Chinese scientists have proved two core conjectures of Keller geometry

Chen Xiuxiong, founding director of the Center for Geometric Physics of the University of Science and Technology of China, and collaborator Cheng Jingrui achieved "landmark results" in the field of partial differential equations and complex geometry. They solved a fourth-order completely nonlinear elliptic equation, successfully proved the "mandatory conjecture" and the "geodesy stability conjecture", two core conjectures that have been pending in the international mathematical community for more than 60 years, and solved several famous problems about the constant scalar curvature measure and the Calabi extreme value measurement on Keller's manifold. Two papers were published in the Journal of the American Mathematical Society.

The existence of constant scalar curvature measures on Keller manifolds has been one of the core problems in geometric research over the past 60 years. There are three well-known conjectures about its existence – the stability conjecture, the coercive conjecture, and the geodetic stability conjecture. The stability conjecture, which is limited to the Keller-Einstein measure, is called the Yau Chengtong conjecture, which was proposed by the famous mathematician Chengtong Yau in the 1990s and was first solved by Chen Xiuxiong, Donaldson, and Sun Song. After nearly 20 years of efforts by many mathematicians, the necessity of the mandatory conjecture and geodetic stability conjecture has become very clear, but the proof of its sufficiency was considered out of reach before Chen's work.

Solving a class of fourth-order completely nonlinear elliptic equations proves the existence of a constant scalar curvature measure. Chen-Cheng's work is precisely to prove the existence of such equation solutions under the assumption that k-energy is mandatory or geodetic stability. The study of such equations is extremely difficult, and previously there were few suitable tools for dealing with such equations. The most important breakthrough of the Chen-Cheng was to give a priori estimates of such equations and the strategy for successfully implementing the new continuous parameters proposed by Chen Xiuxiong.

Experts believe that solving a class of fourth-order completely nonlinear elliptic equations, previously like an invisible curtain wall in front of mathematicians, Chen-Cheng's job is to "dig a hole" in the curtain wall, find a breakthrough without warning, not only find the solution of the equation, but also establish a set of systematic research methods for such equations, providing a new tool for exploring the unknown mathematical world.

In addition, they gave proofs of the stability conjecture on ring-symmetric Keller manifolds, generalized Donaldson's classical theorem on ring-symmetric Keller surfaces to high dimensions, and proposed possible solutions to the proof of the general stability conjecture, making it possible to fully solve the general stability conjecture.

Related paper information:

https://doi.org/10.1090/jams/967

https://doi.org/10.1090/jams/966

Source: China Science Daily