General constraints on growth

Given a system made from material with certain overall properties, one can ask what distributions of growth are consistent with those properties, and what kinds of shapes can be produced. With material that is completely rigid growth can occur only at boundaries. With material where every part can deform arbitrarily any kind of growth can occur. With material where parts can locally expand, but cannot change their shape, page 1007 showed that a 2D surface will remain flat if the growth rate is a harmonic function. The Riemann mapping theorem of complex analysis then implies that even in this case, any smooth initial shape can grow into any other such shape with a suitable growth rate function. In a 3D system with locally isotropic growth the condition to avoid tearing is that the Ricci scalar curvature must vanish, and this is achieved if the local growth rate satisfies a certain partial differential equation. (See also page 1049.)