An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

By Luca Capogna, D. Danielli, Scott D. Pauls, Jeremy T. Tyson. 2007

577 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6f8ad8372ec954e36af6761a9466c44e0cd03814

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Sobolev Mappings between Manifolds and Metric Spaces (2009, methodology): “An important class of spaces that support the p-Poincaré inequality is provided by the so-called Carnot groups [23, 68, 9] and more general Carnot– Carathéodory spaces [22, 23].”
Sharp constants in several inequalities on the Heisenberg group (2010, unknown): “1), but we note the subtlety that the level surfaces of H are neither isoperimetric surfaces [CaDaPaTy] nor level surfaces of the homogenous norm |·| appearing in (1.”
Convex isoperimetric sets in the Heisenberg group (2006, methodology): “We refer to the monograph [3] for a more detailed introduction to the isoperimetric problem in the Heisenberg group.”
Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity (2008, background): “The recent monograph by Capogna, Danielli, Pauls and Tyson [7] gives a recent overview of the subject with an exhaustive list of references.”
Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups (2009, unknown): “Open problems and questions 41 References 44”
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, methodology): “Since we are interested in non-characteristic graphs, it is convenient that we use canonical coordinates of the second kind (the so called polarized coordinates [6]) and denote (s, x) the elements of the group, where x = (x 1 , ..., x 2n ).”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, unknown): “The contributions to this field now constitute the corpus of a vast literature, which includes the papers [1,3,5, 9,10,14,15,19,21,23,26,29,30,32,37,39,44,47,48,51,52,54–56,61–63,71,77,86,88,91] and the monographs [16,18,20,45,49,70,80].”
Area-stationary surfaces in the Heisenberg group H1 (2005, background): “For background on H(1) we refer the reader to Section 2 and [8].”
Quantization on Nilpotent Lie Groups (2016, methodology): “We mention another recent book by Calin, Chang and Greiner [CCG07] containing many explicit calculations related to the Heisenberg group and its sub-Riemannian geometry, as well as a sub-Riemannian treatment in Capogna, Danielli, Pauls and Tyson [CDPT07].”
Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs (2009, background): “The following characterization of LipH(H) holds (see, for instance, [8], section 6.”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, background): “For an extensive bibliography we refer the reader to the recent monographs [10,27] (see also the forthcoming book [47]), and to the papers [26,28].”
A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries (2018, background): “Such groups appear in harmonic analysis, in the study of hypoelliptic di erential operators, as boundaries of strictly pseudo-convex complex domains, see the books [26, 96] as initial references.”
Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group (2016, methodology): “It is well known (see, for instance [10]) that the horizontal mean curvature of Σ at a non-characteristic point x coincides, up to a choice of sign, with the signed horizontal curvature of the Legendrian curve γ in Σ through x.”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, background): “To quote a few: H0 can be defined in terms of the first variation of the area functional [20, 29, 9, 36, 40, 31, 7], as horizontal divergence of the horizontal unit normal or as limit of the mean curvatures Hε in the Riemannian metrics gε as ε → 0.”
Characterizations of intrinsic rectifiability in Heisenberg groups (2010, background): “For a general review on Carnot groups, and in particular on Heisenberg groups, we refer to [8, 9, 19, 32] and to [33].”
Geometry of Carnot-Caratheodory Spaces, Differentiability, Coarea and Area Formulas (2009, unknown): “Tyson [30], D.”