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arXiv · 1110.4282

Rigidity of Derivations in the Plane and in Metric Measure Spaces

Jasun Gong
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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.

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Jasun Gong. 2011-10-19. Rigidity of Derivations in the Plane and in Metric Measure Spaces. https://arxiv.org/abs/1110.4282

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