Constant mean curvature surfaces in sub-Riemannian geometry

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In 1996, Garofalo and Nhieu showed the existence of
solutions to the Plateau Problem in the setting of sub-Riemannian spaces, beginning close to a decade of sustained investigation of minimal surfaces in sub-Riemannian spaces.  In this talk, we will focus on a first example, the sub-Riemannian Heisenberg group,
to describe the current state of knowledge.  While smooth minimal surfaces have some remarkable rigidity properties, we will discuss new constructions of minimal surfaces of lower regularity, some of which
can be shown to be minimizers.  If time permits, we will discuss extensions of these ideas to the minimal surface problem in the roto-translation group which has direct application to a model of thefunction of the first layer of the visual cortex (due to G. Citti and
A. Sarti) and to recent digital inpainting algorithms (due to L. Ambrosio and S. Masnou).