Calculus of Variations and Geometric Measure Theory

D. Vittone

Submanifolds in Carnot groups

created by vittone on 07 May 2008
modified on 25 Jul 2017


Ph.D. Thesis

Inserted: 7 may 2008
Last Updated: 25 jul 2017

Journal: Theses of Scuola Normale Superiore di Pisa (New Series). Edizioni della Normale, Pisa.
Volume: 7
Pages: xx+180 pp.
Year: 2008


The thesis is devoted to the study of submanifolds in the setting of Carnot groups equipped with a sub-Riemannian structure; particular emphasis is given to the case of Heisenberg groups. A Geometric Measure Theory viewpoint is adopted, and features as intrinsic perimeters, Hausdorff measures, area formulae, calibrations and minimal surfaces are considered. Area formulae for the measure of submanifolds of arbitrary codimension are obtained in Carnot groups. Intrinsically regular hypersurfaces in the Heisenberg group are extensively studied: suitable notions of graphs are introduced, together with area formulae leading to the analysis of Bernstein type problems. The thesis has been written under the supervision of Prof. L. Ambrosio and defended in December 2006 at SNS, Pisa.

Keywords: Geometric measure theory, Heisenberg group, sub-Riemannian geometry