On the Sharp Lower Bound for Duality of Modulus

TL;DR

This paper proves a precise inequality relating the modulus of connecting curves and separating surfaces in certain metric spaces, answering a specific open question and introducing new approximation methods.

Contribution

It establishes a sharp reciprocity inequality for modulus in compact metric spaces, especially for planar rectangles, and develops novel approximation techniques.

Findings

Proved a sharp inequality between curve and surface moduli in metric spaces.

Answered an open question by K. Rajala and M. Romney for planar rectangles.

Developed new approximation methods for modulus analysis.

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.