Uniform harmonic approximation on Riemannian manifolds
This paper deals with uniform approximation by harmonic functions on closed subsets of a Riemannian manifold. The purpose is to generalize results previously known in the context of Riemann surfaces or Euclidean space \(\mathbb{R}^ n\). After a substantial introduction to harmonic functions on Riemannian manifolds the authors establish several results concerning localization and approximation. Let \(F\) be a closed subset of an orientable Riemannian manifold \(\Omega\), and et \(H(F)\) denote collection of functions on \(F\) which can be extended to a harmonic function on some open set containing \(F\). Further, let \(\overline{H}(F)\) denote the collection of functions in \(C(F)\) which can be uniformly approximated by functions in \(H(F)\), and let \(A(F)= C(F)\cap H(F^ 0)\). The localization theorem is as follows. Let \(f\in C(F)\). Then \(f\in \overline{H}(F)\) if and only if for each point \(p\) in \(F\) there is a compact neighbourhood \(K_ p\) of \(p\) such that the restriction of \(f\) to \(F\cap K_ p\) belongs to \(\overline{H} (F\cap K_ p)\). This is then used to prove that \(A(F)= \overline{H} (F)\) if and only if \(\Omega\setminus F\) and \(\Omega \setminus F^ 0\) are thin at the same points. The paper also considers the possibility of approximation on \(F\) by functions which are harmonic on \(\Omega\) except for isolated singularities lying in a preassigned set.
scientific article; zbMATH DE number 569429
scientific article