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By Gauss C.F.

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Both of these are true for cusp singularities; see [31, 32]. Also, the fibered conical end is conformally equivalent to a fibered cusp end. Because the middle dimensional L2 -cohomology is conformally invariant, the theorem follows. Acknowledgment The second author would like to thank Tamas Hausel, Eugenie Hunsicker, and Rafe Mazzeo for very stimulating conversations. We also thank the referee for useful suggestions. During the preparation of this work, the first author was supported by NSF grant DMS 0105128, while the second author was supported by NSF grant DMS 0707000.

For definiteness, assume that j = m + 1; then, any element of U (m+1) has a unique representative in Cm+1 of the form (w1 , . . , wm , 1), and the m-uple (w1 , . . , wm ) is then uniquely defined up to a am+1 -th root of 1. Moreover, for any m-uple w1 , . . , wm , there exists a unique positive number, s = s(w1 , . . , wm ), such that s · (w1 , . . , wm , 1) = (sa1 w1 , . . , sam wm , sam+1 ) belongs to the unit sphere S2m+1 in Cm+1 , namely the unique positive root of the equation m ∑ s2ai |wi |2 + s2am+1 = 1.

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