On the singular set of the geodesic flow on nonpositively curved surfaces

We prove that the singular locus of the geodesic flow on the unit tangent bundle of a nonpositively curved surface enjoys an additional regularity property: its characteristic function belongs to the geometric anisotropic Sobolev space $H_x^{5\mathrm{/}8-}{H}_{\theta }^{-13\mathrm{/}16}$. In particular, this implies that if the geodesic flow is not ergodic with respect to the Liouville measure, then the zero set of the curvature must be thick in the sense that it supports a nonzero, nonnegative function in $H^{5\mathrm{/}8-}$. The proof relies on semiclassical analysis and a new radial estimate for parabolic flows.