A spectral volume comparison for manifolds with weakly convex boundary
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916973805305856 |
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| author | Li, Jia |
| author_facet | Li, Jia |
| contents | We establish the Bonnet-Myers theorem and the Bishop-Gromov volume comparison theorem in the spectral sense for manifolds with weakly convex boundary. For $n\geq 3$, let $(M^n,g)$ be a simply connected compact smooth $n$-manifold with weakly convex boundary $\partial M$. If there exists a positive function $w\in C^{\infty}(M)$ that satisfies: \begin{equation*}
\begin{cases}
-\frac{n-1}{n-2}Δw+Λ_{\Ric} w\geq (n-1)w, \enspace in \enspace M,
\frac{\partial w}{\partial η}=0, \enspace\enspace\enspace\enspace \enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace \enspace\enspace \enspace\enspace on \enspace\partial M,
\end{cases} \end{equation*} where $Λ_{\Ric}$ denotes the smallest eigenvalue of the Ricci tensor, $η$ is the unit co-normal vector field of $\partial M$ in $M$, then the diameter of $M$ satisfies $\diam(M)\leq (\frac{\max w}{\min w})^{\frac{n-3}{n-1}}π$.\par If, in addition, $w$ attains its minimum on the boundary $\partial M$, we obtain a sharp upper bound for the volume of $M$: $\Vol(M)\leq \Vol(\bS^n_{+})$, with equality holding if and only if $M^n$ is isometric to the unit round hemisphere $\bS^{n}_{+}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03482 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A spectral volume comparison for manifolds with weakly convex boundary Li, Jia Differential Geometry We establish the Bonnet-Myers theorem and the Bishop-Gromov volume comparison theorem in the spectral sense for manifolds with weakly convex boundary. For $n\geq 3$, let $(M^n,g)$ be a simply connected compact smooth $n$-manifold with weakly convex boundary $\partial M$. If there exists a positive function $w\in C^{\infty}(M)$ that satisfies: \begin{equation*} \begin{cases} -\frac{n-1}{n-2}Δw+Λ_{\Ric} w\geq (n-1)w, \enspace in \enspace M, \frac{\partial w}{\partial η}=0, \enspace\enspace\enspace\enspace \enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace\enspace \enspace\enspace \enspace\enspace on \enspace\partial M, \end{cases} \end{equation*} where $Λ_{\Ric}$ denotes the smallest eigenvalue of the Ricci tensor, $η$ is the unit co-normal vector field of $\partial M$ in $M$, then the diameter of $M$ satisfies $\diam(M)\leq (\frac{\max w}{\min w})^{\frac{n-3}{n-1}}π$.\par If, in addition, $w$ attains its minimum on the boundary $\partial M$, we obtain a sharp upper bound for the volume of $M$: $\Vol(M)\leq \Vol(\bS^n_{+})$, with equality holding if and only if $M^n$ is isometric to the unit round hemisphere $\bS^{n}_{+}$. |
| title | A spectral volume comparison for manifolds with weakly convex boundary |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2503.03482 |