Rigidity results on totally real submanifolds in complex space forms

Abstract

We study n-dimensional totally real minimal submanifolds M immersed in complex space forms ${\tilde{M}}^n(c)$ for $c\le 0$. We prove that M is totally geodesic if the $L^{2p}$-norm of the second fundamental form is finite for some $p\in \mathbb{R}$. We also derive vanishing results for the space of harmonic forms on such submanifolds.