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

Abstract

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.