Intermediate curvature and splitting theorem

Abstract

In this paper, we prove several rigidity results for complete noncompact manifolds with nonnegative intermediate curvatures. We show that when either $3\le n\le 5$, $1\le m\le n-1$, or $6\le n\le 7$, $m\in \{1,n-1,n-2\}$, any manifold of the topological type $M^{n-m}\times {\mathbb{T}}^{m-1}\times \mathbb{R}$ with nonnegative $m$-intermediate curvature is isometrically covered by the canonical product $M\times {\mathbb{R}}^m$. We also construct smooth metrics on $M^{n-m}\times {\mathbb{T}}^{m-1}\times \mathbb{R}$ with uniformly positive $m$-intermediate curvature for $6\le n\le 7$, $2\le m\le n-3$. This proves that the algebraic condition $m^2-mn+m+n>0$ from \cite{chenshuli_end} is sharp. The proof is based on a new recursion theorem for spectral intermediate curvatures and cylindrical splitting theorems. In particular, when $m=n-1$, this provides a new proof of some results by Chodosh--Li \cite{chodoshlisoapbubble} and Zhu \cite{zhu-splitting}. Moreover, the recursion theorem can be used to reprove the result of Brendle--Hirsch--Johne \cite{brendlegeroch'sconjecture}.

28 pages, comments welcome