Homomorphism between homotopy groups induced by elements of the Sobolev space \(L^{1,p}(M,N)\)
Let \(M\), \(N\) be Riemannian manifolds. Let \(L^{1,p} (M,N)\) denote the Sobolev space of \(L^ p\)-elements whose derivatives belong also to \(L^ p\) \((p>0)\). \textit{R. Schoen} and \textit{S. T. Yau} [Ann. Math., II. Ser. 110, 127-142 (1979; Zbl 0431.53051)] proved that for any element \(f\in L^{1,2} (M,N)\), the homomorphism \(f_ \#: \pi_ 1(M)\to \pi_ 1(N)\) between the fundamental groups is well-defined. \textit{B. White} [Acta Math. 160, No. 1/2, 1-17 (1988; Zbl 0647.58016)] gave a generalization of this result: Theorem 1. For any element \(f\) of the Sobolev space \(L^{1,p} (M,N)\), the homomorphism \(f_ \#: \pi_ i(M)\to \pi_ i(N)\) is well-defined for \(i=1,2,\dots, [p-1]\), where [ ] denotes the Gauss symbol, i.e., \([p- 1]\) is the largest integer not greater than \(p-1\)). In this note we give a simple proof, different from White's, of Theorem 1 along the lines of the argument of Schoen and Yau.
\(L^ p\)-maps between Riemannian manifolds
scientific article; zbMATH DE number 490666