Non-recursive functions, knots “with thick ropes,” and self-clenching “thick” hyperspheres
Non-recursive functions, knots “with thick ropes,” and self-clenching “thick” hyperspheres
复制标题
非递归函数、“粗绳”结和自紧“粗”超球面
DOI：
10.1002/cpa.3160480402
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发表时间：
2010
影响因子：
3
通讯作者：
A. Nabutovsky
中科院分区：
文献类型：
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作者：
A. Nabutovsky
利用n维球面对n≥5的算法不可识别性，我们给出了一种解决某些几何变分问题的方法。有时，这种方法可以证明一个所考虑的变分问题存在无穷多个解。本文将这种递归方法应用于Rn+1中c1.1-光滑超曲面空间上的一类泛函，其中n是任意固定数≥5.其中，最简单的泛函Kv由公式Kv(Σn)=(Vol(Σn))1/n/r(Σn)定义，其中r(Σn)表示Σn⊂Rn+L的正态指数映射的内射性半径.我们证明了对于任意n≥5，C1.1-光滑超球面空间上Kv的局部极小值的无穷集合的存在性.
泛函KV是在试图推广纽结理论以处理“厚”圆和更一般的“厚”球体在欧氏空间中的嵌入和同位素时自然产生的。我们引入了“粗绳结”类型的概念。“粗绳”结的理论与经典的“粗绳结”理论有很大的不同，其结果是：对于大于或等于5的每一维，存在一个余维为1的非平凡的“粗绳”结的无限集合。
We introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n-dimensional sphere for n ≥ 5. Sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem. This recursion-theoretic approach is applied in this paper to a class of functionals on the space of C1.1-smooth hypersurfaces diffeomorphic to Sn in Rn+1, where n is any fixed number ≥ 5. The simplest of these functionals kv is defined by the formula kv(Σn) = (vol(Σn))1/n/r(Σn), where r(Σn) denotes the radius of injectivity of the normal exponential map for Σn ⊂ Rn+l. We prove the existence of an infinite set of distinct locally minimal values of kv on the space of C1.1-smooth topological hyperspheres in Rn+1 for any n ≥ 5.
The functional kv naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of “thick” circles and, more generally, “thick” spheres into Euclidean spaces. We introduce the notion of knot “with thick rope” types. The theory of knot “with thick rope” types turns out to be quite different from the classical knot theory because of the following result: There exists an infinite set of non-trivial knot “with thick rope” types in codimension one for every dimension greater than or equal to five.