  In contrast to elliptic surfaces, the Fourier restriction problem for
hypersurfaces of non-vanishing Gaussian curvature which admit principal
curvatures of opposite signs is still hardly understood. In fact, even for
2-surfaces, the only case of a hyperbolic surface for which Fourier restriction
estimates could be established that are analogous to the ones known for
elliptic surfaces is the hyperbolic paraboloid or "saddle" z = xy. The bilinear
method gave here sharp results for p > 10/3 (Lee 05, Vargas 05, Stovall 17),
and this result was recently improved to p > 3.25 (Cho-Lee 17, Kim 17). This
paper aims to be a first step in extending those results to more general
hyperbolic surfaces. We consider a specific cubic perturbation of the saddle
and obtain the sharp result, up to the end-point, for p > 10/3. In the
application of the bilinear method, we show that the behavior at small scale in
our surface is drastically different from the saddle. Indeed, as it turns out,
in some regimes the perturbation term assumes a dominant role, which
necessitates the introduction of a number of new techniques that should also be
useful for the study of more general hyperbolic surfaces.
