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2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature |
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Authors: | Marcos Petrúcio Cavalcante Heudson Mirandola Feliciano Vitório |
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Institution: | 1. Instituto de Matemática, Universidade Federal de Alagoas, 57072-900, Maceió, AL, Brazil 2. Instituto de Matemática, Universidade Federal do Rio de Janeiro, 21945-970, Rio de Janeiro, RJ, Brazil
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Abstract: | Let $x:M^{m}\to\bar{M}$ , m≥3, be an isometric immersion of a complete noncompact manifold M in a complete simply connected manifold $\bar{M}$ with sectional curvature satisfying $-k^{2}\leq K_{\bar{M}}\leq0$ , 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 $K_{\bar{M}}\not\equiv0$ , 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 are no nontrivial L 2-harmonic 1-forms on M. |
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