AI 中文总结
本研究针对标准化的闭有向双曲三维流形$M^3$，证明了$M^3 \times \mathbb{S}^1$中满足标量曲率$\operatorname{Sc}_g \geq -6$的黎曼度量下，代表切片类的闭嵌入超曲面体积有精确下界，并完成了等号情形的分类。
AI 中文摘要
设$(M^3, g_{\mathrm{hyp}})$是闭的有向双曲三维流形，标准化为其截面曲率$\operatorname{sec}_{g_{\mathrm{hyp}}} \equiv -1$。我们证明了$M^3 \times \mathbb{S}^1$中代表切片类$[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \mathbb{S}^1; \mathbb{Z})$的超曲面体积的精确下界，并对等号成立的情形进行了分类。若$g$是$M^3 \times \mathbb{S}^1$上的光滑黎曼度量，且标量曲率$\operatorname{Sc}_g \geq -6$，则每个代表切片类$[M^3\times\{\mathrm{pt}\}]$的闭嵌入超曲面$\Sigma$都满足$\operatorname{vol}_g(\Sigma) \geq \operatorname{vol}_{g_{\mathrm{hyp}}}(M^3)$。该下界可由乘积度量$g_{\mathrm{hyp}}+h$达到，其中$h$是$\mathbb{S}^1$上的任意度量。反之，若某个$\Sigma$满足等号，则在保持切片类的微分同胚下，$g=g_{\mathrm{hyp}}+h$且$\Sigma= M^3 \times \{\mathrm{pt}\}$。
英文摘要
Let $(M^3, g_{\mathrm{hyp}})$ be a closed oriented hyperbolic three-manifold normalized so that $\operatorname{sec}_{g_{\mathrm{hyp}}} \equiv -1$. We prove a sharp lower bound for the volume of hypersurfaces in $M^3 \times \mathbb{S}^1$ representing the slice class $[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \mathbb{S}^1; \mathbb{Z})$, and we classify the equality case. If $g$ is a smooth Riemannian metric on $M^3 \times \mathbb{S}^1$ with the scalar curvature $\operatorname{Sc}_g \geq -6$, then every closed embedded hypersurface $Σ$ representing the slice class $[M^3\times\{\mathrm{pt}\}]$ satisfies $\operatorname{vol}_g(Σ) \geq \operatorname{vol}_{g_{\mathrm{hyp}}}(M^3)$. The bound is attained by the product metric $g_{\mathrm{hyp}}+h$, with $h$ any metric on $\mathbb{S}^1$. Conversely, if equality holds for some $Σ$, then up to a diffeomorphism preserving the slice class, $g=g_{\mathrm{hyp}}+h$ and $Σ= M^3 \times \{\mathrm{pt}\}$.
Comments15 pages
