Scott D. Pauls

Minimal Surfaces in the Heisenberg Group

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 provide a link between our minimal surfaces and Riemannian constant mean curvature surfaces in H equipped with different Riemannian metrics approximating the Carnot–Carathéodory metric. We generate a large library of examples of minimal surfaces and use these to show that the solution to the Dirichlet problem need not be unique. Moreover, we show that the minimal surfaces we construct are in fact X-minimal surfaces in the sense of Garofalo and Nhieu.

Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model

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 non-characteristic minimal surfaces as ruled surfaces, we give a method to compute a minimal spanning surface given fixed boundary data presuming such a surface exists. Moreover, we describe a number of obstructions to existence and uniqueness but also show that under suitable conditions, smooth minimal spanning surfaces with good properties exist. Not only does this provide an explicit realization of the disocclusion process for the neurobiological model, but it also has application to constructing disocclusion algorithms in digital image processing.

H-minimal graphs of low regularity in H1

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharacteristic

Differential contributions of intra‐cellular and inter‐cellular mechanisms to the spatial and temporal architecture of the suprachiasmatic nucleus circadian circuitry in wild‐type, cryptochrome‐null and vasoactive intestinal peptide receptor 2‐null mutant mice

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Topological structures in the equities market network

H-minimal graphs whose components of the unit horizontal Gauss map are in

Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups

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Measures of centrality based on the spectrum of the Laplacian.

are ruled surfaces with

Deconstructing Circadian Rhythmicity with Models and Manipulations

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A new framework for dynamical models on multiplex networks

seed curves. Moreover, in light of a structure theorem of Franchi, Serapioni and Serra Cassano, we see that any H-minimal graph is, up to a set of perimeter zero, composed of such pieces. Along these lines, we investigate ways in which patches of

Coevolution of Cooperation and Partner Rewiring Range in Spatial Social Networks

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