Lipschitz and biLipschitz Maps on Carnot Groups

Details Lipschitz and biLipschitz Maps on Carnot Groups Lipschitz and biLipschitz Maps on Carnot Groups

2010-03-19

arxiv.org/abs/1003.3723v5

Mathematics

Metric Geometry

Scientific paper

Suppose A is an open subset of a Carnot group G, where G has a discrete analogue, and H is another Carnot group. We show that a Lipschitz function from A to H whose image has positive Hausdorff measure in the appropriate dimension is biLipschitz on a subset of A of positive Hausdorff measure. We then construct Lipschitz maps from open sets in Carnot groups to Euclidean space that do not decrease dimension. Finally, we discuss two counterexamples to explain why Carnot group structure is necessary for these results.

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