Sobolev Critical Exponents of Rational Homotopy Groups
Sobolev Critical Exponents of Rational Homotopy Groups
复制标题
有理同伦群的索博列夫临界指数
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
T. Rivière
中科院分区：
文献类型：
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作者：
T. Rivière
摘要：给定闭单连通流形Nn，整数p和元素z ∈（πp（N），定义了z的代数和解析Sobolev临界指数νz和μz. z的代数Sobolev临界指数νz与使用Sullivan的最小模型构造表示z的树中所需的最小分支数有关。z的解析Sobolev临界指数μz与当z（u）趋于+∞时，从p−球面Sp到Nn的映射u中的最小W1，p范数<$u <$W1，p的对数增加有关。我们研究了νz和μz之间的关系，并描述了这些指数在光滑映射逼近流形之间的Sobolev映射问题中所起的关键作用。这些指数及其与Sobolev空间中逼近问题的联系最初是由Robert Hardt和作者在[HR 3]中引入的。
Abstract : A closed simply connected manifold Nn, an integer p and an element z ∈ (πp(N) ⊗ Q)∗ being given, we define the algebraic and the analytic Sobolev critical exponent νz and μz of z. The algebraic Sobolev critical exponent νz of z is related to the minimal number of branchs needed in trees representing z using the minimal model construction of Sullivan . The analytic Sobolev critical exponent μz of z is related to the logarithmic increase of the minimal W 1,p norm ‖u‖W 1,p among the maps u from the p−sphere Sp into Nn as z(u) goes to +∞. We study the relation between νz and μz and we describe the crucial role played by these exponents in the problem of approximating Sobolev maps between manifolds by smooth maps. These exponents and their connection with the approximation problem in Sobolev Spaces were originally introduced by Robert Hardt and the author in [HR3].