Affiliation:
1. Department of Mathematics, University of Toronto , Toronto , ON M5S 2E4 , Canada
2. Gran Sasso Science Institute, Viale F. Crispi 7 , 67100 L’Aquila , Italy
Abstract
Abstract
A natural higher-order notion of
C
1
,
α
{C}^{1,\alpha }
-rectifiability,
0
<
α
≤
1
0\lt \alpha \le 1
, is introduced for subsets of the Heisenberg groups
H
n
{{\mathbb{H}}}^{n}
in terms of covering a set almost everywhere with a countable union of
(
C
H
1
,
α
,
H
)
\left({{\bf{C}}}_{H}^{1,\alpha },{\mathbb{H}})
-regular surfaces. Using this, we prove a geometric characterization of
C
1
,
α
{C}^{1,\alpha }
-rectifiable sets of low codimension in Heisenberg groups
H
n
{{\mathbb{H}}}^{n}
in terms of an almost everywhere existence of suitable approximate tangent paraboloids.
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