The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

中文翻译：

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渐近锥中的Dehn函数和Hölder扩展

Dehn函数测量填充空间中闭合曲线的最小圆盘的面积；它在分析，几何和几何群论中是一个重要的不变式。有几种等效的方法来定义Dehn函数，具体取决于所使用的光盘类型。在本文中，我们介绍了Dehn函数的新定义，并用它来证明几个定理。首先，我们将Dehn函数的拟等距不变性推广到宽泛的一类空间。其次，我们证明了具有二次Dehn函数及其渐近锥的空间的Hölder扩展性质。最后，我们证明具有二次Dehn函数的空间的超极限和渐近锥也具有二次Dehn函数。我们的结果证明依赖于最近存在和规律性结果，以求度量空间中的Sobolev映射最小化。