The question asks whether area-minimizing cones and tensor varieties, when lifted to the universal cover of a closed hyperbolic manifold, satisfy a uniform comparison theorem relating the volume of metric balls centered on these varieties to the corresponding volumes in a fixed model space, with constants depending only on dimension and not on the particular hyperbolic manifold chosen. Concretely, one wants to know if the Bishop-Gromov volume comparison framework, which governs hyperbolic ball volumes on universal covers as in the first paper, can be extended so that it controls the induced volumes on embedded area-minimizing subvarieties in a dimension-independent and manifold-independent way.
This problem is genuinely open because the two ingredients, namely the global volume comparison on hyperbolic universal covers and the local stability theory of area-minimizing tensor varieties, come from largely separate toolkits. The Bishop-Gromov inequality exploits global curvature bounds in an ambient sense, while the regularity and volume estimates for area-minimizing varieties rely on monotonicity formulas that are typically stated for Euclidean ambient spaces or, at best, for manifolds with bounded sectional curvature. Combining them requires controlling how the curvature of the hyperbolic ambient space distorts the monotonicity ratio of the area-minimizing subvariety as one moves through the universal cover, and it is not clear that any such distortion remains uniformly bounded independent of the choice of hyperbolic manifold or the combinatorial complexity of the tensor variety.
Resolving this would have significant consequences. A positive answer would give a new family of uniformly controlled geometric objects in hyperbolic geometry, potentially leading to compactness theorems for sequences of area-minimizing subvarieties in degenerating hyperbolic manifolds, with applications to the geometry of moduli spaces and arithmetic hyperbolic manifolds. A negative answer, exhibiting families where no such uniform comparison exists, would sharply delineate the boundary between what hyperbolic curvature controls globally and what it fails to control along minimizing submanifolds, illuminating obstructions in geometric measure theory on negatively curved spaces.
The connection to the source papers is direct and threefold. The first paper establishes the uniform comparison of hyperbolic ball volumes on universal covers, providing the ambient geometric framework. The fourth paper constructs explicit families of area-minimizing tensor varieties and studies their stability, providing the subvarieties whose volume behavior is in question. The fifth paper on conic Laplacians is relevant because singular points of tensor varieties, where the geometry becomes cone-like, are precisely where the standard monotonicity arguments break down and where a new analytic framework analogous to conic spectral theory would be needed to restore uniform estimates.