Curve packing and modulus estimates

Fässler, Katrin; Orponen, Tuomas

doi:10.1090/tran/7175

Curve packing and modulus estimates
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by Katrin Fässler and Tuomas Orponen PDF

Trans. Amer. Math. Soc. 370 (2018), 4909-4926 Request permission

Abstract:

A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $\mathbb {R}^{2}$ of length one. The classical “worm problem” of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family always has area at least $c$ for some small absolute constant $c > 0$. We strengthen Marstrand’s result by showing that for $p > 3$, the $p$-modulus of a Moser family of curves is at least $c_{p} > 0$.

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