Restriction estimates for surfaces with negative curvature in ${\mathbb{R}}^3$

In ${\mathbb{R}}^3$, we prove that $L^q\to L^p$ restriction estimates associated with smooth surfaces with negative Gaussian curvature hold for all $p>\frac{22}{7}$ and $q^{\mathrm{\prime }}<\frac{p}{2}$. Building on Demeter–Wu’s work for the model hyperbolic paraboloid, we introduce a geometric propagation principle for good lines, which controls the degenerate directions arising from straight-line segments on the surface. This overcomes a key difficulty in the general case, where such directions may vary with the geometry rather than being fixed by the coordinate axes.