L-p harmonic 1-forms on totally real submanifolds in a complex projective space

Abstract

Let pi : S2n+1 -> CPn be the Hopf map and let phi be a totally real immersion of a k(>= 3)-dimensional simply connected manifold Sigma into CPn. It is well known that there exists an isotropic lift (phi) over bar into S2n+1 preserving the second fundamental form. Using this isotropic lift, we obtain a vanishing theorem for of L-p harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space provided the L-k norm of the traceless second fundamental form phi is sufficiently small. Moreover, we prove that if the L-k norm of phi is finite, then the dimension of L-p harmonic 1-forms on a complete noncompact totally real submanifold in a complex projective space is finite. As consequences, we obtain a vanishing theorem and a finiteness result for L-2 harmonic 1-forms on a complete noncompact minimal Lagrangian submanifold in a complex projective space.