L² Harmonic 1-forms on submanifolds with finite total curvature

2012/01/25 by Marcos P. Cavalcante, Cavalcante, Marcos P., Heudson Mirandola +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1201.5392

17 pages, to appear in Journal of Geometric Analysis

arxiv created 2012/06/05 · arxiv updated 2012/06/07

Abstract

Let x:M^m→ M, with m≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply-connected manifold M with sectional curvature satisfying -c²≤ K _M≤ 0, for some constant c. Assume that the immersion has finite total curvature. If c≠ 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 L² 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 L²-harmonic 1-forms on M.

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