  We obtain an improved pseudolocality result for Ricci flows on
two-dimensional surfaces that are initially almost-hyperbolic on large
hyperbolic balls. We prove that, at the central point of the hyperbolic ball,
the Gauss curvature remains close to the hyperbolic value for a time that grows
exponentially in the radius of the ball. This two-dimensional result allows us
to precisely conjecture how the phenomenon should appear in the higher
dimensional setting.
