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Let (M, g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the 3-sphere S-3 with canonical metric. In this work the global well-posedness problem for the quintic nonlinear Schrodinger equation i partial derivative(t)u + Delta u = +/-vertical bar u vertical bar(4)u, u vertical bar(t=0) = u(0) is solved for small initial data u(0) in the energy space H-1(M), which is the scaling-critical space. Further, local well-posedness for large data, as well as persistence of higher initial Sobolev regularity is obtained. This extends previous results of Burq-Gerard-Tzvetkov to the endpoint case.

In this paper we continue the study (c. f. [C. -K.], [S. -T. 1, 2]) of special Cremona transformations, i.e. those whose base scheme Y is smooth and connected. Crauder and Katz have completely classified these when dim Y < 2. Our main results classify them when they are quadro-quadric (Theorem 2.6) and when codim Y = 2 (Theorem 3.2). In the first case they are just the maps given by systems of quadrics through Severi varieties (by a result of Zak there are just four of these: the Veronese surface V C P5, the Segre embedding of P2 x P2 in P8, the Grassmannian G(2, 6) in p14 and the 16dimensional E6-variety in p26), and in the second either Y is a quintic elliptic scroll in P4 or Y is defined in pI by the vanishing of the n x n minors of