Let \((M^n,g)\) be simply connected, complete, with non-positive sectional curvatures, and \(\Sigma \) 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 \(\partial S = \Sigma \). We use a weak mean curvature flow, obtained via elliptic regularisation, starting from \(\Sigma \), to show that S satisfies the optimal Euclidean isoperimetric inequality: \( 6 \sqrt{\pi }\, \mathbf {M}[S] \le (\mathbf {M}[\Sigma ])^{3/2} \). 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 \(L^2\).

中文翻译：

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Cartan-Hadamard流形中任何维数的曲面的最佳等式不等式

让\（（M ^ n，g）\）简单地连接起来，具有非正截面曲率，并且\（\ Sigma \）二维封闭积分电流（或扁平链模2），具有紧凑的支撑中号。令S为最小化积分3电流的面积（分别为扁平链mod 2），使得\（\ partial S = \ Sigma \）。我们使用从\（\ Sigma \）开始的通过椭圆正则化得到的弱平均曲率流来证明S满足最优的欧氏等距不等式：\（6 \ sqrt {\ pi} \，\ mathbf {M} [ S] \ le（\ mathbf {M} [\ Sigma]）^ {3/2} \）。如果M的截面曲率从上方受限制，我们还可以获得最佳估计\（-\ kappa <0 \）并刻画相等的情况。证明是基于合适的等渗差沿更高的一维沿近似流动的几乎单调性和二维积分杂波的Willmore能量的最优估计，其第一变化可累加为（（L ^ 2 \））。