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arXiv · 2512.24649

Periodic Beurling-Ahlfors Extensions and Quasisymmetric Rigidity of Carpets

Fan Wen
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Abstract

We establish periodic quasiconformal extension theorems for periodic orientation-preserving quasisymmetric self homeomorphisms of quasicircles or quasi-round carpets. As applications, we prove that, if $f$ is a periodic orientation-preserving quasisymmetric self homeomorphism of a quasi-round carpet $S$ of measure zero in $\mathbb{C}$, which has a fixed point in the outer peripheral circle of $S$, then $f$ is the identity on $S$. Moreover, we prove that, if $f$ is a quasisymmetric self homeomorphism of a square carpet $S$ of measure zero in a rectangle ring, which fixes each of the four vertices of the outer peripheral circle of $S$, then $f$ is the identity on $S$. An analogous rigidity problem for the $\mathbb{C}^*$-square carpets is discussed.

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Fan Wen. 2025-12-31. Periodic Beurling-Ahlfors Extensions and Quasisymmetric Rigidity of Carpets. https://arxiv.org/abs/2512.24649

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