Research Projects of Yuguang Zhang

My research is in Kähler geometry. The main interests are metric geometric properties of Calabi-Yau manifolds, and related topics such as Einstein metric, Kähler-Ricci flow, and Yang-Mills gauge theory.

A Riemannian metric on a manifold is called an Einstein metric if the Ricci curvature is a constant. The study of Einstein metrics is a main theme in the research of differential geometry. In the 1970s, Yau and Aubin proved the famous Calabi conjecture, which asserts the existence of Einstein metrics on compact Kähler manifolds with the first Chern class zero and negative. Such metrics are called Kähler-Einstein metrics, and the Ricci-flat Kähler-Einstein metrics are called Calabi-Yau metrics. Much progress has been made, including: the applications in algebraic geometry, e.g. Bogomolov decomposition theorem, and the discovery of the connection with the superstring theory in physics etc. The other storyline is that Gromov introduced the notion of Gromov-Hausdorff topology in the 1980s, which played an essential role in solving famous conjectures, for example the proofs of the Poincaré conjecture and the Yau-Tian-Donaldson conjecture etc.

My focus is to study the interaction between the behaviours of Calabi-Yau metrics, or general Kähler-Einstein metrics, in the Gromov-Hausdorff sense and the degenerations of complex/Kähler structures of underlying manifolds, and the applications to the moduli spaces of Einstein metrics. Another topic is the interaction between Einstein metrics and other geometric objects, e.g. the dimension reduction of 4D Yang-Mills connections under the collapsing of Calabi-Yau metrics etc.