Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric

By Zoltán M. Balogh, Matthieu Rickly, Francesco Serra Cassano. 2003

49 citations recorded by the source corpus.

We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreover a comparison between Euclidean and H-rectifiability is given.

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/e16df16e6340610e2b65811a80e3bea7333f9128

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In Applied Mathematics

In Astronomy and Astrophysics

Citations in the corpus

A Notable Family of Entire Intrinsic Minimal Graphs in the Heisenberg Group which are not Perimeter Minimizing (2006, unknown): “The development of geometric measure theory in sub-Riemannian spaces has received a strong impulse over the past decade, see [Pa1], [Pa2], [CDG], [KR], [E1], [E2], [E3], [Gro], [GN], [Be], [DS], [DGN1], [AK1], [AK2], [CS1], [A], [FSS1], [Ma1], [FSS2], [Ma2], [CMS], [FSS3], [ BRS ], [DGN4], [DGN5], [”
Absolute Continuity of Quasiconformal Mappings on Curves (2007, background): “Assume that there exists H ≥ 1 such that for each Bi = B(xi, ri) ∈ B we have that B ( f(xi), 12H diam(f(Bi)) ) ⊂ f(Bi) and B satisfies condition (ii) of Lemma 2.2.”
On the structure of finite perimeter sets in step 2 Carnot groups (2003, result): “The same results in the special case of the Heisenberg group have been proved recently in [ 8 ].”
Regularity of convex functions on Heisenberg groups (2003, result): “However, the Hausdorff measures induced by dE and dH are quite different; we refer to the recent results in [4] on this subject.”
Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups (2009, background): “It continues our program in this area [ 5 , 4, 7], which is one component in a worldwide endeavor investigating sub-Riemannian geometric measure theory, including theories of rectiability and perimeter [14], [26], [27], [17], [41], [3], [38]; fractals and tilings [55], [56]; geometric analysis of no”
Intrinsic regular hypersurfaces in Heisenberg groups (2006, result): “The notions of Euclidean and H-rectifiability have been compared in [ 7 ]; generalizations of this notion of rectifiability have been studied by V. Magnani in [42] for general Carnot groups (see also [41] for a general account of GMT in Carnot groups).”
Projection and slicing theorems in Heisenberg groups (2012, unknown): “…in Mathematics 231 (2012) 569–604 Acknowledgments.............................................................................................................603 References .......................................................................................................................603”
The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations (2006, unknown): “On the other hand since S2n+1 ∞ H2n (see [6]) and ||∂E||H S2n+1 ∞ (see [24]) we get ||∂E||H( \ ̃) = 0, (2.”
Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group (2004, unknown): “1 in [5] which states that for every α ≥ 0 Hmin{α,1+ α 2 } << H d on R(3) it follows 5 2 is the smallest possible jump between the Euclidean and Carnot-Carathéodory Hausdorff dimension of such subsets of R3.”
Characterizations of intrinsic rectifiability in Heisenberg groups (2010, background): “On the contrary, for 1 ≤ d ≤ n, a (d,H)-rectifiable set is d-dimensional Euclidean rectifiable while the opposite is false (see [7], [15] and [17]).”
On the Hausdorff volume in sub-Riemannian geometry (2010, background): “A deep study of Hausdorff measure for hypersurfaces in sub-Riemannian geometry, in particular in the context of Carnot groups, can be found in [8, 10, 11, 14, 19, 22, 32, 36] and references therein.”