Research Problems in Function Theory — Problem 6.62

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Jordan domain has bounded boundary rotation in the sense of Paatero if its conformal parametrization $u:\mathbb D\to D$ satisfies $\sup_{0<r<1}\int_0^{2\pi}|\operatorname{Re}(1+re^{it}u''(re^{it})/u'(re^{it}))|\,dt<\infty$. This expresses bounded total turning, including limits at nonsmooth boundary points. Let $D_1,D_2$ be bounded Jordan domains of this type, whose rectifiable boundaries have the same length. Identify the boundaries by an orientation-reversing arclength-preserving homeomorphism, and uniformize the resulting conformally sewn sphere. Let $C$ be the seam in a planar coordinate with its pole outside $C$.

Statement recorded in the supplied source.

Give a function-theoretic proof that $C$ has bounded boundary rotation. The conclusion itself is recorded as following from the geometric results of Aleksandrov and Reshetnyak; the requested proof uses the analytic functions and conformal sewing rather than that geometric route. [S2]

Short English statement

Find an analytic proof that conformal arclength sewing preserves bounded boundary rotation.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2306062 (AMR-022-6062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 6, Problem 6.62 and Update 6.62; Paatero boundary-rotation convention and Noonan reference in Problem 6.62. https://arxiv.org/abs/1809.07200.