Curve packing and modulus estimates

Fässler, Katrin; Orponen, Tuomas

doi:10.1090/tran/7175

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Journal article

Curve packing and modulus estimates

Fässler, Katrin Department of Mathematics, University of Fribourg, Switzerland - Department of Mathematics and Statistics, University of Jyväskylä, Finland
Orponen, Tuomas Department of Mathematics and Statistics, University of Jyväskylä, Finland

2018

Published in:

Transactions of the American Mathematical Society. - 2018, vol. 370, no. 7, p. 4909–4926

English 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$

Faculty

Faculté des sciences et de médecine

Department

Département de Mathématiques

Language