The Non-existence Theorems Of L~2 Harmonic Forms In Hadamard Manifolds

In this paper,we mainly study non-existence problem of L2 harmonic form on a complete noncompact submainfold Mn(n ≥ 3)in Hadamard mainfolds Nn+m.Let Φ,H,λ1(M)are the traceless second fundamental form,mean curvature vector,the first,eigenvalue of the La.place operators.We get the following results:(1)Let Mn(n≥ 3)be a complete noncompact immersed submainfold in a Hadamard mainfold Nn+m.Assume that the sectional curvature of Nn+m has the non positive and lower bounds,if the ‖Φ‖LnLn(M),‖H‖Ln(M),λ1(M)in Mn satisfy certain restrictions,then there exist no nontrivial L2 harmonic:1-form on Mn.(2)Let Mn(n ≥ 3)be a complete noncompact immersed hypersurface in a.Hadamard mainfold Nn+1.Assume that the sectional curvature of Nn+1 has the non positive and lower bounds,if the ‖Φ‖Ln(M),‖H‖Ln(M),λ1(M)in Mn satisfy certain restrictions,then there exist no nontrivial L2 harmonic 2-forms on Mn.