  We prove that the moduli of continuity of viscosity solutions to fully
nonlinear parabolic partial differential equations are viscosity subsolutions
of suitable parabolic equations of one space variable. As applications, we
obtain sharp Lipschitz bounds and gradient estimates for fully nonlinear
parabolic equations with bounded initial data, via comparison with
one-dimensional solutions. This work extends multiple results of Andrews and
Clutterbuck for quasilinear equations to fully nonlinear equations.
