Optimal isoperimetric inequalities for 2-dimensional surfaces in Hadamard-Cartan manifolds in any codimension

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\mathrm{\Sigma }$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp.~flat chain mod 2) such that $\partial S=\mathrm{\Sigma }$. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $\mathrm{\Sigma }$, to show that S satisfies the optimal Euclidean isoperimetric inequality: $6\sqrt{\pi }\mathbf{M}[S]\le (\mathbf{M}[\mathrm{\Sigma }]{)}^{3/2}$. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by $-\kappa <0$ and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in $L^2$.