Ancient Solutions of Mean Curvature Flow


Ancient solutions are important in studying singularities of mean curvature flows (MCF). So far most rigidity results about ancient solutions are modeled on shrinking spheres or spherical caps. In this talk, I will describe the behavior of MCF for a class of submanifolds, called isoparametric submanifolds, which have more complicated topological type. We can show that all such solutions are in fact ancient solutions, i.e. they exist for all time which goes to negative infinity. Similar results also hold for MCF of regular leaves of polar foliations in simply connected symmetric spaces with non-negative curvature. I will also describe our conjectures proposed together with Terng on rigidity of ancient solutions to MCF for hypersurfaces in spheres. These conjectures are closely related to Chern’s conjecture for minimal hypersurfaces in spheres. This talk is based on joint works with Chuu-Lian Terng and Marco Radeschi.


Xiaobo Liu is a Chair Professor at Peking University, the Deputy Director of the Beijing International Center for Mathematical Research, and the President of the Beijing Mathematical Society. He previously served as a professor at the University of Notre Dame in the United States and received a Research Fellowship from the Sloan Foundation. In 2006, he was invited to deliver a 45-minute lecture at the International Congress of Mathematicians in Madrid. His main research areas include the theory of Gromov-Witten invariants and the theory of isoparametric submanifolds. He has published numerous high-quality papers in prestigious international journals such as Annals of Mathematics, Duke Mathematical Journal, Communications in Mathematical Physics, and Journal of Differential Geometry.