报告题目：Integral-Einstein hypersurfaces and Simons-type inequalities in spheres

 报告人:  葛建全  (北京师范大学)

 时间：2021年12月14日（周二） 10:00-11:00

 腾讯会议ID：781 764 219

摘要：We introduce a generalization, the so-called Integral-Einstein (IE) submanifolds, of Einstein manifolds by combining intrinsic and extrinsic invariants of submanifolds in Euclidean spaces, in particular, IE hypersurfaces in unit spheres. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in unit spheres, which is the main object of the Chern conjecture: such hypersurfaces are isoparametric. For these hypersurfaces, we obtain some integral inequalities with the bounds characterizing exactly the totally geodesic hypersphere, the non-IE minimal Clifford torus $S^{1}(\sqrt{\frac{1}{n}})\timesS^{n-1}(\sqrt{\frac{n-1}{n}})$ and the IE minimal CSC hypersurfaces. Moreover, if further the third mean curvature is constant, then it is an IE hypersurface or an isoparametric hypersurface with $g\ge2$ principal curvatures. In particular, all the minimal isoparametric hypersurfaces with $g\geq3$ principal curvatures are IE hypersurfaces. As applications, we also obtain some spherical Bernstein theorems. A universal lower bound for the average of squared lenth of second fundamental form of non-totally geodesic minimal hypersurface in unit spheres is established, partially proving the Perdomo Conjecture.

报告人简介：葛建全教授主要研究微分几何，特别是子流形的几何拓扑及其应用。其代表性研究成果主要集中在如下两个方面：DDVV 猜想的解决及其推广应用；等参理论在怪球和 4 维流形等方面的发展及应用。至今已在 Adv.Math., J.Reine Angew.Math., Math.Ann., J.Funct.Anal., Int.Math.Res.Not., Trans.AMS 等国际著名数学期刊上接受发表了27 篇论文，合作组织主办了微分几何青年论坛、等参理论国际会议、北京几何日会议等多次学术会议。2011 年获得中国数学会钟家庆数学奖和德国洪堡基金。2015 年获得国家自然科学优秀青年基金，2019 年作为主持人获得北京市自然科学基金重点研究专题项目。   

 邀请人：王 奎