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

报告时间：5月17日 10:00-11:00

报告人：葛建全（北京师范大学）

报告地点：2507

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 $S^{1}(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})$ 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 $g\leq2$ 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.