Global Simply Connected Weak Solutions

The major assertion (Theorem 5.1) in this chapter is that starting Laplacian growth from any simply connected planar domain with smooth boundary, the solution, in a weak form, can be continued forever as an evolution of simply connected domains. The price for keeping the domains simply connected is that the solution must be allowed to go up on a branched Riemann surface above the complex plane, a Riemann surface which is not known in advance and has to be created along with the solution. The proof of Theorem 5.1 depends on several technical lemmas, one of which (Lemma/Conjecture 5.3) is stated as a conjecture. There is no real doubt concerning the validity of that lemma/conjecture, but at present a rigorous proof is missing.