Stability of Fourier Restriction for a Class of Finite Type Surfaces in ${\mathbb{R}}^3$

We establish the stability of the optimal non-endpoint Fourier restriction range under smooth flat perturbations of curved-root surfaces in ${\mathbb{R}}^3$. Specifically, we consider graphs of $$\phi (x_1,x_2)=(x_2-x_1^m{)}^n+R(x_1,x_2),$$ where $m,n\ge 2$ are integers and $R$ is smooth and flat at the origin. On a sufficiently small neighborhood depending on the perturbation, we prove the local $L^p({\mathbb{R}}^3)\to L^2(S,d\sigma )$ restriction estimate for $1\le p<p_c$, where $$p_c=\frac{q_c}{q_c-1},q_c=\max \{2+\frac{2mn}{m+1},2n\}\mathrm{.}$$ If the nonnegative localizing density is positive at the origin, an adapted-box example shows that the estimate fails for $p>p_c$. Thus arbitrary smooth flat perturbations preserve the optimal restriction range away from the critical endpoint. For $n\ge 3$, the main local estimate combines planar curve decoupling with a one-dimensional finite-type extension estimate, whose principal scale factors cancel exactly. For $n=2$, we prove a uniform $C^4$ perturbation estimate using a weighted convolution bound and an anisotropic Whitney decomposition. Anisotropic rescaling and summation complete the proof.