A notion of rectifiability modeled on Carnot groups

By Scott D. Pauls. 2000

63 citations recorded by the source corpus.

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, for E a subset of a Carnot group M and N a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and the existence of approximate tangent cones isometric to N almost everywhere in E. Second, we prove that, under a stronger condition concerning the existence of approximate tangent cones isomorphic to N almost everywhere in a set E, that E is N-rectifiable. Third, we investigate the rectifiability properties of level sets of C 1 N functions, f : N → R, where N is a Carnot group. We show that for almost every t ∈ R and almost every noncharacteristic x ∈ f -1 (t), there exist a subgroup T x of H and r > 0 so that f -1 (t) n B H (x, r) is T x -approximable at x and an approximate tangent cone isomorphic to T x at x.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2859de3e8b6ae0d8051266c63f0dd228d1d890cc

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs (2009, background): “In fact, in [17] classical De Giorgi’s rectiability and divergence theorem for sets of nite perimeter have been fully extended to a Carnot group of step 2 (see also [ 23 ]).”
The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations (2006, methodology): “Moreover, introducing a suitable intrinsic notions of rectifiability in a Carnot group of step 2, a counterpart of De Giorgi’s result on the structure of sets of finite Euclidean perimeter (see, for instance, [30]) has been obtained in [24] (see also [3], [ 41 ],[34],[35],).”
Characterizations of intrinsic rectifiability in Heisenberg groups (2010, methodology): “As an example, Pauls in [28] considered images in Hn of Lipschitz maps defined not on Rk but on subgroups of Hn; in [18] a different notion of intrinsic Lipschitz sub-manifold is considered.”
Geometry of Carnot-Caratheodory Spaces, Differentiability, Coarea and Area Formulas (2009, methodology): “Pauls [122] for Lipschitz mappings of”
Differentiability of Intrinsic Lipschitz Functions within Heisenberg Groups (2011, unknown): “, [1, 2, 17, 21, 29, 34] and the references therein).”
Regular submanifolds, graphs and area formula in Heisenberg groups (2007, background): “• Is there a unifying approach for low dimensional and low codimensional H-regular surfaces? May be looking at images of subgroups of Hn or, dually, at level sets of group valued functions defined on Hn? (see the approach of Scott Pauls in [ 24 ])”
Unrectifiability and rigidity in stratified groups (2004, methodology): “The next theorem states the area formula for Lipschitz mappings between stratified groups, [9], [12], [17].”
Intrinsic regular hypersurfaces in Heisenberg groups (2006, background): “Theorems 1.2 and 1.3 give some partial answers to the problem of the good parameterization of H-regular surfaces proposed in [27]: To find out a model metric space such that each H-regular surface can be locally viewed as its image through a bi-lipschitz continuous map (see also [55, 56] for similar”