In collaboration with Academician TIAN Gang of the Beijing International Center for Mathematical Research, Peking University, and Professor Min-chun HONG of the University of Queensland, Australia, Professor FU Yu from the School of Data Science and Artificial Intelligence, Dongbei University of Finance and Economics (DUFE), has published an online academic paper in the Journal of the European Mathematical Society (JEMS), a top-tier comprehensive mathematics journal. The paper is entitled “The biharmonic hypersurface flow and the Willmore flow in higher dimensions”.

Research on high-order curvature flows stands as a core topic in geometric analysis, which, after half a century of development, still faces numerous fundamental challenges, and a complete theoretical framework has yet to be established even in low-dimensional cases. Studies on fourth-order curvature flows have advanced from the evolution of smooth geometric objects to in-depth explorations of the evolutionary behavior of more general geometric objects that may have singular structures—prominent examples include the Willmore flow and diffusion flow. However, research on the Willmore flow has been confined to low dimensions, leaving high-dimensional curvature flows largely uncharted.

Led by Professor TIAN Gang, an Academician of the Chinese Academy of Sciences, the research team developed analytical tools on manifolds, including the Michael–Simon Sobolev inequality, and innovatively established a novel Gagliardo–Nirenberg inequality to construct an analytical framework for high-dimensional fourth-order flows. By employing delicate covering arguments, the team successively derived maximal time estimates and global smooth solutions for biharmonic heat flows and Willmore flows in arbitrary dimensions. This methodology not only holds profound theoretical significance but also represents a critical breakthrough in revealing the nature of higher-order curvature flows and resolving the biharmonic conjecture, thereby paving new avenues for investigating classical problems such as high-dimensional Willmore and diffusion flows.

This work is supported by the National Natural Science Foundation of China (Grant No. 12571051), led by Professor FU Yu. Marking the first publication from Liaoning Province to appear in the JEMS—the official journal of the European Mathematical Society and a premier comprehensive mathematics journal with an outstanding global reputation—this work serves as a major milestone for the international research team formed by DUFE, Peking University, and the University of Queensland. It underscores a significant breakthrough for DUFE in the fields of mathematics and applied mathematics, reflecting DUFE’s growing international academic influence.

Written by: Xin Hua

Source: School of Data Science and Artificial Intelligence