Rigidity of Derivations in the Plane and in Metric Measure Spaces

Details Rigidity of Derivations in the Plane and in Metric Measure Spaces Rigidity of Derivations in the Plane and in Metric Measure Spaces

2011-10-19

arxiv.org/abs/1110.4282v1

Mathematics

Metric Geometry

33 pages

Scientific paper

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.

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