Pointwise convergence of solution to Schrodinger equation on manifolds

Abstract

Let $(M^n,g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity is required so that the solution to free Schr\"{o}dinger equation converges pointwisely to its initial data. Assume the initial data is in $H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the 2 dimensional Torus, we prove that $\alpha>\frac{1}{2}$ is enough. For general compact manifolds, due to lacking of local smoothing effect, it is hard to beat the bound $\alpha>1$ from interpolation. We managed to go below 1 for dimension $\leq 3$. The more interesting thing is that, for 1 dimensional compact manifold, $\alpha>\frac{1}{3}$ is sufficient.