Cecilia Wang

Positive harmonic functions and biharmonic degeneracy

The class 0HP of Riemann surfaces or Riemannian manifolds which do not carry (nonconstant) positive harmonic functions is the smallest harmonically or analytically degenerate class. In particular, it is strictly contained in the classes 0HB and 0HD of Riemann surfaces or Riemannian manifolds without bounded or Dirichlet finite harmonic functions, and in the classes 0AB and 0AD of Riemann surfaces without bounded or Dirichlet finite analytic functions. In the present paper we ask: Are there any relations between 0HP and the classes 0H2B and 0HzD of Riemannian manifolds without bounded or Dirichlet finite nonharmonic biharmonic functions? We shall show that the answer is in the negative. Explicitly, if 0 is a null class of AT-dimensional manifolds, and 0 its complement, then all four classes

Quasiharmonic functions on the Poincaré

Endowing an abstract Riemann surface with a conformai metric does not affect the harmonicity or the Dirichlet integral of a function on it. A fortiori, the classes 0G, 0HP, 0HB, 0HD of Riemannian 2-manifolds which do not carry Greens functions or harmonic functions which are positive, bounded, or Dirichlet finite, are invariant under varying conformai metrics. Here the harmonicity is defined by Aw = (dö + bd)u = 0, with d the exterior derivative, b the coderivative. In sharp contrast with the harmonic functions, the quasiharmonic functions [8], i.e., solutions of Aw = 1, on a Riemannian manifold are greatly affected by conformai metrics, and consequently so are the corresponding null classes 0QP, 0QB, and 0QD. A deep and interesting problem is to determine this dependence. We study the problem in a concrete setup which (a) permits explicit results and (b) yields applications to the general biharmonic classification theory of Riemannian manifolds. The present work is devoted to the former aspect. The latter aspect, an elaborate question in its own right, will be discussed in later studies (e.g. [1], [2], [10]-[16]). Among the phenomena that will be encountered we mention here the following striking result (Hada-Sario-Wang [1]): On the iV-ball \x\ < 1 with the metric ds — (1 — r) \dx\ there exist Dirichlet finite nonharmonic biharmonic functions if and only ii N ^ 10. The Riemannian manifold we choose for our present study of quasiharmonic functions is the iV-ball B% with the generalized Poincaré metric

N

The first purpose of this paper is to determine those values of the parameter a for which the class HD(B%), N ^ 3, of Dirichlet finite nonharmonic biharmonic functions on B% is nonvoid. In Sario-Wang [3] it was proved that HD(B%) # 0 for AT = 3ifandonlyifa > 3 / 5 , and the question was raised whether the same is true for every iVif and only if a > — 3/(N + 2). We show that this is indeed so if 3 :g N :g 6. However, quite unexpectedly, for N > 6 it turns out that HD(Ba) # 0 if and only if a e ( 3/( JV + 2), 5/(N 6)). The above result has interesting applications to the classification theory. Let Q be the class of quasiharmonic functions w, defined by Aw = 1, and denote by QD the subclass of Dirichlet finite functions in Q. The classes 0 G , 0QD, and 0HiD of Riemannian manifolds without Greens functions, gD-functions, and i/D-functions, respectively, have the following properties: (i) For every N, the classes 0QD and 0H2D decompose the totality of Riemannian iV-manifolds into three nonempty disjoint subclasses. (ii) For every TV, the class 0G — 0H2D is nonvoid. (iii) For N > 6, the classes 0G and 0H2D decompose the totality of Riemannian JV-manifolds into four nonempty disjoint subclasses. (iv) The unit N-ball with the natural metric (1 — |x|) \dx\ belongs to 0H2D if and only if N > 10. The proofs will appear in [1].

-ball

From the class A of analytic functions we proceed to the class H of harmonic functions. The latter are in a sense more flexible than the former and thus easier to treat. In particular the solvability of the Dirichlet problem makes it possible to obtain detailed information on the causes of degeneracy. On the other hand the lack of rigidity results in a great diversity of degeneracy phenomena. To subject them to a systematic treatment it is convenient to start with the class HD of harmonic functions with finite Dirichlet integrals and the corresponding null class 0 HD The close connection with Dirichlet’s principle makes the class 0 HD the most significant one among degeneracy classes related to H.