Rigidity of minimal submanifolds in hyperbolic space

Abstract

We prove that if an $n$-dimensional complete minimal submanifold $M$ in hyperbolic space has sufficiently small total scalar curvature then $M$ has only one end. We also prove that for such $M$ there exist no nontrivial $L^2$ harmonic 1-forms on $M$

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

“…Later Yun [17] proved that if M ⊂ R n+1 is a complete minimal hypersurface with sufficiently small total scalar curvature M |A| n dv, then there is non nontrivial L 2 harmonic 1-form on M . Yun's result still holds for any complete minimal submanifold with sufficiently small total scalar curvature in hyperbolic space ( [11]). Recently the second author [12] showed that if M is an n-dimensional complete stable minimal hypersurface in hyperbolic space satisfying (2n − 1)(n − 1) < λ 1 (M ), then there is no nontrivial L 2 harmonic 1-form on M .…”

“…Yun's result still holds for any complete minimal submanifold with sufficiently small total scalar curvature in hyperbolic space ( [11]). Recently the second author [12] showed that if M is an n-dimensional complete stable minimal hypersurface in hyperbolic space satisfying (2n − 1)(n − 1) < λ 1 (M ), then there is no nontrivial L 2 harmonic 1-form on M . We generalize this result to a complete noncompact Riemannian manifold with sectional curvature bounded below by a nonpositive constant.…”

L2HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY

“…In [24], Yun proved that if M ⊂ R n+1 is a complete minimal hypersurface with sufficiently small total scalar curvature ||A|| 2 L n , then there is no nontrivial L 2 harmonic 1-form on M . Later, Seo [19] proved this result is valid for complete minimal hypersurface in hyperbolic space. Thereafter, it turned out that these vanishing theorems hold for more general submanifolds.…”

Rigidity of Immersed Submanifolds in a Hyperbolic Space

“…For example, in [13], Lei Ni proved that if M n , n ≥ 3 is a complete minimal immersed hypersurface in R n+1 , then M does not admit any non-trivial L 2 harmonic one-form, consequently, M has only one end. 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 ).…”