International Congress of Mathematicians

then eitherorDifferential Geometry via Harmonic Functions 297(1) M has no non-constant L 2 -harmonic 1-forms, i.e.,H 1 (L 2 (M)) = 0;(2) M = R x N with the warped product metricds 2 M = dt 2 + cosh" tds 2 N,where (N,ds 2 N) is a compact manifold with RìCM > —(m — 2). Moreover, Ai (M) =m — 2.To see that this is indeed a generalization of the theorems of Witten-Yau andCai-Galloway, one uses a theorem of Mazzeo [16] asserting that on a conformallycompact manifoldH 1 (L 2 (Mj)-H 1 (M,dM).By a standard exact sequence argument, the conclusion that H 1 (L 2 (Mj) = 0 impliesthat M has only 1 end. In addition to this, one also uses a theorem of Lee [10] givinga lower bound on Ai for conformally compact, Einstein manifold with non-negativeYamabe constant on dM.Theorem (Lee). Let M be a conformally compact, Einstein manifold withRìCM = — (TO — 1).Suppose that dM has non-negative Yamabe constant, thenXi(M)>^l.! — 1 ^2Sincem 4 ' > m — 2, Wang's theorem implies the theorems of Witten-Yauand Cai-Galloway. Observe that the warped product case in Wang's theorem hasnegative Yamabe constant on dM.At this point, let us also recall a theorem of Cheng [5] stating that:Theorem (Cheng). Let M be a complete manifold withRìCM > —(TO — 1),then(m - l) 2Ai(Af)