Rigidity of immersed submanifolds in a hyperbolic space

Abstract

Let \(M^{n}\), \(n \geq 5\) be a complete noncompact sub-manifold immersed in \(\mathbb{H}^{n+p}\). We will prove that there exist certain positive constants \(\alpha\), \(C\) such that if \(||H|| \leq \alpha\) and the total scalar curvature \(||A||_{n} < C\) then \(M\) does not admit any nonconstant harmonic function \(u\) with finite energy. Excepting these two conditions, there is no more additional condition on the curvature. Moreover, in the lower dimensional case, namely, \(2 \leq n \leq 5\), we show that there exist two certain positive constants \(0 < \delta \leq 1), and \(\beta\) depending only on \(\delta\) and the first eigenvalue \(\lambda_{1}(M)\) of Laplacian acting on \(M\) such that if \(M\) satisfies a (\(\delta\)-SC) condition and \(\lambda_{1}(M)\) has a lower bound then \(H^{1}(L^{2\beta}(M)) = 0\). Again, we do not need to have any additional condition on the curvature.