Chinese scientists have successfully proved the two core conjectures of Keller geometry

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

The existence of constant scalar curvature measures on Keller manifolds has been one of the central problems in geometry 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. After nearly 20 years of work by many famous mathematicians, the necessity of mandatory conjectures and geodetic stability conjectures has become completely clear, but the proof of their sufficiency was considered out of reach before Chen Cheng's work, as difficult as climbing the peak without any equipment.

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 for a long time industry experts generally did not believe that there would be a satisfactory theory of existence. So that industry 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.

Reviewers commented, "Chen's groundbreaking work is extremely original and technically difficult, not only solving major problems in Keller geometry, but also providing profound insights into such nonlinear equations." It is foreseeable that this series of papers will become classics in the field of geometry and partial differential equations. ”

Wen/Science and Technology Daily reporter Wu Changfeng

Photo / Beijing Youth Daily reporter Fan Hui