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We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge-Ampère type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge-Ampère measures for convex functions in the Heisenberg group.

We present a graphene-based metasurface that can be actively tuned between different regimes of operation, such as anomalous beam steering and focusing, cloaking and illusion optics, by applying electrostatic gating without modifying the geometry of the metasurface. The metasurface is designed by placing graphene nano-ribbons (GNRs) on a dielectric cavity resonator, where interplay between geometric plasmon resonances in the ribbons and Fabry-Perot resonances in the cavity is used to achieve 2π phase shift. As a proof of the concept, we demonstrate that wavefront of the field reflected from a triangular bump covered by the metasurface can be tuned by applying electric bias so as to resemble that of bare plane and of a spherical object. Moreover, reflective focusing and change of the reflection direction for the above-mentioned cases are also shown. arXiv:1712.04111v1 [physics.optics]