Abstract

This paper is a continuation of the study of some rigidity or non-existence issues discussed in [An1, x6]. The results obtained here also play a signicant role in the approach to geometrization of 3-manifolds discussed in [An4]. Let N be an oriented 3-manifold and consider the functional R 2 (g) = Z N jr g j 2 dV g ; (0.1) on the space of metrics M on N where r is the Ricci curvature and dV is the volume form. The Euler-Lagrange equations for a critical point of R 2 read rR 2 = D Dr +D 2 s 2 R r 1 2 (s jrj 2 ) g = 0; (0.2) s = 1 3 jrj 2 : (0.3) Here s is the scalar curvature, D 2 s the Hessian of s, s = trD 2 s the Laplacian, and R the action of the curvature tensor R on symmetric bilinear forms, c.f. [B, Ch.4H] for further details. The equation (0.3) is just the trace of (0.2). It is obvious from the trace equation (0.3) that there are no non-at R 2 critical metrics, i.e. solutions of (0.2)-(0.3), on compact manifolds N ; this follows immediately by integrating (0.3) over N . Equivalently, since the functional R 2 is not scale invariant in dimension 3, there are no critical metrics g with R 2 (g) 6= 0. To obtain non-trivial critical metrics in this case, one needs to modify R 2 so that it is scale-invariant, i.e. consider v 1=3 R 2 ; where v is the volume of (N; g)

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