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Japan 37 (1985), 391413. [Ka2] M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986), 227-238. [Ka3] M. Kanai, Analytic inequalities and rough isometries between non-compact Riemannian manifolds, In 'Curvature and topology of Riemannian manifolds' (eds. K. Shiohama, T. Sakai, and T. Sunada), Lecture Notes in Mathematics 1201 (Springer, Berlin, 1986), pp. 122-137. [Kar] H. Karcher, Riemannian comparison constructions. In 'Global Differential Geometry' (Ed.

We first fix two real numbers 6 and E such that 0 < 3E < 6 < l. In the wall Hn we are only going to use the vertices which have y-coordinates at most n 1+3E and z-coordinate at most nO. The active vertices through which we are going to send the flow are therefore all contained in this wedge subgraph of H. 48 Steen Markvorsen We are going to construct the flow such that all the flow values in the edges in B which go from L n- 1 to Ln are at most n- 1 - 2E . This will guarantee, that the contribution to the flow energy W(f) by the edges from L n- 1 to Ln is at most (n1+3E) .

Since O(m) has two connected components, we can restrict to the pre-image of the identity component SO(m) to obtain the spinor group Spin(m). , and Spin(m) is generated by products of an even number of unit vectors in the Clifford algebra. In fact: Spin(m) = {Vl ... V21 E Clml q(Vi' Vi) = ±1}. We can also complexify the Clifford algebra CI~ = Clm 0C and define the complex spinor group as SpinC(m) = Spin(m) 0Z/2 Sl. One basic property of the spinor group is that there exists (half-integral) representations which do not descend to SO(n).

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