On the structure of submanifolds in the hyperbolic space (Q530750)

Suppose \(M\) is an immersed submanifold of the hyperbolic space \(H^{n+m}\). (a) If \(M\) is compact and \(n^{2}|H|^{2}>\max\{p,n-p\}|A|^{2}+2p(n-p)\) for \(1\leq p \leq n-1\), then the Betti number is \(\beta_{p}(M)=0\). (b) If \(M\) is complete, non-compact with two upper bounds conditions on \(n|H|\) and total curvatures, then \(H^{p}(L^{2q+2}(M))={0}\). Also, if the first eigenvalue of the Laplace-Beltrami operator of \(M\) satisfies \[ \lambda_{1}(M) \geq p(n-p)-\frac{n^{2}}{2}\mathrm{inf }|H|^{2} \] for \(1\leq p\leq n-1\) with extra inequality condition, then \(H^{p}(L^{2}(M))=\{0\}\). Furthermore, \(M\) has only one end.