Linear restriction estimates for Schroedinger equation on metric cones

Abstract

In this paper, we study some modified linear restriction estimates of the dynamics generated by Schroedinger operator on metric cone $M$, where the metric cone $M$ is of the form $M=(0,\infty)_r\times\Sigma$ with the cross section $\Sigma$ being a compact $(n-1)$-dimensional Riemannian manifold $(\Sigma,h)$ and the equipped metric is $g=\mathrm{d}r^2+r^2h$. Assuming the initial data possesses additional regularity in angular variable $\theta\in\Sigma$, we show some linear restriction estimates for the solutions. As applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initial data scattering theory for the mass-critical nonlinear Schroedinger equation on two-dimensional metric cones.