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.
