Solution for a Special Case of Calabi's Conjecture

In two recent papers published in Journal of the American Mathematical Society, a research group led by Prof. CHEN Xiuxiong from University of Science and Technology of China (USTC) of the Chinese Academy of Sciences, in cooperation with CHENG Jingrui, solved various problems about constant scalar curvature Kähler metrics on a compact Kähler manifold. Their work was highly commented as ‘a remarkable, fundamental, and completely unexpected advance in complex differential geometry’ by American mathematician Claude LEBRUN. 

The long-standing subject is influential in both maths and physics, starting with the conjecture made by Calabi in 1954 about the existence of constant scalar curvature Kähler metrics in any Kähler class on a compact Kähler manifold. This conjecture later turned out to be untrue in the full generality of its original proposal, and additional stability hypotheses must be added to ensure the validity of the results. 

Researches on specific cases of this conjecture have achieved a lot, typically, including Shing-Tung YAU’s Fields winning theorem on the existence of Ricci-flat Kähler metrics in the case of Kähler-Einstein metric, with these metrics being solutions of a variant in Einstein’s equation of general relativity. 

However, no significant progresses have been made, but the isolated existential results until the breakthrough made by Prof. CHEN Xiuxiong and CHENG Jingrui. 

The core problem of the conjecture is proving the existence of solutions to a fourth-order nonlinear elliptic equation. The equation is difficult to solve with few tools available. 

In this study, researchers proved an optimal version of Calabi’s flawed conjecture by adding one version of the missing stability proposal. By introducing a series of subtle propositions, CHEN Xiuxiong and CHENG Jingrui obtained ‘priori estimates’ for the solutions, which was a crucial step in proving existential results. Their work offered a new approach towards the core equation and created new methods to deal with similar difficulties.

Results of the study were published in Journal of Differential Geometry in September, 2020.

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