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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.10809 |
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| _version_ | 1866916866511863808 |
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| author | Ketterer, Christian |
| author_facet | Ketterer, Christian |
| contents | 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 $K_F\in \mathbb R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10809 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Warped products over one-dimensional base spaces and the RCD condition Ketterer, Christian Differential Geometry Metric Geometry 51M15, 53C21, 49Q22 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 $K_F\in \mathbb R$. |
| title | Warped products over one-dimensional base spaces and the RCD condition |
| topic | Differential Geometry Metric Geometry 51M15, 53C21, 49Q22 |
| url | https://arxiv.org/abs/2506.10809 |