Positive biharmonic functions
Bradley Beaver, Leo Sario, Cecilia Wang · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1982
Let Qbe the class of quasiharmonic functions 4, defined by Aq:1, Å:dö*åd, on a Riemannian manifold.The best source for counterexamples in the Q-classi- fication of Riemannian manifolds has been the Poincar6 tr[-ball A!, that is, the unit N-ball {x:(xl,...,xil;llx1=t} endowed with the Riemannian metric fuo: (l-lxl2)ldxl, a€R.In Sario-Wang [3] it was shown that for each of the classes QP, QB, QD, and QC of Q-functions which are positive, bounded, Dirichlet finite, or bounded Dirichlet finite, respectively, values d can be found for which these clas- ses are void.By contrast, the class QN of negative Q-functions on B| is not void for any a.Even the Euclidean plane R2, which is void of any other functions considered in classification theory, trivially carries QN-functions.It was, therefore, long thought that there may exist no Riemannian manifolds which carry no QN-functions.That this is, however, not the case, was shown in Nakai Sario [1].