Regular Hypersurfaces, Intrinsic Perimeter and Implicit Function Theorem in Carnot Groups

By Francesco Serra Cassano, Bruno Franchi, Raul Paolo Serapioni. 2003

161 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b829679bc1c23392798fbd572d0c7a82f7af7e81

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Regular submanifolds, graphs and area formula in Heisenberg groups (2007, background): “folds have topological dimension n, and 1-codimensional H-regular surfaces have topological dimension 2n (the systematic specification ‘topological’ is not pointless, because, as already noticed in [10], [ 11 ], other different dimensions play a role in the geometry of Carnot groups).”
Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups (2009, background): “[27],[26].”
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.”
Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group (2004, unknown): “This notion of intrinsically regular surface was introduced in order to study in the setting of Carnot groups, of which H1 is the simplest example, the classical problem in Geometric Measure Theory (GMT) of defining regular hypersurfaces (i.e. topological submanifold of codimension 1) and different…”
Harnack inequality for fractional sub-Laplacians in Carnot groups (2012, background): “ve h(XIf)∗ g|ψi = hXIf|ψ∗wgi = (−1)|I|hf|ψ∗ (XI wg)i = (−1)|I|hf∗wXI wg|ψi. (8) If E ⊂ G is a mesurable set, a notion of G-perimeter measure |∂E|G has been introduced in [20]. We refer to [20], [17], [19], [18] for a detailed presentation. For our needs, we restrict ourselves to remind that, if Ehas”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, background): “intrinsic graph (see [24, 25 ]) is a (non-characteristic) graph of the form”
On the structure of finite perimeter sets in step 2 Carnot groups (2003, methodology): “If in the spirit of De Giorgi's theorem we want to describe the structure of sets of finite intrinsic perimeter in a Carnot group G, we need a natural notion of rectifiable subsets, and in this perspective, the crucial missing point is--as already pointed in [32] and in [ 33 ]--the correct definitio”
Differentiability of Intrinsic Lipschitz Functions within Heisenberg Groups (2011, background): “Indeed, implicit function theorems for groups [10, 11, 17, 18, 21] state precisely that these level sets are always, locally, intrinsic graphs.”
Intrinsic regular hypersurfaces in Heisenberg groups (2006, methodology): “we infer that S is a topological submanifold of dimension 2n. Besides, by using again the Implicit Function Theorem and the Blow-Up Theorem (see Theorem 2.17), an area type formula for the (Q - 1)-dimensional spherical Hausdorff measure S O-1 induced in (H n, d,,~) and the existence of the tangent g”
Characteristic points, rectifiability and perimeter measure on stratified groups (2006, methodology): “The systematic study of Geometric Measure Theory on strati ed groups has begun only very recently with a continuously growing literature and it is now the subject of several contributions [1], [5], [8], [12], [15], [17], [26], [27], [28], [29], [30], [32], [41], [45], [46], [47], [48], [57], [54], [”