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- SCOTT D. PAULS
- 2008

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 of the classical theorem of Bernstein stating that the only complete minimal graphs in R are planes.

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)

- SCOTT D. PAULS
- 2006

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C H-minimal graphs whose components of the unit horizontal Gauss map are in W 1,1 are ruled surfaces with C seed curves. In a different direction, we investigate ways in which patches of C H-minimal graphs can be glued together to form continuous piecewise C… (More)

- Robert K. Hladky, Scott D. Pauls
- Journal of Mathematical Imaging and Vision
- 2009

We investigate solutions to the minimal surface problem with Dirichlet boundary conditions in the roto-translation group equipped with a sub-Riemannian metric. By work of G. Citti and A. Sarti, such solutions are completions of occluded visual data when using a model of the first layer of the visual cortex. Using a characterization of smooth… (More)

- SCOTT D. PAULS
- 1998

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 CAT0 metric space or an Alexandrov metric space. The main technical aspect… (More)

One of the most celebrated problems in geometry and calculus of variations is the Bernstein problem, which asserts that a C2 minimal graph in R3 must necessarily be an affine plane. Following an old tradition, here minimal means of vanishing mean curvature. Bernstein [Be] established this property in 1915. Almost fifty years later a new insight of Fleming… (More)

- SCOTT D. PAULS
- 2004

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 Nrectifiable 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)

We derive a formula for the first variation of horizontal perimeter measure for C hypersurfaces of completely general sub-Riemannian manifolds, allowing for the existence of characteristic points. For C hypersurfaces in vertically rigid sub-Riemannian manifolds we also produce a second variation formula for variations supported away from the characteristic… (More)