Existence shown for hypersurfaces with constant (n-2)-curvature

What the study found: The authors report existence of smooth complete hypersurfaces in hyperbolic space with constant $(n-2)$-curvature and a prescribed asymptotic boundary at infinity. They say this existence result now covers all possible curvature values.

Why the authors say this matters: The study suggests that the existence question is no longer limited to a restricted range of curvature values. The authors conclude that deriving curvature estimates was enough to extend the result to the full range.

What the researchers tested: The paper investigates hypersurfaces in hyperbolic space, a non-Euclidean space of constant negative curvature, under a constant $(n-2)$-curvature condition with a prescribed boundary at infinity. The approach is based on deriving curvature estimates.

What worked and what didn't: The authors state that the new curvature estimates allowed them to deduce existence for all possible curvature values. The abstract does not describe any failed cases or negative results.

What to keep in mind: The abstract gives no further technical details, and it does not describe limitations beyond noting that earlier work covered only a restricted range of curvature values.

Key points

• The paper reports existence of smooth complete hypersurfaces in hyperbolic space.
• These hypersurfaces have constant $(n-2)$-curvature.
• A prescribed asymptotic boundary at infinity is part of the problem setup.
• The result is said to cover all possible curvature values, not just a restricted range.
• The abstract says the conclusion was obtained by deriving curvature estimates.