Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres

Delaunay-type hypersurfaces in cohomogeneity one manifolds

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction specializes to the classical examples in round spheres, and allows to obtain Delaunay-type hypersurfaces in many other ambient spaces, ranging from complex and quaternionic projective spaces to Kervaire exotic spheres.

New Examples of Obstructions to Non-Negative Sectional Curvatures in Cohomogeneity One Manifolds

K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.

具有两个求和项的余齐次1 Einstein度量的不存在性

Non-existence of cohomogeneity one Einstein metrics of two summands

针对主轨道含两个不等价不可约求和项的双圆盘丛紧流形，本文利用相空间势垒论证证明余齐次1 Einstein度量不存在，并提出新的存在性障碍。

低维等余一流形上的正中间 Ricci 曲率

Positive intermediate Ricci curvature on cohomogeneity one manifolds in low dimensions

研究闭低维等余一流形上具有正中间 Ricci 曲率的不变度量的存在性，对特定作用构造了正第4中间 Ricci 曲率度量并证明不能有正第3中间 Ricci 曲率度量，同时建立了类似对称性障碍。

Isoparametric functions on exotic spheres

This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy $n$-sphere ($n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres \cite{St96} ( only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres ). As an application, we improve a beautiful result of Bérard-Bergery \cite{BB77} ( s…

Equivariant cohomology of cohomogeneity one actions

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce as corollaries several results concerning the usual (de Rham) cohomology of $M$, such as the following obstruction to the existence of a cohomogeneity one action: if $M$ admits a cohomogeneity one action, then $χ(M)>0$ if and only if $H^{\rm odd}(M)=\{0\}$.