L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

Abstract

Let x : M m →M , m ≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply-connected manifoldM with sectional curvature satisfying −k 2 ≤ KM ≤ 0, for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L m -norm. If KM ≡ 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 Λ > 0, explicitly computed, such that if the total curvature is bounded from above by Λ then there is no nontrivial L 2 -harmonic 1-forms on M .

Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold

“…In this section, we prove a similar vanishing theorem for L 2 harmonic 1-forms on complete noncompact submanifolds under the same assumptions as in [2] except that the lower bound of λ 1 (M ) depends on φ n . More precisely, we prove In the case k = 0, assume further that the first eigenvalue λ 1 (M ) of M satisfies…”

“…More generally, let M be an n-dimensional complete noncompact submanifold in a complete simplyconnected Riemannian manifold N with sectional curvature K N satisfying −k 2 ≤ K N ≤ 0 for some constant k. In [2], Cavalcante, Mirandola and Vitório recently proved that if the L n norm of the traceless second fundamental form φ is sufficiently small and if the first eigenvalue λ 1 (M ) of the Laplacian satisfies…”

“…In [32], Yun proved that if M ⊂ R n+1 is a complete minimal hypersurface with sufficiently small total scalar curvature M |A| n , then there is no nontrivial L 2 harmonic 1-form on M . Yun's result has been generalized into various ambient spaces [2,6,[23][24][25].…”

L2HARMONIC 1-FORMS ON SUBMANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY

“…More precisely, we have the following theorem which is a generalization of Theorem 1.2 in [3]. Theorem 1.5.…”

“…After that, Dung and Seo [6] proved a similar vanishing theorem for L 2 harmonic 1-forms on complete noncompact submanifolds under the same assumption as in [3] except that the lower bound of…”