Scott D. Pauls

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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)
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)
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)
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)
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)