Higher-dimensional Willmore-type inequality in Cartan–Hadamard manifolds

Determine whether every closed hypersurface of a Cartan–Hadamard n-manifold with n≥4 satisfies the inequality ∫_Γ (|H|/(n−1))^{n−1} ≥ |S^{n−1}|, where H is the sum of the principal curvatures and |S^{n−1}| is the Euclidean unit-sphere area.

Background

The paper proves a sharp constant-mean-curvature inequality in dimension five: a closed embedded hypersurface with constant positive mean curvature H in a Cartan–Hadamard 5-manifold has area at least (4/H)^4 times the area of the Euclidean unit 4-sphere. For such hypersurfaces, this is equivalent to the corresponding integral inequality with exponent four.

The authors note that the analogous integral inequality for arbitrary closed hypersurfaces is known in dimension three but remains unresolved in dimensions n≥4. Their dimension-five CMC result settles only the constant-mean-curvature case, not the general hypersurface problem.