Warped products over one-dimensional base spaces and the RCD condition

Abstract

We prove the Riemannian curvature-dimension condition $\mathsf{RCD}(KN,N+1)$ for an $N$-warped product $B{\times }_f^{N}F$ over a one-dimensional base space $B$ with a Lipschitz function $f:B\to {\mathbb{R}}_{\ge 0}$, provided (1) $f$ is a $Kf$-concave function, (2) $f$ satisfies a sub-Neumann boundary condition $\frac{\partial f}{\partial n}\ge 0$ on $\partial B\backslash f^{-1}(0)$ and $F$ is a compact metric measure space satisfying (3) the condition $\mathsf{RCD}(K_F(N-1),N)$ with $K_F:={\sup }_B\{(Df{)}^2+K{f}^2\}$. The result is sharp, i.e. we show that (1), (2) and (3) are necessary for the validity of statement provided $K_F\ge 0$. In general, only a weaker statement is true. If $f$ is assumed to be $Kf$-affine, then the condition $\mathsf{RCD}(KN,N+1)$ for the $N$-warped product holds if and only if the condition $\mathsf{RCD}(K_F(N-1),N)$ holds for $F$ for any $K_F\in \mathbb{R}$.

arXiv comment52 pages, minor changes