Biharmonic classification of Riemannian manifolds

Abstract

In the study of the structure of the space of biharmonic functions it is often necessary to impose some nondegeneracy condition on the base manifold with respect to quasiharmonic functions (cf. [2], [4]). For this reason it is useful to introduce various quasiharmonically degenerate classes of Riemannian manifolds and to investigate relations among them. This is the purpose of the present note. 1. Quasiharmonic degeneracy. Let R be a noncompact orientable C?-manifold of dimension m _ 2 with C?? Riemannian metric ds2 = gij dxidxj. The corresponding Laplace-Beltrami operator is (1) A = g1/2jL()g1/2gii.) i=l Ax j=1 AXi where g = det (gj) and (gij) = (gij)-l. We call a function u quasiharmonic if Au = const $ 0, and denote by Q = Q(R) the class of quasiharmonic functions u on R normalized by Au = 1. Such functions are superharmonic on R. Following [5], we denote the classes of nonnegative, bounded, and Dirichlet finite functions by P, B, and D respectively, and we set BD = B n D. Similarly we write QX (X = P, B, D, or BD) for Q n X. We are interested in the question as to when QX = 0. 2. Characterization of null classes. We denote by OQx the class of Riemannian manifolds R on which QX(R) = 0, and by OG the class of parabolic manifolds. Let G(x, y) = GR(X, y) be the harmonic Green's function on R ? OG, and set G(x, y) = oo on R E OG. Denote by dy the Riemannian volume element (g(y))1/2 dyl ... dym. Received by the editors September 11, 1970. AMS 1970 subject classifications. Primary 31B30. 1 The work was sponsored by the U.S. Army Research Office-Durham, Grant DA-ARO-D-31-124-70-G7, University of California, Los Angeles. ? American Mathematical Society 1972 165 This content downloaded from 157.55.39.177 on Tue, 15 Nov 2016 03:53:17 UTC All use subject to http://about.jstor.org/terms 166 MITSURU NAKAI AND LEO SARIO [January THEOREM 1. The classes ?QX are characterized in terms of G(x, y) as follows: R e OQP if and only if fG(x, Y) dy = oo; R eOQB if and only if a = sup G(x, y) dy = oo; XER (2) x ReOQD if and only if b= G(x,y)dxdy = oo; R XR ReOQBD if andonlyif a=oo or b=oo. 3. Proof. Let u e Q(R) and take a regular subregion Q of R containing a given point x. Denote by HQ the harmonic solution of the Dirichlet problem on Q with boundary values u. By Stokes' formula, f [(u(y) Hu(y)) A8GQ(x, y) Gn(x, y) A/\g(u(y) H'(y))] dy (3) [(U(Y)HU(y)) * d Gn(x, y) AQ-aB -Gn(x, y) * dj(u(y) Hu(y))] with B a small geodesic ball about x of radius 8. On letting e -+ 0, we obtain the Riesz representation (4) u(x) = H(x) + Gn(x, y) dy on Q (cf., e.g., [1]). Again by Stokes' formula, D({ GQ(., y) dy) =f (fGn(x, y) dy /Xx{Gn(x, y) dy dx =f Gn(x, y) dx dy. We conclude that (6) DQ(u) = Dn(H) + f GQ(x, y) dx dy. nxn If SR G(x, y) dy exists, then it is of class C2, and AX G(x, y) dy = 1 (cf., e.g., [3]). On letting Q -+ R in (5), we obtain (7) DR G(, Y) dy) f?G(X, y) dx dy < o0. This content downloaded from 157.55.39.177 on Tue, 15 Nov 2016 03:53:17 UTC All use subject to http://about.jstor.org/terms 1972] QUASIHARMONIC CLASSIFICATION 167 Suppose there exists a u E QP. Since u is positive and superharmonic on R, h(x) = limn,R Hn(x) exists and a fortiori G(x, y) dy = yim GQ(X, Y) dy = u(x) h(x) < oo. If u E QB, then since IH(x)l < sup0n lul for every x E? Q, (4) implies {GQ(,Y)dY < 2supJul R and consequently a < oo. If u E QD, then (6) and (7) give b < oo. A fortiori, if u E QBD, then a < xo and b < oo. Conversely let v(x) = fR G(x, y) dy. If it is finite or bounded on R, then v E QP or QB. If b < oo, then v(x) is finite and (7) implies v E QD. Consequently, if a < oo and b < oo, then v E QBD. 4. Strict inclusion relations. By means of the characterizations in Theorem 1 we shall prove: THEOREM 2. The following strict inclusion relations are valid: