A research group led by Prof. CHEN Xiuxiong and WANG Bing from University of Science and Technology of China (USTC) of the Chinese Academy of Sciences, proved the Hamilton-Tian conjecture, which says that each Kahler Ricci flow on a Fano manifold must converge to a limit Kahler Ricci Shrinking soliton with mild singularities.
This problem was listed as the next target problem of Perelman in his paper resolving the famous Poincare conjecture. The resolution of this conjecture has many consequences. For example, it implies the partial-C0-conjecture for Kahler manifolds whose Ricci curvature is bounded from below.
In their study, researchers set up a structure theory for polarized K？hler Ricci flows with proper geometric bounds, based on the compactness of the moduli of non-collapsed Calabi–Yau spaces with mild singularities. Their theory is a generalization of the structure theory of non-collapsed K？hler Einstein manifolds.
The researchers develop new tools like “canonical radius”, “Riemannian conifolds” to bridge the gap from traditional Kahler Einstein manifolds to Kahler Ricci flow with bounded scalar curvature.
Furthermore, as application of this result, the team provide a concise Ricci flow proof of Yau’s stability conjecture.
The result was commented as “a big event in recent Ricci flow development” by Fields medal winner S.K. Donaldson.
Results of the study were published in Journal of Differential Geometry in September, 2020.