A sharp hyperbolic volume bound for hypersurfaces in $M^3\times {\mathbb{S}}^1$

Abstract

Let $(M^3,g_{\mathrm{hyp}})$ be a closed oriented hyperbolic three-manifold normalized so that ${\sec }_{g_{\mathrm{hyp}}}\equiv -1$. 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 ${\mathrm{Sc}}_g\ge -6$, then every closed embedded hypersurface $\Sigma$ representing the slice class $[M^3\times \{\mathrm{pt}\}]$ satisfies ${\mathrm{vol}}_g(\Sigma )\ge {\mathrm{vol}}_{g_{\mathrm{hyp}}}(M^3)$. 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 $\Sigma$, then up to a diffeomorphism preserving the slice class, $g=g_{\mathrm{hyp}}+h$ and $\Sigma =M^3\times \{\mathrm{pt}\}$.

15 pages