Before revision: The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed for over three decades , and its properties and universality classes are well established. The vast majority of all the past studies were, however, concerned with surface growth that started from flat substrates. In several natural phenomena, as well as technological processes, interface growth occurs on curved surfaces. Examples include tumour and bacterial growth, as well as the interface between two fluid phases during injection of a fluid into a porous medium in which it moves radially. Since in growth on flat substrates the linear size of the system remains constant, whereas it increases in the caseof growth on curved substrates, the universality class of{\bf  [  in the long-time limit.
After revision: The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed for over three decades . Some experiments indicated the power law for the interface width, w(t)\sim t^\beta, remains the same as in growth on planar surfaces. Escudero (Phys. Rev. Lett.{\bf [ in the long-time limit.