Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rigidity of Derivations in the Plane and in Metric Measure Spaces   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.

25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn