Minimizing the Elastic Energy of Growing Leaves by Conformal Mapping

During morphogenesis, the shape of living species results from growth, stress relaxation, and remodeling. When the growth does not generate any stress, the body shape only reflects the growth density. In two dimensions, we show that stress free configurations are simply determined by the time evolution of a conformal mapping which concerns not only the boundary but also the displacement field during an arbitrary period of time inside the sample. Fresh planar leaves are good examples for our study: they have no elastic stress, almost no weight, and their shape can be easily represented by holomorphic functions. The growth factor, isotropic or anisotropic, is related to the metrics between the initial and current conformal maps. By adjusting the mathematical shape function, main characteristics such as tips (convex or concave or sharp-pointed), undulating borders, and veins can be mathematically recovered, which are in good agreement with observations. It is worth mentioning that this flexible method allows us to study complex morphologies of growing leaves such as the fenestration process in Monstera deliciosa, and can also shed light on many other 2D biological patterns.