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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  =  x  y   . The bilinear method gave here sharp results for    p  >  10  /  3   , and this result was recently improved to    p  >  3.25   . This paper aims to be the 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 endpoint, for    p  >  10  /  3   . In the application of the bilinear method, we show that the behavior at small scales 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. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid