We show that geometric shielding of the reactive flux in chemical vapor deposition by tall neighboring structures obtained by deep substrate patterning, along with short surface diffusion lengths, can provide nearly space filling arrays of high-quality epitaxial crystals despite large mismatches of lattice parameters and thermal expansion coefficients. The density of extended defects is strongly reduced by the method, and wafer bowing and crack formation largely inhibited. The concept is shown to be valid for SiGe/Si heterostructures ranging from pure Si to pure Ge both on Si (001) and Si (111) substrates. Here, dislocations are efficiently eliminated from three-dimensional faceted crystals with high-aspect ratios on top of micron-sized Si pillars. The application to 3C-SiC/Si (001) ridges, characterized by a lattice mismatch of nearly 20%, provides significantly lower stacking fault densities compared with layers grown on …