Biharmonic and quasiharmonic degeneracy

Abstract

Among the vast complex of problems on inclusion relations between biharmonic and quasiharmonic null classes of Riemannian manifolds, we consider in the present paper perhaps the most intriguing case: Are there inclusion relations between Of 2c and OξLP? Here H , C, Q, L are the classes of functions which are nonharmonic biharmonic, bounded Dirichlet finite, quasiharmonic, or of finite L norm, respectively; a function u is biharmonic or quasiharmonic according as Δu=Q or Δu=l, with Δ the Laplace-Beltrami operator dδ+δd; for any two classes X, Y of functions, XY stands for Xr\Y, and 0%γ for the class of Riemannian iV-manifolds on which XY—φ. The classes H, Q, and L are not meaningful on Riemann surfaces, but are of great interest on Riemannian manifolds. It is known that both Of 2C and OQLP are strictly contained in Oξc, but whether or not there is an inclusion relation between O^c and OQLP ha-s been an open question. The purpose of the present paper is to show that the answer is in the negative. In particular, for any Λfe2 and any p^l, there exist Riemannian Λf-manifolds which carry QL functions but nevertheless fail to carry HC functions. For any null class 0^ of Riemannian Λf-manifolds, denote by 0 the complementary class. In Nos. 1 and 2, it is readily verified that the classes Of 2C Γ\OQLP, O%2cΓλθQLP, and 0%2CΓ\0$LP are all nonvoid. The interesting relation is OH^CΓΛOQLVΦΦ, for which we use two approaches, one in Nos. 3-6, the other in Nos. 7-10.