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

Let $$\pi : {\mathbb {S}}^{2n+1}\rightarrow {\mathbb {C}}P^n$$ be the Hopf map and let $$\phi$$ be a totally real immersion of a $$k(\ge 3)$$-dimensional simply connected manifold $$\Sigma$$ into $${\mathbb {C}}P^n$$. It is well known that there exists an isotropic lift $${\overline{\phi }}$$ into $${\mathbb {S}}^{2n+1}$$ 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 $$\Phi$$ 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.