A Sharp Hyperbolic Volume Bound for Hypersurfaces in M3 × 𝕊1

Let [Formula: see text] be a closed oriented hyperbolic three-manifold normalized so that [Formula: see text]. We prove a sharp lower bound for the volume of hypersurfaces in [Formula: see text] representing the slice class [Formula: see text], and we classify the equality case. If [Formula: see text] is a smooth Riemannian metric on [Formula: see text] with scalar curvature [Formula: see text], then every closed embedded hypersurface [Formula: see text] representing the slice class [Formula: see text] satisfies [Formula: see text]. The bound is attained by the product metric [Formula: see text], with [Formula: see text] any metric on [Formula: see text]. Conversely, if equality holds for some [Formula: see text], then up to a diffeomorphism preserving the slice class, [Formula: see text] and [Formula: see text].