The Loewner driving function of trajectory arcs of quadratic differentials

Normalized in this way ft is unique and is said to be hydrodynamically normalized. The function C(t) is positive and strictly increasing: it is called the half-plane capacity of γ((0, t]), denoted by cap H (γ((0, t]). Thus we can reparameterize γ such that C(t) = 2t for all t, we will call this parameterization by half-plane capacity. With this normalization and parameterization, the function ft satisfies the differential equation ∂ft ∂t = − 2f ′ t(z) z − ξ(t) (1)