Minimal-volume status of extremal minimal-complexity manifolds

Determine whether, for each even integer k, the unique manifold in the set M_{k,k} of minimal combinatorial complexity hyperbolic 3-manifolds with k torus cusps and connected totally geodesic boundary of genus k attains the minimal volume among all hyperbolic 3-manifolds with k cusps and connected totally geodesic boundary, and whether this minimizer is unique. Additionally, determine whether any manifold in the set M_{k+1,k} of minimal combinatorial complexity hyperbolic 3-manifolds with k torus cusps and connected totally geodesic boundary of genus k+1 attains the minimal volume within the class of hyperbolic 3-manifolds with k cusps and connected totally geodesic boundary.

Background

The paper studies hyperbolic 3-manifolds that are minimal with respect to a combinatorial complexity measure defined via ideal triangulations, focusing on the extremal cases with the smallest possible genus of totally geodesic boundary for a given number of cusps. The authors completely classify the manifolds in M_{k,k} (for k even) and M_{k+1,k}, describe their isometry groups, relate them via certain Dehn fillings, and compute commensurability invariants.

Using prior work on canonical decompositions and uniqueness of hyperbolic structures, they show that these minimal-complexity triangulations determine unique hyperbolic manifolds. They also give volume computations for the M_{k,k} manifolds (with linear growth in k). Motivated by known minimal volume results in related settings (e.g., Kojima–Miyamoto for compact manifolds with geodesic boundary and the first author's prior work for the 1-cusped case), the authors raise the explicit question of whether the extremal minimal-complexity manifolds also minimize volume in the corresponding classes.