Minkowski Inequality in Cartan–Hadamard Manifolds.

The article focuses on establishing a sharp Minkowski-type lower bound for the total mean curvature of convex surfaces with a given area in Cartan–Hadamard 3-manifolds, which are complete simply connected Riemannian spaces of nonpositive curvature. Using harmonic mean curvature flow, the authors prove that for a smooth strictly convex surface \(\Gamma\) in such a manifold with curvature \(K \leq a \leq 0\), the total mean curvature \(\mathcal{M}(\Gamma)\) satisfies \(\mathcal{M}(\Gamma) \geq \sqrt{16\pi|\Gamma| - 2a|\Gamma|^2}\), with equality only if the enclosed domain is isometric to a Euclidean ball. This result generalizes Minkowski's classical inequality from Euclidean space and improves known estimates in hyperbolic space. Additionally, the authors derive a Bonnesen-style isoperimetric inequality for surfaces with convex distance functions in nonpositively curved 3-manifolds, relating volume, surface area, and inradius, and extend these connections to higher-dimensional Cartan–Hadamard manifolds under suitable assumptions. The work employs geometric flow techniques, approximation arguments for nonsmooth convex surfaces, and curvature comparison theorems, providing new insights into curvature inequalities and isoperimetric problems in nonpositively curved spaces.