The curvature-improved multilinear restriction conjecture

Let $S_i\subset\mathbb{R}^{n+1}$ be hypersurfaces, and let $\mathcal R^*(2\times\cdots\times 2\rightarrow p)$ denote the $k$-linear restriction estimate associated with the surfaces $S_i$. Assume that the surfaces satisfy the appropriate transversality and curvature conditions. Curvature-improved multilinear restriction conjecture. The estimate $\mathcal R^*(2\times\cdots\times 2\rightarrow p)$ should hold for every

$$p\geq p(k)=\frac{2(n+1+k)}{k(n+k-1)}.$$

This conjecture proposes the optimal exponent in the cases $3\leq k\leq n$, improving on the generic threshold $\frac{2}{k-1}$; the paper proves the trilinear case for double-conic surfaces, while the general assertion remains open.