Vanishing theorems on complete manifolds with weighted Poincaré inequality and applications

Rigidity of Immersed Submanifolds in a Hyperbolic Space

“…When the ambient space N is a hyperbolic space, Seo [14] proved that there are non L 2 harmonic one form on a complete super stable minimal hypersurface in a hyperbolic space if the first eigenvalue λ 1 (M ) of Laplacian is bounded from below by a certain positive number depending only on the dimension of M . Later, Fu and Yang [7] improved the result of Seo by giving a better lower bound of λ 1 (M ). Recently, in [9], Kim and Yun studied complete oriented noncompact hypersurface M n in a complete Riemannian manifold of nonnegative sectional curvature.…”

“…Note that if δ = 1, then the condition (1.1) means the index of the operator ∆ + (−n + |A| 2 ) is zero (see [7]). In this case, we also say that M satisfies a (SC) condition or M is stable.…”