Homotopy classes in Sobolev spaces and energy minimizing maps. (English) Zbl 0595.58011
The author, firstly, points out that for those M such that \(\pi_ 1(M)\) and \(\pi_ 2(M)\) are both trivial, the identity map is homotopic to maps of arbitrarily small energy. Secondly, the author considers an energy functional like \(\Phi (f)=\int_{M}| Df|^ p\) and proves the following results: The infimum of \(\Phi\) (g) among Lipschitz maps \(g: M\to N\) homotopic to a given Lipschitz map \(f: M\to N\) is equal to the infimum of \(\Phi\) (g) among all Lipschitz maps that are merely [p]- homotopic to f. In particular, the infimum is 0 if and only if the restriction of f to the [p]-skeleton of M is homotopically trivial. The infimum of \(\Phi\) (g) among all maps \(g\in L^{1,p}(M,N)\) with a given [p-1]-homotopy type is attained.
Reviewer: Shen Yibing
MSC:
| 58E20 | Harmonic maps, etc. |
| 55P10 | Homotopy equivalences in algebraic topology |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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