Research

Tian Gang makes progress on the structure of almost Einstein manifolds

JUL . 13 2015
Peking University, July 13, 2015: Professor Tian Gang recently published his paper, "On the structure of almost Einstein manifolds" (joint with Bing Wang), in Journal of the American Mathematical Society (JAMS). JAMS is considered as one of the four top journals in mathematics, while the other three are Annals of Mathematics, Inventiones Mathematicae and Acta Mathematica.
 
From late 1990s, the structure and regularity theory about Einstein manifolds has been one of the central problems in differential geometry. They have close relation with other problems in differential geometry, for example, the existence of canonical metrics on Kahler geometry. In 1997, the famous American mathematicians Cheeger and Colding analyzed the singularities of the limit space of non-collapsed Riemannian manifolds with Ricci curvature bounded below and they proved that the singular set has a tangent cone structure. After their seminal work, the regularity of the limit space has become an active problem.
 
The paper by Tian and Wang studied the Gromov-Hausdorff limit space of a sequence of almost Einstein manifolds. They proved a deep structure theorem, which asserts that the regular part of the limit space is a smooth, convex open Riemannian manifold while the singular part has codimension bigger than 2. The theorem has many importance applications in Kahler geometry. It was used to solve the famous Yau-Tian-Donaldson conjecture on the existence of Kahler-Einstein metrics. In the proof, they obtained a pseudo-locality property and a delicate bound of the Gromov-Hausdorff distance between metrics along the Ricci flow. These will have important implications in geometric analysis and metric geometry.
 
Professor Tian now serves the chair of the School of Mathematical Sciences and the director of the Beijing International Center for Mathematical Research. He has devoted to the study of differential geometry and mathematical physics for many years. He solved a series of important problems, in particular, his breakthrough work in the study of Kahler-Einstein metrics. The current work of Tian and his collaborator on the structure of almost Einstein manifolds will have more applications in differential geometry and other fields.



Edited by: Zhang Jiang