Upper and lower bounds and modulus of continuity of decomposed Mobius energies
Upper and lower bounds and modulus of continuity of decomposed Mobius energies
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分解莫比乌斯能量的上下界和连续模
DOI：
10.1007/s12220-020-00496-x
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Takeyuki Nagasawa
中科院分区：
文献类型：
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作者：
Aya Ishizeki;Takeyuki Nagasawa
Möbius能量是结能之一，因其Möbius不变性而得名。众所周知，它有几种不同的表达方式。一个是用保角余弦来表示的，叫做余弦公式。另一种是分解为Möbius不变部分，称为分解的Möbius能量。因此余弦公式是分解能量的和。这就提出了一个问题。每个分解的能量可以用余弦公式来估计吗？这里我们给出一个肯定的答案：分解部分的上界、下界和连续模可以用余弦公式求出。此外，我们还提供了两条曲线之间的分解能量差的估计，以Möbius不变量的形式。
The Möbius energy is one of the knot energies, and is named after its Möbius invariant property. It is known to have several different expressions. One is in terms of the cosine of conformal angle, and is called the cosine formula. Another is the decomposition into Möbius invariant parts, called the decomposed Möbius energies. Hence the cosine formula is the sum of the decomposed energies. This raises a question. Can each of the decomposed energies be estimated by the cosine formula? Here we give an affirmative answer: the upper and lower bounds, and modulus of continuity of decomposed parts can be evaluated in terms of the cosine formula. In addition, we provide estimates of the difference in decomposed energies between the two curves in terms of Möbius invariant quantities.