Robert K. Hladky

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.

Non-compactness of the Neumann operator on the Heisenberg ball

For , we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on -forms in on the unit ball in -dimensional Heisenberg space.