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Calabi flows with unbounded curvature

Periodic Reporting for period 1 - CFUC (Calabi flows with unbounded curvature)

Reporting period: 2016-05-26 to 2018-05-25

The Calabi flow was introduced by Calabi in the 1950s, aiming to find constant scalar curvature Kähler (cscK) metrics. An important class of cscK metrics is the Ricci flat Kähler metrics. The Calabi conjecture was about the existence of such metrics and its resolution was part of Yau's Fields medal work.

Nowadays, the existence of cscK metrics has been intensively studied and its development reveals the link between differential geometry and algebraic geometry, which is now known as the Yau-Tian-Donaldson conjecture.

Geometric flows have been successful in finding canonical metrics. The theory of Ricci flow developed by Hamilton and Perelman has successfully solved the conjecture of Poincare and Thurston, one of the seven $1 million Clay Mathematics Institute Millennium Prizes.

This proposal concerns singularity analysis of the Calabi flow, when the curvature is unbounded. The cscK metrics are the stationary solutions of the Calabi flow and the Kähler metrics with cone singularities generally do not have bounded curvature. Consequently, cscK metrics with cone singularities serve as a natural singularity model of the Calabi flow with unbounded curvature. During the fellowship, the researcher pioneered the development and established a theory of cscK metrics with cone singularities.

The primary impact of the project is knowledge exchange between the researcher and his collaborators. The broad development of fundamental mathematics such as the research in this proposal is giving new ways to understand our universe and is leading to applications that affect our everyday life. For example, geometric flows, as studied in this proposal, are being used for image processing.
The researcher mainly worked on two problems concerning constant scalar curvature Kähler (cscK) metrics during the fellowship:

1. Donaldson's conjectural picture on the asymptotic behaviours of the Calabi flow.

With H. Li and B. Wang, we confirm the conjecture in complex dimension 2 for cscK metrics. In high dimensions, a condition is required, that is the scalar curvature is bounded.
The Calabi flow is a 4th order flow. It is believed to be very difficult to analyse, because there is a lack of maximum principle, which is a key property of solutions to partial differential equations. However, we adapted Perelman's brilliant ideas of the Ricci flow to the Calabi flow and finally proved a regularity estimate that plays a key role in the proof of our results. These results have been published in an international research journal (JGA, 2017).

2. The construction, asymptotic behaviour, uniqueness and existence of cscK metrics with cone singularities.

The theory of cscK metrics with cone singularities has been built during the fellowship.
The results on these topics are published in international research journals (CVPDE 2018 with L. Li, PLMS 2018 with J. Keller, Math. Ann. 2017 with L. Li). More recent results have been released in the arxiv (arXiv:1609.03111 with H. Yin, arXiv:1709.09616). In arXiv:1803.09506 the properness conjecture and Donaldson's geodesic conjecture in the context of cscK metrics with cone singularities have been proved; these provide the equivalent criteria for the existence of such metrics. As an application, the researcher derived two important properties of the path deforming the cone angles of cscK cone metrics: openness and approximation.
The techniques developed in the study of the asymptotic behaviours of the Calabi flow have further potential applications to other geometric flows, for example the mean curvature flow in higher codimensions. In the long term, two properties of the path deforming the cone angles of cscK cone metrics provide a new point of view to attempt the Yau-Tian-Donaldson conjecture. The impact of this project will influence research in geometric analysis, PDE theory and geometry. The socio-economic impact of this project will also be generated via subsequent academic research in other areas, including physics, biology, chemistry, computer science and engineering. The PDE techniques growing from this project could ultimately be useful beyond the immediate field of research, e.g. for the financial sector, pricing derivatives and in optimal decision making.
A wedge with cone angle beta