A rectifiability result for finite-perimeter sets in Carnot groups
| Autor: | Sebastiano Don, Enrico Le Donne, Terhi Moisala, Davide Vittone |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Mathematics - Differential Geometry
semigroups of horizontal normal intrinsic Lipschitz graphs General Mathematics 53C17 22E25 49Q15 28A75 rectifiability outer cone property Metric Geometry (math.MG) Group Theory (math.GR) filiform groups Sets of finite perimeter Carnot groups rectifiability filiform groups outer cone property semigroups of horizontal normal intrinsic Lipschitz graphs abnormal directions Mathematics - Analysis of PDEs Differential Geometry (math.DG) Mathematics - Metric Geometry Carnot groups abnormal directions FOS: Mathematics Mathematics::Metric Geometry Sets of finite perimeter Mathematics - Group Theory Analysis of PDEs (math.AP) |
| Popis: | In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $\Gamma$ that, similarly to intrinsic Lipschitz graphs, have a cone property: there exists an open dilation-invariant subset $C$ whose translations by elements in $\Gamma$ don't intersect $\Gamma$. However, a priori the cone $C$ may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second type. Comment: 19 pages |
| Databáze: | OpenAIRE |
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