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BIT Has Achieved Research Results in the Classification of Steady Ricci Solitons

Several days ago, Professor Deng Yuxing from the School of Mathematics and Statistics of Beijing Institute of Technology published a research paper entitled "Higher dimensional steady Ricci solitons with linear curvature decay" in the top international academic journal "Journal of the European Mathematical Society". The paper proved that when the dimensionality is not less than 4, the volume curvature decays linearly and the volume non-collapsing steady Kaehler-Ricci soliton with non-negative curvature operator must be rotationally symmetric. In other words, such solitons are either isomorphic to Euclidean space, or isomorphic to Bryant solitons. When the dimensionality is 4, the non-negative condition of the curvature operator can be reduced to a non-negative section curvature.

From 2002 to 2003, Perelman (Fields Medal winner) introduced Ricci Flow to solve the three-dimensional Poincare conjecture with a history of more than 100 years and a more extensive geometric conjecture. In his proof, singularity analysis is vital. How to promote Perelman's work to study the geometry and topology of four-dimensional manifolds has become an issue of concern. However, the singularity of the four-dimensional Ricci Flow is more complicated. Mathematicians such as Freedman (Fields Medal winner) and Bennett Chow speculate that the singularity classification of four-dimensional Ricci Flows can be attributed to the classifications of steady Ricci soliton and shrinking Ricci soliton. Regarding the steady Ricci solitons, Perelman made a famous conjecture in his paper on solving the three-dimensional Poincare conjecture. This conjecture is that the steady Ricci soliton with non-trivial three-dimensional and non-collapsed volume must be rotationally symmetric, that is, it must be a Bryant soliton. In 2012, Simon Brendle (Bocher Prize winner) proved this conjecture. For higher-dimensional steady Ricci solitons, Brendle proved the rotational symmetry of solitons with normal section curvature under the condition of asymptotically columnar structure. The asymptotic columnar condition introduced by Brendle consists of two parts: linear decay of quantity curvature and dimensionality reduction condition. The definition of the dimensionality reduction assumption is complicated. When the dimensionality is 3, from the non-collapsing volume condition, the linear attenuation of curvature and the dimensionality reduction condition can be deduced. However, when the dimensionality is 4, under the condition of positive curvature, whether the linear attenuation of curvature and the dimensionality reduction condition can be derived from the volume non-collapsing condition is still unknown. In other words, he believes that the curvature decay condition cannot be removed.

Deng Yuxing and Professor Zhu Xiaohua of Peking University (Chen Xingshen Award winner) proved that when the dimensionality is greater than or equal to 4, the volume curvature decays linearly and the volume non-collapsing steady Ricci soliton with non-negative curvature operator must be rotationally symmetric. When the dimensionality is 4, the non-negative condition of the curvature operator can be reduced to a non-negative section curvature. Since the solitons appearing in the singularity analysis of the Ricci flow automatically meet the volume non-collapsing condition, everyone is more concerned about the steady Ricci solitons that meet the volume non-collapsing condition. For the soliton that everyone cares about most, such as volume non-collapse solitons, the classification results of Deng Yuxing and others remove the dimensionality reduction assumption in Brendle's theorem, which essentially improves Brendle's work. The reviewers of the Journal of the European Mathematical Society unanimously evaluated the result as the best classification result for steady Ricci solitons with four or more dimensions.

This research work was completed by Professor Deng Yuxing and Professor Zhu Xiaohua of Peking University. Professor Deng Yuxing was the first author. This work was supported by the National Natural Science Foundation of China.

Link to the paper: https://www.emsph.org/journals/show_abstract.php?issn="1435-9855&vol=22&iss=12&rank=7&p403=1


Attachment of the research team and personal profile:

Deng Yuxing, professor, main member of the geometry team of the School of Mathematics and Statistics, BIT. Graduated from Beijing Normal University with a bachelor's degree and a Ph.D. from Peking University. He has long been engaged in the research of differential geometry, especially Ricci Flow, and won the National Outstanding Youth Fund (Project No. 12022101) in 2020, and has presided over two projects funded by the National Natural Science Foundation of China. As the first author, he published 8 SCI papers in comprehensive journals such as Journal of the European Mathematical Society, Mathematische Annalen, Transactions of the American Mathematical Society, International Mathematics Research Notices, Mathematische Zeitschrift.