## Final Report Summary - GEODESICRAYS (From geodesic rays in spaces of Kähler metrics to the Hele-Shaw flow)

Complex Monge-Ampère equations play a fundamental role in complex geometry. The Einstein equation for Kähler metrics reduces to a complex Monge-Ampère equation, and this fact underpins the successful theory of Kähler-Einstein metrics. The geodesic equation on the space of Kähler metrics is an example of a homogeneous complex Monge-Ampère equation (HCMAE), which connects to the study of constant scalar curvature Kähler metrics and the Yau-Tian-Donaldson conjecture. In such applications the regularity of the solution is often essential. However, our knowledge of the regularity of solutions to the HCMAE is still very incomplete, and that ignorance is a serious obstacle to the successful use of HCMAEs in complex geometry.

In 2012, the scientist in charge, Dr. Julius Ross at the University of Cambridge, and I found a striking connection between certain geodesics of Kähler metrics (i.e. solutions to the HCMAE) and the Hele-Shaw flow. The overall objective of the project GEODESICRAYS was to use this connection to develop the regularity theory of the Hele-Shaw flow and the HCMAE in conjuction.

First, using the conceptual connection to the HCMAE allowed us to apply the theory of moduli spaces of embedded holomorphic curves to get new short time regularity results for the Hele-Shaw flow. In contrast to older results no restrictive real analyticity assumptions were needed.

Then, a similar analysis yielded a very strong local regularity result for an interesting class of geodesics of Kähler metrics. We then used this regularity to construct canonical tubular neighbourhoods in Kähler manifolds, adapted to both the holomorphic and the symplectic structure. Another interesting application was our proof of optimal regularity of certain extremal envelopes, which the work of Berman has shown to be intimately connected to Bergman kernels with vanishing and slope stability.

Next, we studied the process in which solutions to the HCMAE such as geodesics of Kähler metrics develop singularities. Again this was done using connection with the Hele-Shaw flow. First of all we showed that there is in fact a duality between Hele-Shaw flows and certain solutions to the HCMAE on the Cartesian product of the Riemann sphere and the unit disc. When the solution is perfectly regular one gets so called harmonic discs which foliate the whole space. The partial regularity theory for solutions to the HCMAE developed by X.X. Chen and G. Tian said that even if these harmonic discs might not foliate the whole space, they always almost do. This strong regularity result was used by Chen and Tian to prove uniqueness up to automorphisms of constant scalar curvature Kähler metrics. We could show using our duality that harmonic discs of the solutions to the HCMAE corresponded exactly to the simply connected Hele-Shaw domains. This then allowed us to construct examples of solutions whose harmonic discs were very far from foliating the space, hence contradicting the earlier results of Chen and Tian.

The abundance or sparsity 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. Again this is very subtle and not much is known. E.g. it was only recently that Lempert and Vivas showed that geodesic segments of Kähler metrics with regular endpoints can fail to be C^2. In those examples however it is not known exactly where the singularities appear, we only know that they have to happen somewhere. Ross and I continued our analysis of the solutions that are dual to Hele-Shaw flows, and using our earlier short time regularity results we were able to construct solutions that fail to be twice differentiable at specified sets such as a curve segment. Importantly the corresponding Hele-Shaw flow gives an explicit picture of how exactly the singularities appear.

The main impact of the project has been the contribution to the regularity theory for the HCMAE. Perhaps most importantly the results prove that the earlier partial regularity theory of Chen and Tian is fundamentally flawed. This changes the our understanding of the HCMAE dramatically. Furthermore, the Hele-Shaw duality provides us with a rich nontrivial class of examples which can guide us towards the true picture. As a confirmation of its importance this work was recently published in the very prestigious journal Publication mathematiques de l'IHES.

In 2012, the scientist in charge, Dr. Julius Ross at the University of Cambridge, and I found a striking connection between certain geodesics of Kähler metrics (i.e. solutions to the HCMAE) and the Hele-Shaw flow. The overall objective of the project GEODESICRAYS was to use this connection to develop the regularity theory of the Hele-Shaw flow and the HCMAE in conjuction.

First, using the conceptual connection to the HCMAE allowed us to apply the theory of moduli spaces of embedded holomorphic curves to get new short time regularity results for the Hele-Shaw flow. In contrast to older results no restrictive real analyticity assumptions were needed.

Then, a similar analysis yielded a very strong local regularity result for an interesting class of geodesics of Kähler metrics. We then used this regularity to construct canonical tubular neighbourhoods in Kähler manifolds, adapted to both the holomorphic and the symplectic structure. Another interesting application was our proof of optimal regularity of certain extremal envelopes, which the work of Berman has shown to be intimately connected to Bergman kernels with vanishing and slope stability.

Next, we studied the process in which solutions to the HCMAE such as geodesics of Kähler metrics develop singularities. Again this was done using connection with the Hele-Shaw flow. First of all we showed that there is in fact a duality between Hele-Shaw flows and certain solutions to the HCMAE on the Cartesian product of the Riemann sphere and the unit disc. When the solution is perfectly regular one gets so called harmonic discs which foliate the whole space. The partial regularity theory for solutions to the HCMAE developed by X.X. Chen and G. Tian said that even if these harmonic discs might not foliate the whole space, they always almost do. This strong regularity result was used by Chen and Tian to prove uniqueness up to automorphisms of constant scalar curvature Kähler metrics. We could show using our duality that harmonic discs of the solutions to the HCMAE corresponded exactly to the simply connected Hele-Shaw domains. This then allowed us to construct examples of solutions whose harmonic discs were very far from foliating the space, hence contradicting the earlier results of Chen and Tian.

The abundance or sparsity 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. Again this is very subtle and not much is known. E.g. it was only recently that Lempert and Vivas showed that geodesic segments of Kähler metrics with regular endpoints can fail to be C^2. In those examples however it is not known exactly where the singularities appear, we only know that they have to happen somewhere. Ross and I continued our analysis of the solutions that are dual to Hele-Shaw flows, and using our earlier short time regularity results we were able to construct solutions that fail to be twice differentiable at specified sets such as a curve segment. Importantly the corresponding Hele-Shaw flow gives an explicit picture of how exactly the singularities appear.

The main impact of the project has been the contribution to the regularity theory for the HCMAE. Perhaps most importantly the results prove that the earlier partial regularity theory of Chen and Tian is fundamentally flawed. This changes the our understanding of the HCMAE dramatically. Furthermore, the Hele-Shaw duality provides us with a rich nontrivial class of examples which can guide us towards the true picture. As a confirmation of its importance this work was recently published in the very prestigious journal Publication mathematiques de l'IHES.