Walk dimension and vanishing curve modulus in metric measure spaces

Aobo Chen

Walk dimension and vanishing curve modulus in metric measure spaces

Metric Geometry 2026-06-26 v1

Authors: Aobo Chen

Abstract

We prove that, on a regular local $p$ -Dirichlet space supporting a $p$ -Poincar\'e inequality, if the $p$ -walk dimension is strictly greater than $p$ , then every curve family has zero $p$ -modulus. As a consequence, we show that no Ahlfors-regular metric space equipped with a sub-Gaussian heat kernel is minimal for its Ahlfors-regular conformal dimension.

Keywords

Walk dimension Curve modulus Metric measure space p-Dirichlet space Poincare inequality Ahlfors-regular Sub-Gaussian heat kernel

Cite

@article{arxiv.2606.27869,
  title  = {Walk dimension and vanishing curve modulus in metric measure spaces},
  author = {Aobo Chen},
  journal= {arXiv preprint arXiv:2606.27869},
  year   = {2026}
}

Comments

29 pages; comments are welcome