Rodin's formula in arbitrary codimension

TL;DR

This paper generalizes Rodin's formula for p-modulus from curves to higher codimension families in Euclidean space, using level set and Jacobian matrix techniques.

Contribution

It extends Rodin's formula to arbitrary codimension, providing a new theoretical framework for p-modulus calculations in higher dimensions.

Findings

Derived a generalized formula for p-modulus in arbitrary codimension.

Utilized level set and Jacobian matrix methods for the proof.

Presented examples illustrating the extended formula.

Abstract

We extend the Rodin's formula for $p$--modulus of the family of curves in $\mathbb{R}^n$ to arbitrary codimension. The proof relies on the formula for the $p$--modulus of family of level sets of a submersion and an algebraic lemma relating Jacobi matrices of considered maps. We state appropriate examples.