How to weld: Energies, weldings, and driving functions

Abstract

We prove a variant of the welding zipper algorithm converges for curves $\gamma \subset \nH \cup \{0\}$ that have Loewner driving functions $\xi \in C^{3/2+\epsilon}$. Convergence holds whether one ``zips up'' with straight line segments, circular arc segments orthogonal to $\mathbb{R}$, or either of two energy-minimizing curve families, or any combination of these. One of the energy-minimizing families is new, and we also prove some new properties of the known minimizing family. We furthermore show the Loewner energy of a curve can be computed by means of the conformal welding through the \emph{zipper welding energy}. Lastly, we generalize a result of Bishop from $T_2$ Weil-Petersson quasicircles to the $p$-integrable Teichmuller space $T_p$, showing $\gamma \in T_p$ if and only if $\gamma$ has $p$-summable $\beta$-numbers, for $p > 2$.