Abstract. Let M n , 2 ≤ n ≤ 6 be a complete noncompact hypersurface immersed in H n+1 . We show that there exist two certain positive constants 0 < δ ≤ 1, and β depending only on δ and the first eigenvalue λ 1 (M ) of Laplacian such that if M satisfies a (δ-SC) condition and λ 1 (M ) has a lower bound then H 1 (L 2 (M )) = 0. Excepting these two conditions, there is no more additional condition on the curvature.
In this paper, we will count the number of cusps of complete Riemannian manifolds M with finite volume. When M is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume V of M if some geometric conditions hold true. Moreover, we use the nonlinear theory of the p-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.2010 Mathematics Subject Classification. Primary 53C23; Secondary 53C24, 58J0.
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