On the Sharp Lower Bound for Duality of Modulus

2021/02/05 by Sylvester Eriksson-Bique, Eriksson-Bique, Sylvester, Pietro Poggi-Corradini +1 · 1 citation
Mathematics · #49N15 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 30L15 #Secondary: 28A75 #math.MG #msc:28A75 #msc:30L15 #msc:49N15

paper · pdf · doi:10.48550/arxiv.2102.03035

12 pages. Comments are welcome

arxiv created 2021/02/05 · arxiv updated 2021/02/08

Abstract

We establish a sharp reciprocity inequality for modulus in compact metric spaces X with finite Hausdorff measure. In particular, when X is also homeomorphic to a planar rectangle, our result answers a question of K. Rajala and M. Romney. More specifically, we obtain a sharp inequality between the modulus of the family of curves connecting two disjoint continua E and F in X and the modulus of the family of surfaces of finite Hausdorff measure that separate E and F. The paper also develops approximation techniques, which may be of independent interest.

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• Uniformization of metric surfaces: A survey

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