Math. Z. 179, 535-554 (1982) Mathematische Zeitschrift 9 Springer-Verlag 1982 Subharmonic Functions on Real and Complex Manifolds Leon Karp* School of Mathematics, Institute for Advances Study, Princeton, New Jersey 08540, U.S.A. 1. Introduction In recent years a number of new results have been obtained concerning the global properties of harmonic, subharmonic, and holomorphic functions on noncompact manifolds. In particular, Greene and Wu [19] (cf. [17]) proved the following Theorem (Greene-Wu). Let M be a complete noncompact Riemannian manifold of positive sectional curvature. If u is a continuous nonnegative subharmonic function (for a definition see below, Sect. 2) on M, then for any p > 1 there exist positive constants C and ro such that uPdvol> C(r-ro) for all r>r o. (1.1) B(xo;r) Here r is geodesic distance from a fixed point x o and B(xo; r) is the geodesic ball of radius r and center x o. A similar divergence rate was obtained by Yau [43] without curvature assumptions but under the restrictions that p > 1, u ~ constant and u~ C 2. More- over, it is remarked in the appendix to [43] that, by applying the approxima- tion theorems of Greene-Wu [18], the smoothness restriction, ueC 2, may be removed. We present (see below, Sect. 2) new and essentially optimal growth rates under these same assumptions. As a special case of our results we have the following quantitative version of the triviality of Lp harmonic or non- negative subharmonic functions on a Riemannian manifold: Theorem A. If u is a nonconstant (a) harmonic or (b) nonnegative continuous subharmonic function on a complete noncompact Riemannian manifold M, then for any p>l and xo~M 1 lim supr~ r 2 logr B( J';r)~o ]u]Pd vol= + oo * Research supported in part by National Science Foundation grant MCS 76-23465 0025- 5874/82/0179/0535/$04.00