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

Let π : 𝕊^2n+1→ℂP^n be the Hopf map and let ϕ be a totally real immersion of a k(≥ 3) -dimensional simply connected manifold Σ into ℂP^n . It is well known that there exists an isotropic lift ϕ into 𝕊^2n+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 Φ is sufficiently small. Moreover, we prove that if the L^k norm of Φ 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.