论文标题
m {Ö}在任意维度的非平滑套件的Bius-invariant自避免能量
M{ö}bius-invariant self-avoidance energies for non-smooth sets in arbitrary dimensions
论文作者
论文摘要
在本文中，我们调查了O'Hara的MöbiusEnergy在曲线上的概括{ohara_1991a}，以对$ \ r^n $的$ \ r^n $的非平滑度子集进行任意维度和共同维度和共同维度的非平滑度子集。特别是，我们在轻度的假设下对可接受的可能无限制的$σ\ subset \ r^n $的局部平坦性表示，这些局部有限的能量意味着$σ$实际上是嵌入式的Lipschitz submanifold的$ \ r^n $ - 有时甚至是先前给出的其他常规设置）。另一方面，我们还证明，在集合$σ$上的低分数Sobolev规则性的本地图结构已经足以保证$σ$的有限能量。鉴于Blatt的特征\ Cite {Blatt_2012a}的oble {Blatt_2012a}，这种类型的Sobolev规律性正是封闭曲线上MöbiusEnergy的正确能量空间的{Blatt_2012a}。我们的结果尤其适用于Kusner和Sullivan的余弦能量$ e_ \ textnormal {ks} $ \ cite {kusner-sullivan_1997}，因为此处考虑的一种能量等同于$ e_ \ e_ \ textnormal {ks} $。
In the present paper we investigate generalizations of O'Hara's Möbius energy on curves \cite{ohara_1991a}, to Möbius-invariant energies on non-smooth subsets of $\R^n$ of arbitrary dimension and co-dimension. In particular, we show under mild assumptions on the local flatness of an admissible possibly unbounded set $Σ\subset \R^n$ that locally finite energy implies that $Σ$ is, in fact, an embedded Lipschitz submanifold of $\R^n$ -- sometimes even smoother (depending on the a priorily given additional regularity of the admissible set). We also prove, on the other hand, that a local graph structure of low fractional Sobolev regularity on a set $Σ$ is already sufficient to guarantee finite energy of $Σ$. This type of Sobolev regularity is exactly what one would expect in view of Blatt's characterization \cite{blatt_2012a} of the correct energy space for the Möbius energy on closed curves. Our results hold in particular for Kusner and Sullivan's cosine energy $E_\textnormal{KS}$ \cite{kusner-sullivan_1997} since one of the energies considered here is equivalent to $E_\textnormal{KS}$.
