Minkowski Inequality in Cartan-Hadamard Manifolds

Using harmonic mean curvature flow, we establish a sharp Minkowski-type lower bound for total mean curvature of convex surfaces with a given area in CartanHadamard 3-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic 3-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved 3-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan-Hadamard manifolds of any dimension.