L^2 Harmonic 1-forms on submanifolds with finite total curvature

Abstract

Let $x:M^m\to \bar{M}$, with $m\ge 3$, be an isometric immersion of a complete noncompact manifold $M$ in a complete simply-connected manifold $\bar{M}$ with sectional curvature satisfying $-c^2\le K_{\bar{M}}\le 0$, for some constant $c$. Assume that the immersion has finite total curvature. If $c\ne 0$, assume further that the first eigenvalue of the Laplacian of $M$ is bounded from below by a suitable constant. We prove that the space of the $L^2$ harmonic 1-forms on $M$ has finite dimension. Moreover there exists a constant $\La>0$, explicitly computed, such that if the total curvature is bounded from above by $\La$ then there is no nontrivial $L^2$-harmonic 1-forms on $M$.

17 pages, to appear in Journal of Geometric Analysis

References in corpus (2)

• Rigidity of minimal submanifolds in hyperbolic space
• On submanifolds with tamed second fundamental form