proceedings of the american mathematical society Volume 112, Number 2, June 1991 HARMONIC TWO-FORMS IN FOUR DIMENSIONS WALTER SEAMAN (Communicated by Jonathan M. Rosenberg) Abstract. Conformai invariance of middle-dimensional harmonic forms is used to improve Kato's inequality for four-manifolds. An application to posi- tively curved four-manifolds is given. 0. Introduction The purpose of this paper is to prove the following: Theorem 1. Let (M , g) be a four-dimensional Riemannian manifold. Let co be a harmonic two-form on (M, g). Then co satisfies the pointwise inequality: (0.1) \Vco\2>\\d\co\\2. Kato's inequality [1, p. 130], states that if E is a Riemannian vector bundle with connection V over a Riemannian manifold M, then any smooth section s of E, satisfies the pointwise inequality: (0.2) \VS\2 >\d\s\\2. Now by definition, if s(co) vanishes at p e M, then d\s\(d\co\) = 0 at p. Thus, (0.1) and (0.2) are automatically valid at such a point. At points where co does not vanish (0.1) can be thought of as a quantitative improvement of (0.2), for the case of harmonic two-forms on four-dimensional manifolds. As an application of the above theorem, we prove: Theorem 2. Let (M , g) be a compact, connected four-dimensional Riemannian manifold whose sectional curvature K(g) satisfies 1 > K(g) > ô. If (0.3) Ô > 1/(3(1 + 3 • 2l/4/51/2)1/2 + 1) « .1714 then M is definite. This theorem represents an improvement of results starting with [2] followed by [4, 7, 6]. The relevance of Theorems 1 and 2 stems from the following facts Received by the editors November 6, 1989 and, in revised form, July 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C20; Secondary 57N13. Key words and phrases. Harmonic forms, Weitzenbock operator, curvature, four-manifolds. ©1991 American Mathematical Society 0002-9939/91 $1.00+ $.25 per page 545 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use