abstract: In this talk we consider the problem of characterizing gauge balls in the Heisenberg group by prescribing their (non-constant) horizontal mean curvature. We discuss two uniqueness results: in the lowest dimensional case under an assumption on the location of the singular set, and in higher dimensions in the restricted class of horizontally umbilical hypersurfaces. We focus on the
degenerate ellipticity of the underlying operators, and on the presenceabsence of H ̈ormander- type properties. This is a joint work with C. Guidi and V. Martino.