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

TL;DR

This paper establishes conditions under which warped products over one-dimensional bases satisfy the Riemannian curvature-dimension condition, providing sharp criteria involving the warping function and the fiber space.

Contribution

It proves the RCD condition for warped products over one-dimensional bases with specific curvature and boundary conditions, and shows these conditions are necessary and sufficient in certain cases.

Findings

The RCD condition holds under specified concavity and boundary conditions on the warping function.

The criteria are sharp, meaning necessary and sufficient in particular cases.

The results extend the understanding of curvature conditions in warped product spaces.

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 $f: B\rightarrow \mathbb R_{\geq 0}$, provided (1) $f$ is a $Kf$-concave function, (2) $f$ satisfies a sub-Neumann boundary condition $\frac{\partial f}{\partial n}\geq 0$ 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 $K_F:= \sup_B \{ (Df)^2 + Kf^2\}$. The result is sharp, i.e. we show that (1), (2) and (3) are necessary for the validity of statement provided $K_F\geq 0$. In general, only a weaker statement is true. If $f$ is assumed to be $Kf$-affine, then the condition $\mathsf{RCD}(K N, N+1)$ for the $N$-warped product holds if and only if the condition $\mathsf{RCD}(K_F(N-1), N)$ holds for $F$ for any $K_F\in \mathbb R$.