Quasikonforme

Quasikonforme is a term used in mathematics, specifically within the field of geometric function theory. It refers to mappings, or functions, that are "almost" conformal. Conformal mappings are those that preserve angles locally, meaning that the angle between any two intersecting curves at a point is the same as the angle between their images under the mapping. This angle-preserving property is very strong and leads to many nice geometric properties.

A quasikonforme mapping, on the other hand, only approximately preserves angles. More formally, a quasikonforme mapping is a homeomorphism (a continuous bijective function with a continuous inverse) whose partial derivatives satisfy a certain boundedness condition. This condition, often expressed in terms of the Beltrami equation, quantifies how much the mapping distorts angles and shapes. While not perfectly angle-preserving, quasikonforme mappings are still very well-behaved geometrically.

The study of quasikonforme mappings is crucial for understanding how shapes can be deformed in a controlled manner. They have significant applications in various areas of mathematics and physics, including complex analysis, differential geometry, and the theory of partial differential equations. For instance, they play a role in understanding the geometry of surfaces and manifolds, and in solving certain types of physical problems involving fluid dynamics or elasticity. The theory of quasikonforme mappings provides a framework to analyze transformations that are not strictly conformal but still possess important geometric properties.