Kquasiconformal

K-quasiconformal (often written K-quasiconformal) refers to a class of maps that generalize conformal maps by allowing controlled distortion. In the plane (and more generally in n-dimensional spaces), a sense-preserving homeomorphism f between domains is K-quasiconformal if it has locally square-integrable first derivatives and its differential distorts infinitesimal shapes by at most a factor K ≥ 1.

A common analytic formulation uses the Beltrami equation. There exists a measurable function μ with ||μ||∞ ≤ k < 1 (the complex dilatation) such that f satisfies ∂̄f = μ ∂f almost everywhere. The distortion constant K is then given by K = (1 + k)/(1 − k). Equivalently, at almost every point z, the Jacobian J_f(z) and the differential satisfy an inequality that makes the image of a small circle an ellipse with eccentricity bounded by K.

Key properties include: K = 1 corresponds to conformal maps; the inverse of a K-quasiconformal map is K-quasiconformal; the composition of a K1-quasiconformal map with a K2-quasiconformal map is K1K2-quasiconformal. They also preserve orientation when required. Infinitesimal circles map to ellipses whose eccentricity is uniformly bounded by K, and moduli of curve families are distorted by at most a factor K.

Regularity aspects include Hölder continuity with exponent 1/K in the plane, with sharpness in general. K-quasiconformal mappings are central in geometric function theory, Teichmüller theory, and deformation problems of complex structures, providing a robust framework for controlled deformation beyond conformal mappings.