Minimal-volume status and uniqueness for minimal-complexity manifolds in M_{k,k} and M_{k+1,k}

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

Background

The paper classifies compact, connected, orientable hyperbolic 3-manifolds of minimal combinatorial complexity with connected totally geodesic boundary and k cusps, focusing on the extremal cases M_{k,k} (even total complexity) and M_{k+1,k} (odd total complexity). It proves that, for k even, M_{k,k} contains a unique homeomorphism type and determines its isometry group and commensurability invariants; for M_{k+1,k}, it classifies all homeomorphism types and their isometry groups.

Motivated by observed coincidences between minimal combinatorial complexity and minimal volume in known examples (e.g., results of Kojima–Miyamoto and related work showing minimal volume in small cases such as M_{2,0} and M_{2,1}), the authors raise the question of whether these minimal-complexity manifolds also minimize volume in the broader class of hyperbolic 3-manifolds with k cusps and connected totally geodesic boundary. The question is posed explicitly for both the unique element of M_{k,k} and for any of the manifolds in M_{k+1,k}.

An affirmative resolution would extend known volume-minimization phenomena beyond previously studied low-volume regimes to the entire families M_{k,k} and M_{k+1,k}, and would address uniqueness of minimizers in these classes.