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

TL;DR

This paper proves that for certain complete submanifolds with finite total curvature in negatively curved spaces, the space of square-integrable harmonic 1-forms is finite-dimensional, and under specific bounds, trivial.

Contribution

It establishes finiteness and triviality conditions for the space of $L^2$ harmonic 1-forms on submanifolds with finite total curvature in negatively curved ambient spaces.

Findings

Finite-dimensionality of $L^2$ harmonic 1-forms under finite total curvature.

Existence of an explicit curvature bound ensuring no nontrivial $L^2$ harmonic 1-forms.

Results depend on bounds of sectional curvature and the first eigenvalue of the Laplacian.

Abstract

Let $x:M^m\to \bar M$, with $m\geq 3$, be an isometric immersion of a complete noncompact manifold $M$ in a complete simply-connected manifold $\bar M$ with sectional curvature satisfying $-c^2\leq K_{\bar M}\leq 0$, for some constant $c$. Assume that the immersion has finite total curvature. If $c\neq 0$, 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 $M$.