We express two CR invariant surface area elements in terms of quantities in
pseudohermitian geometry. We deduce the Euler-Lagrange equations of the
associated energy functionals. Many solutions are given and discussed. In
relation to the singular CR Yamabe problem, we show that one of the energy
functionals appears as the coefficient (up to a constant multiple) of the log
term in the associated volume renormalization.Comment: 50 page
