Very recently Dr. Julius Ross at the Univ. of Cambridge and I found a striking connection between geodesic rays in spaces of Kähler metrics and the Hele-Shaw flow (Laplacian growth). By this connection both have a similar interpretation as certain families of embedded holomorphic curves attached along their boundaries to a Lagrangian submanifold.The first objective is to develop the regularity theory of the Hele-Shaw flow (Laplacian growth) using techniques from the theory of moduli spaces of embedded holomorphic curves. These are powerful techniques used with great success in e.g. Gromov-Witten Theory and various Floer theories in symplectic topology. I thus hope to extend the short-time regularity result of Kufarev and Vinogradov, and also gain new insights as to how and when singularities occur.The second objective is to develop the regularity theory for (weak) geodesic rays in spaces of (cohomologically equivalent) Kähler metrics, using the Hele-Shaw flow as a one (complex) dimensional model case. Donaldson, and later Chen and Tian, have successfully applied techniques from the theory of moduli spaces of embedded holomorphic curves to a related problem connected to the regularity of geodesic segments rather than rays. Dr. Ross and I have a preliminary method to adapt some of these techniques to the setting of geodesic rays.