle dependence of the matter terms in the four-loop cusp anomalous dimension. We find that the conjecture does not hold for two of the seven fermionic color structures, but passes all tests for the remaining terms. This provides strong support for the validity of the corresponding conjectured expressions with full angle dependence. Taking the limit of large Minkow…

Constraints on Conformal Windows from Holographic Duals

2009

We analyze a beta function with the analytic form of Novikov-Shifman-Vainshtein-Zakharov result in the five dimensional gravity-dilaton environment. We show how dilaton inherits poles and fixed points of such beta function through the zeros and points of extremum in its potential. Super Yang-Mills and supersymmetric QCD are studied in detail and Seiberg's electric-magnetic duality in the dilaton potential is explicitly demonstrated. Non-supersymmetric proposals of similar functional form are tested and new insights into the conformal window as well as determinations of scheme-independent value of the anomalous dimension at the fixed point are presented.

Computing black hole entropy in loop quantum gravity from a conformal field theory perspective

2009

Motivated by the analogy proposed by Witten between Chern-Simons and conformal field theories, we explore an alternative way of computing the entropy of a black hole starting from the isolated horizon framework in loop quantum gravity. The consistency of the result opens a window for the interplay between conformal field theory and the description of black holes in loop quantum gravity.

Properties of some conformal field theories with M-theory duals

2007

24 pages.-- ISI Article Identifier: 000245078200049.-- ArXiv pre-print available at: http://arxiv.org/abs/hep-th/0611219

Weyl Invariance and Black Hole Evaporation

1995

We consider the semiclassical dynamics of CGHS black holes with a Weyl-invariant effective action for conformal matter. The trace anomaly of Polyakov effective action is converted into the Virasoro anomaly thus leading to the same flux of Hawking radiation. The covariance of semiclassical equations can be restored through a non-local redefinition of the metric-dilaton fields. The resulting theory turns out to be equivalent to the RST model. This provides a mechanism to solve semiclassical equations of 2D dilaton gravity coupled to conformal matter for classically soluble models.