Quasiconformal deformations of chordal Loewner chains

Abstract

We establish first-order variational formulas for the driving function and the half-plane capacity of a chordal Loewner chain under quasiconformal deformations. Under the standard normalization fixing $0,1$ and $\infty$, the formulas hold for every bounded Beltrami differential. If the curve has positive area, the formulas contain additional integrals over the curve; these terms vanish when the curve has zero area or when the differential is extended by zero on the curve. The proof uses the normalized Ahlfors--Bers variation together with a Stoilow factorization adapted to the slit domain. We then consider the asymptotic normalization for differentials whose symmetric extensions to $\mathbb{C}$ have the form $ν(z)=a(\bar z/z)^{n-1}+ν_0(z)$, where $a\in(-1,1)$, $n\ge1$ is an integer, and $ν_0\in L^r(\mathbb{C})$ for $0<r<2$. The asymptotically normalized map differs from the standardly normalized map by a positive dilation. Using this relation, the variational formulas in the second normalization follow directly from the standardly normalized case.