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An interface description and numerical simulations of model A kinetics are used for the first time to investigate the intrasurface kinetics of phase ordering on corrugated surfaces. Geometrical dynamical equations are derived for the domain interfaces. The dynamics is shown to depend strongly on the local Gaussian curvature of the surface, and can be fundamentally different from that in flat systems: dynamical scaling breaks down despite the persistence of the dominant interfacial undulation mode; growth laws are slower than t1/2 and even logarithmic; a new very-late-stage regime appears characterized by extremely slow interface Motion; finally, the zero-temperature fixed point no longer exists, leading to metastable states. Criteria for the existence of the latter are derived and discussed in the context of more complex systems.