A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces

• 成果类型: Article
• 署名作者: Bethuel, Fabrice
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00911-3
• 发表日期: 2020
• 页码: 507-651
• 关键词: dirichlet energy; HARMONIC MAPS; approximation; dimension; TOPOLOGY; mappings

摘要

The present paper presents a counterexample to the sequential weak density of smooth maps between two manifolds M and N in the Sobolev space W-1,W-p(M, N), in the case p is an integer. It has been shown (see e.g. Bethuel in Acta Math 167:153-206, 1991) that, if p < dim M is not an integer and the [p]-th homotopy group pi([p])(N) of N is not trivial, [p] denoting the largest integer less then p, then smooth maps are not sequentially weakly dense in W-1,W-p(M, N). On the other hand, in the case p < dim M is an integer, examples of specific manifolds M and N have been provided where smooth maps are actually sequentially weakly dense in W-1,W-p(M, N) with pi(p)(N) not equal 0, although they are not dense for the strong convergence. This is the case for instance for M = B-m, the standard ball in R-m, and N = S-p the standard sphere of dimension p, for which pi(p)(N) = Z. The main result of this paper shows however that such a property does not holds for arbitrary manifolds N and integers p. Our counterexample deals with the case p = 3, dim M >= 4 and N = S-2, for which the homotopy group pi(3)(S-2) = Z is related to the Hopf fibration. We explicitly construct a map which is not weakly approximable in W-1,W-3(M, S-2) by maps in C-infinity(M, S-2). One of the central ingredients in our argument is related to issues in branched transportation and irrigation theory in the critical exponent case, which are possibly of independent interest. As a byproduct of our method, we also address some questions concerning the S-3-lifting problem for S-2-valued Sobolev maps.