Laplacian growth on a branched Riemann surface (Analysis on Shapes of Solutions to Partial Differential Equations)
Björn Gustafsson · Kyoto University Research Information Repository (Kyoto University) · 2018
Laplacian growth refers to domain evolution driven by harmonic gradients, for example the gradient of the Green's function with a fixed pole. It makes sense on Riemannian manifolds of arbitrary dimension, and there is a notion of weak solution which allows for changes of topology of the domain during the evolution. However, here we discuss the possibility, in the case of two dimensions, of avoiding changes of topology at the price of allowing the evolution to go up on a branched Riemann covering surface of the original surface. If the initial domain is simply connected one can then describe the evolution by means of conformal mappings from the unit disk, a kind of Loewner evolution. There appear difficulties which are not yet completely solved. Preliminary results can be found in joint work arXiv:1411.1909 with Yu-Lin Lin, which presently is under further progress with also Joakim Roos as an author.