On Kuiper's conjecture

Dupin 情形下的 Chern 猜想

Chern's Conjecture in the Dupin case

本文在 Dupin 超曲面情形下研究 Chern 猜想，证明了在 $g=3,4,6$ 等条件下具有常数曲率的闭正常 Dupin 超曲面是等参的，并运用拓扑与几何方法减少了代数论证所需的假设。

Compact Dupin Hypersurfaces

A hypersurface $M$ in ${\bf R}^n$ is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on $M$, i.e., each continuous principal curvature function has constant multiplicity on $M$. These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in ${\bf R}^n$ or $S^n$. The theory of compact proper Dupin hypersurfaces in $S^n$ is closely related to the theory of isoparametric hypersurface…

Möbius and Laguerre geometry of Dupin Hypersurfaces

In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al on the classification of Dupin hypersurfaces.

具有非常数Lie曲率的紧Dupin超曲面的构造

Constructions of Compact Dupin Hypersurfaces with Non-constant Lie Curvatures

本文综述了单位球面中两类紧真Dupin超曲面的构造，证明其具有非常数Lie曲率，无法通过Lie球变换等价于等参超曲面，成为Cecil和Ryan1985年相关猜想的反例。