Regularity in mathematics of space and flow
Complex Monge-Ampère equations play a fundamental role in complex geometry. The geodesic equation on the space of Kähler metrics is an example of a homogeneous complex Monge-Ampère equation (HCMAE). This equation connects the study of constant scalar curvature Kähler metrics to the Yau-Tian-Donaldson conjecture. In such applications the regularity of the solution is often essential. Our incomplete knowledge of the regularity of solutions for HCMAE is a serious obstacle to the successful use of HCMAEs in complex geometry. There is a striking connection between certain geodesics of Kähler metrics (i.e. solutions to the HCMAE) and the Hele-Shaw flow of fluids between two plates. The overall objective of the GEODESICRAYS (From geodesic rays in spaces of Kahler metrics to the Hele-Shaw flow) project was to use this connection to develop the regularity theory for Hele-Shaw flow and HCMAE. The project studied the process in which solutions to the HCMAE such as geodesics of Kähler metrics develop singularities. It was demonstrated that there is a duality between Hele-Shaw flows and certain solutions to the HCMAE on the Cartesian product of the Riemann sphere and the unit disc. It was also shown using the duality that harmonic discs of the solutions to the HCMAE correspond exactly to the simply connected Hele-Shaw domains. This allowed the construction of examples whose harmonic discs were very far from foliating the space, hence contradicting the earlier results of Chen and Tian. The abundance of harmonic discs is one way to measure the regularity of a solution to the HCMAE. Another is to consider the set where the solution fails to be twice differentiable. Using the earlier short time regularity results, it was possible to construct solutions that fail to be twice differentiable at specified sets such as a curve segment. The main impact of the project has been the contribution to the regularity theory for HCMAEs. The results prove that the earlier partial regularity theory of Chen and Tian is fundamentally flawed. This changes the understanding of the HCMAE dramatically.