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We show that, for all nonnegative integers k, Г, m and n, theYamabe invariant of #k(RP3)#Г(RP2 】 S1)#m(S2 】 S1)#n(S2. 】S1) is equal to the Yamabe invariant of RP3, provided k + Г ∶ 1. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of RP3. More precisely, we show that such manifolds are either S3 or finite connected sums #m(S2 】 S1)#n(S2. 】S1),where S2. 】S1 is the nonorientable S2-bundle over S1. A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yam- abe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green’s function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.

Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group Uq(g)are constructed in[1]and one can use them to define a two-sided ideal of the modified quantized enveloping algebra. It is shown that the quotient algebra inherits a canonical basis from the modified quantized enveloping algebra and is dual to the quantum coordinate algebra defined by Kashiwara for a symmetrizable Kac–Moody algebra g.

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No abstract uploaded!