Loewner chain

A chordal Loewner chain is an increasing family of compact H-hulls, usually parameterized by half-plane capacity, whose mapping-out functions evolve according to the Chordal Loewner equation.

The local growth property says that, after mapping out the hull at time $t$, the new hull grown during a short interval has diameter tending uniformly to zero with the interval length. It ensures that the growth is described by one continuous boundary point.

The Loewner differential equation describes a growing family of simply connected planar domains through ordinary differential equations for their normalized conformal maps.

For a capacity-parameterized locally growing hull family,where the continuous real function $U$ is the Loewner driving function.

The Loewner driving function $U_t$ is the real boundary point at which the mapped-out hull grows. Scaling the hulls by $a^{-1}$ and time by $a^2$ transforms it to $a^{-1}{U}_{a^{2}t}$.