Countable rectifiability in the higher-dimensional length-distortion theorem

Determine whether the higher-dimensional length-distortion conclusion holds for a volume-preserving Lipschitz map when the domain is countably n-rectifiable, rather than lower Ahlfors n-regular.

Background

The paper proves that if X and Y are metric spaces with locally finite Hausdorff n-measure and X satisfies a lower Ahlfors n-regularity bound, then every volume-preserving L-Lipschitz map f from X to Y distorts the length of almost every curve by a uniform multiplicative factor. The lower Ahlfors regularity assumption is used in the higher-dimensional adaptation of the intrinsic proof to supply lower bounds for the measure of small balls.

The authors note that, in the special case L=1, lower Ahlfors regularity can already be replaced by countable n-rectifiability of X. The unresolved question is whether the full theorem, including the case of arbitrary Lipschitz constant L, remains valid under this weaker countable n-rectifiability assumption.

References

Does Theorem \ref{thm:main-higher-dim} hold if, instead of eq:lower-Ahlfors, we assume that $X$ is countably $n$-rectifiable?

eq:lower-Ahlfors:

n(B(x,r))≥c rnfor every x∈X and 0<r<diam(X).^n(B(x,r))\ge c\, r^n \quad \text{for every } x \in X \text{ and } 0<r<diam(X).

— Length distortion of volume-preserving Lipschitz mappings  (2609.09842 - Meier et al., 9 Sep 2026) in Question following Theorem 1.2, Section “Intrinsic proof of Theorem 1.1 and higher dimensions”