## 2021年微分几何会议

时间：2021年7月13日

地点：后主楼1124

报告摘要

华波波（复旦大学）

Title：Steklov eigenvalues on graphs

Abstract：The eigenvalues of the Dirichlet-to-Neumann operator are called Steklov eigenvalues, which are well studied in spectral geometry. In this talk, we introduce Steklov eigenvalues on graphs, and estimate them using geometric quantities, based on joint works with Wen Han, Zunwu He, Yan Huang, and Zuoqin Wang.

王作勤（中国科学技术大学）

Title: On the remainders in the two-term Weyl law of planar disks and annuli

Abstract: Weyl laws relate the asymptotic behaviors of the eigenvalues of certain geometric operators with the geometric/analytic/dynamical properties of the underlying space. In this talk I will briefly describe these connections, with an emphasis on the relation between the eigenvalue counting problem for special planar domains with integrable billiard flows and the classical lattice points counting problem. This talk is mainly based on joint works with Jingwei Guo, Wolfgang Muller and Weiwei Wang.

刘博（华东师范大学）

Title：Introduction to Bismut-Cheeger eta form

Abstract：Last month, the 2021 Shaw Prize in Mathematical Sciences was awarded to Bismut and Cheeger. The press release says that "Bismut and Cheeger have also worked together, and are particularly celebrated for their extension of a famous invariant, the so-called eta invariant, from manifolds to families of manifolds, which allowed them to compute explicitly the limit of the eta invariant along a collapsing sequence of spaces." Now this extension is called the Bismut-Cheeger eta form.

In this talk, we will introduce the celebrated Bismut-Cheeger eta form and discuss some recent progresses.

李发贵（北京师范大学）

Title: Integral-Einstein hypersurfaces and rigidity of scalar curvature of minimal hypersurfaces

Abstract: 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 length of the second fundamental form of non-totally geodesic minimal hypersurface in unit spheres is established, partially proving the Perdomo Conjecture.