  Nucleation and growth is the dominant relaxation mechanism driving first
order phase transitions. In two-dimensional at systems nucleation has been
applied to a wide range of problems in physics, chemistry and biology. Here we
study nucleation and growth of two-dimensional phases lying on curved surfaces
and show that curvature modify both, critical sizes of nuclei and paths towards
the equilibrium phase. In curved space nucleation and growth becomes inherently
inhomogeneous and critical nuclei form faster on regions of positive Gaussian
curvature. Substrates of varying shape display complex energy landscapes with
several geometry-induced local minima, where initially propagating nuclei
become stabilized and trapped by the underlying curvature.
