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In contrast to elliptic surfaces, the Fourier restriction problem forhypersurfaces of non-vanishing Gaussian curvature which admit principalcurvatures of opposite signs is still hardly understood. In fact, even for2-surfaces, the only case of a hyperbolic surface for which Fourier restrictionestimates could be established that are analogous to the ones known forelliptic surfaces is the hyperbolic paraboloid or "saddle" z = xy. The bilinearmethod 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). Thispaper aims to be a first step in extending those results to more generalhyperbolic surfaces. We consider a specific cubic perturbation of the saddleand obtain the sharp result, up to the end-point, for p > 10/3. In theapplication of the bilinear method, we show that the behavior at small scale inour surface is drastically different from the saddle. Indeed, as it turns out,in some regimes the perturbation term assumes a dominant role, whichnecessitates the introduction of a number of new techniques that should also beuseful for the study of more general hyperbolic surfaces.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid