Extrema of Curvature Functionals on the Space of Metrics on 3-manifolds, Ii
نویسنده
چکیده
Here s is the scalar curvature, D2s the Hessian of s, ∆s = trD2s 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-flat R2 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 R2 is not scale invariant in dimension 3, there are no critical metrics g with R2(g) 6= 0. To obtain non-trivial critical metrics in this case, one needs to modify R2 so that it is scale-invariant, i.e. consider v1/3R2, where v is the volume of (N, g). Nevertheless, it is of course apriori possible that there are non-trivial solutions of (0.2)-(0.3) on non-compact manifolds N .
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