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R2 (with smoothness the same as that of the flow) such that Sp(m) = (0,0) E R2 and Sp(> n U) = {0} x [-1, 1] C R2 (we remark that Sp carries the trajectories of f t in U into the lines y = const).

T (z) lie on the other component of the boundary. 23 If in addition we identify the points (x, 1) and (T (x), 0) in k for x e S2 \ 0 {a1, ... , ar }, then we get a manifold MT of genus p > 2, with punctures p1, ... , Ps (the punctures correspond to the points a1, ... , ar, b1, ... , bT; if these points are distinct, then there are s = 2r punctures, but in the general case r < s < 2r). 2. 24 0 Let MT be the compactification of MT. 24). We now construct the flow sus (T) on MT. [(x- 1)2 +y2], with two impassable grains at the points (0,0) and (1,0).

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An introduction to Lorentz surfaces by Tilla Weinstein

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