有限几何曲率能集的相切性质
Tangency properties of sets with finite geometric curvature energies
摘要：
我们研究了逆厚度$1/Δ$和积分Menger曲率能$\mathcal{U}_{p}^α$，$\mathcal{I}_{p}^α$和$\mathcal{M}_{p}^α$，发现当$p\geq α$和$\mathcal{I}_{p}^α$或$\mathcal{M}_{p}^α$分别表示$p\geq 2α$或$p\geq 3α$时，有限$1/Δ$或$\mathcal{U}_{p}^α$意味着在集合的所有点处存在一个近似$α$-切线，有限$\mathcal{I}_{p}^α$或$\mathcal{M}_{p}^α$意味着在集合的每个点处存在一个弱近似$α$ -切线，如果一些附加的密度属性成立的话。这包括尺度不变情况$p=2$用于$\mathcal{I}_{p}^{1}$和$p=3$用于$\mathcal{M}_{p}^{1}$，据我们所知，迄今为止还没有建立正则性。此外，我们证明，$α=1$这些指数是尖锐的，即，如果$p$低于尺度不变性的阈值，则存在一个包含没有（弱）近似1-切线的点的集合，但相应的能量仍然有限。对于$\mathcal{I}_{p}^{1}$和$\mathcal{M}_{p}^{1}$，我们给出了一个例子，其中一个点，没有近似1-切线，但有限的能量为所有的$p\in (0,\infty)$，从而表明存在弱近似1-切线是我们所能期望的，换句话说，我们的结果在这方面也是最优的。
英文摘要：
We investigate inverse thickness $1/Δ$ and the integral Menger curvature energies $\mathcal{U}_{p}^α$, $\mathcal{I}_{p}^α$ and $\mathcal{M}_{p}^α$, to find that finite $1/Δ$ or $\mathcal{U}_{p}^α$ implies the existence of an approximate $α$-tangent at all points of the set, when $p\geq α$ and that finite $\mathcal{I}_{p}^α$ or $\mathcal{M}_{p}^α$ implies the existence of a weak approximate $α$-tangent at every point of the set for $p\geq 2α$ or $p\geq 3α$, respectively, if some additional density properties hold. This includes the scale invariant case $p=2$ for $\mathcal{I}_{p}^{1}$ and $p=3$ for $\mathcal{M}_{p}^{1}$, for which, to the best of our knowledge, no regularity properties are established up to now. Furthermore we prove that for $α=1$ these exponents are sharp, i.e., that if $p$ lies below the threshold value of scale innvariance, then there exists a set containing points with no (weak) approximate 1-tangent, but such that the corresponding energy is still finite. For $\mathcal{I}_{p}^{1}$ and $\mathcal{M}_{p}^{1}$ we give an example of a set which possesses a point that has no approximate 1-tangent, but finite energy for all $p\in (0,\infty)$ and thus show that the existence of weak approximate 1-tangents is the most we can expect, in other words our results are also optimal in this respect.
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