Intermediate curvature and splitting theorem
In this paper, we prove several rigidity results for complete noncompact manifolds with nonnegative intermediate curvatures. We show that when either $3\leq n\leq 5$, $1\leq m\leq n-1$, or $6\leq n\leq 7$, $m\in \{1,n-1,n-2\}$, 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 $6\leq n\leq 7$, $2\leq m\leq n-3$. 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}.