Corni, Francesca
(2021)
Low codimensional intrinsic regular submanifolds in the Heisenberg group H^n, [Dissertation thesis], Alma Mater Studiorum Università di Bologna.
Dottorato di ricerca in
Matematica, 33 Ciclo. DOI 10.48676/unibo/amsdottorato/9800.
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Abstract
The thesis mainly concerns the study of intrinsically regular submanifolds of low codimension in the Heisenberg group H^n, called H-regular surfaces of low codimension, from the point of view of geometric measure theory. We consider an H-regular surface of H^n of codimension k, with k between 1 and n, parametrized by a uniformly intrinsically differentiable map acting between two homogeneous complementary subgroups of H^n, with
target subgroup horizontal of dimension k. In particular the considered submanifold is the intrinsic graph of the parametrization. We extend various results of Ambrosio, Serra Cassano and Vittone, available for the case when k = 1. We prove that the uniform intrinsic differentiability of the parametrizing map is equivalent to the existence and continuity of its intrinsic differential, to the local existence of a suitable approximating family of Euclidean regular maps, and, when the domain and the codomain of the map are orthogonal, to the existence and continuity of suitably defined intrinsic partial derivatives of the function. Successively, we present a series of area formulas, proved in collaboration with V. Magnani. They allow to compute the (2n+2−k)-dimensional spherical Hausdorff
measure and the (2n+2−k)-dimensional centered Hausdorff measure of the parametrized H-regular surface, with respect to any homogeneous distance fixed on H^n.
Furthermore, we focus on (G,M)-regular sets of G, where G and M are two arbitrary Carnot groups. Suitable implicit function theorems ensure the local existence of an intrinsic parametrization of such a set, at any of its points. We prove that it is uniformly
intrinsically differentiable.
Finally, we prove a coarea-type inequality for a continuously Pansu differentiable function acting between two Carnot groups endowed with homogeneous distances. We assume
that the level sets of the function are uniformly lower Ahlfors regular and that the Pansu
differential is everywhere surjective.
Abstract
The thesis mainly concerns the study of intrinsically regular submanifolds of low codimension in the Heisenberg group H^n, called H-regular surfaces of low codimension, from the point of view of geometric measure theory. We consider an H-regular surface of H^n of codimension k, with k between 1 and n, parametrized by a uniformly intrinsically differentiable map acting between two homogeneous complementary subgroups of H^n, with
target subgroup horizontal of dimension k. In particular the considered submanifold is the intrinsic graph of the parametrization. We extend various results of Ambrosio, Serra Cassano and Vittone, available for the case when k = 1. We prove that the uniform intrinsic differentiability of the parametrizing map is equivalent to the existence and continuity of its intrinsic differential, to the local existence of a suitable approximating family of Euclidean regular maps, and, when the domain and the codomain of the map are orthogonal, to the existence and continuity of suitably defined intrinsic partial derivatives of the function. Successively, we present a series of area formulas, proved in collaboration with V. Magnani. They allow to compute the (2n+2−k)-dimensional spherical Hausdorff
measure and the (2n+2−k)-dimensional centered Hausdorff measure of the parametrized H-regular surface, with respect to any homogeneous distance fixed on H^n.
Furthermore, we focus on (G,M)-regular sets of G, where G and M are two arbitrary Carnot groups. Suitable implicit function theorems ensure the local existence of an intrinsic parametrization of such a set, at any of its points. We prove that it is uniformly
intrinsically differentiable.
Finally, we prove a coarea-type inequality for a continuously Pansu differentiable function acting between two Carnot groups endowed with homogeneous distances. We assume
that the level sets of the function are uniformly lower Ahlfors regular and that the Pansu
differential is everywhere surjective.
Tipologia del documento
Tesi di dottorato
Autore
Corni, Francesca
Supervisore
Co-supervisore
Dottorato di ricerca
Ciclo
33
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
Carnot groups; Heisenberg group; intrinsic graphs; Pansu differentiability; intrinsic Lipschitz continuity; intrinsic differentiability; spherical Hausdorff measure, area formula; coarea formula
URN:NBN
DOI
10.48676/unibo/amsdottorato/9800
Data di discussione
27 Maggio 2021
URI
Altri metadati
Tipologia del documento
Tesi di dottorato
Autore
Corni, Francesca
Supervisore
Co-supervisore
Dottorato di ricerca
Ciclo
33
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
Carnot groups; Heisenberg group; intrinsic graphs; Pansu differentiability; intrinsic Lipschitz continuity; intrinsic differentiability; spherical Hausdorff measure, area formula; coarea formula
URN:NBN
DOI
10.48676/unibo/amsdottorato/9800
Data di discussione
27 Maggio 2021
URI
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