Rigidity of Derivations in the Plane and in Metric Measure Spaces

TL;DR

This paper investigates the structure of metric derivations in Euclidean and metric measure spaces, establishing rigidity results that connect measures and derivations, and applies these findings to confirm a conjecture related to Lipschitz images.

Contribution

It proves that measures inducing rank-k derivation modules are absolutely continuous, extending these rigidity results to metric spaces and confirming a case of Cheeger's conjecture.

Findings

Measures with rank-k derivation modules are absolutely continuous to Lebesgue measure.

Results extend to metric spaces supporting doubling measures and Poincaré inequalities.

Confirmed the 2-dimensional case of Cheeger's conjecture on Lipschitz images.

Abstract

Following Weaver we study generalized differential operators, called (metric) derivations, and their linear algebraic properties. In particular, for k = 1, 2 we show that measures on k-dimensional Euclidean space that induce rank-k modules of derivations must be absolutely continuous to Lebesgue measure. An analogous result holds true for measures concentrated on k-rectifiable sets with respect to k-dimensional Hausdorff measure. Though formulated for Euclidean spaces, these rigidity results also apply to the metric space setting and specifically, to spaces that support a doubling measure and a p-Poincar\'e inequality. Using our results for the Euclidean plane, we prove the 2-dimensional case of a conjecture of Cheeger, which concerns the non-degeneracy of Lipschitz images of such spaces.