Indiana University Mathematics Journal
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing C∞ regularity of the sub-Riemannian minimal surface along its Legendrian foliation.
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Capogna, L., Citti, G., & Manfredini, M. (2009). Regularity of non-characteristic minimal graphs in the Heisenberg group H1. Indiana University Math Journal, 58(5), 2115–2160. https://dx.doi.org/10.1512/iumj.2009.58.3673