Diagonal Fourier extension for positively curved surfaces in three dimensions

We prove the diagonal Fourier extension conjecture for compact smooth positively curved surfaces $\Sigma \subset {\mathbb{R}}^3$, including surfaces with smooth boundary. For every $3<p<\infty$, the extension operator maps $L^p(\Sigma )$ boundedly into $L^p({\mathbb{R}}^3)$. The compact paraboloid case also yields free Schrödinger local smoothing in two spatial dimensions for every $3<p<\infty$ and Sobolev order $s>2-6/p$.