We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Carathéodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic partial differential equation and prove an existence result for the Plateau problem in this setting. Further, we… (More)
We establish a sub-Riemannian version for the first Heisenberg group H 1 of the classical theorem of Bernstein stating that the only complete minimal graphs in R 3 are planes.
We investigate solutions to the minimal surface problem with Dirichlet boundary conditions in the roto-translation group equipped with a subRiemannian metric. By work of G. Citti and A. Sarti, such solutions are amodal completions of occluded visual data when using a model of the first layer of the visual cortex. Using a characterization of smooth minimal… (More)
In this paper, we prove results concerning the large scale geometry of connected, simply connected nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT 0 metric space or an Alexandrov metric space. The main technical aspect… (More)
We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M , we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N. First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot… (More)
We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated… (More)
In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C 1 H-minimal graphs whose components of the unit horizontal Gauss map are in W 1,1 are ruled surfaces with C 2 seed curves. In a different direction, we investigate ways in which patches of C 1 H-minimal graphs can be glued together to form continuous… (More)
We present a new method for articulating scale-dependent topological descriptions of the network structure inherent in many complex systems. The technique is based on " Partition Decoupled Null Models, " a new class of null models that incorporate the interaction of clustered partitions into a random model and generalize the Gaussian ensemble. As an… (More)
We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli–Garofalo– Nhieu or Lu–Manfredi–Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph.
We introduce a family of new centralities, the k-spectral centralities. k-spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, k-spectral centralities have various interpretations in terms of spectrally determined information. We explore this centrality… (More)