Classification of Riemannian Manifolds in Conformally Invariant Potential Theory

In this survey, the classification theory of Riemann surfaces is generalized to Riemannian n–manifolds in the conformally invariant case. This leads to the study of the existence of https://www.w3.org/1998/Math/MathML">Ahttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq1814.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>–harmonic functions of type n with various properties and to an extension of the definition of the classical notions with inclusions O_G ⊂ O_HP ⊂ O_HB ⊂ O_HD. In the classical case the properness of the inclusions were proved rather late, in the 1950s by Ahlfors and Tôki. Here we present several examples that show such inclusions are proper also in the generalized case.