  In this work, we consider the area non-increasing map between manifolds with
positive curvature. By exploring the strong maximum principle along the
graphical mean curvature flow, we show that an area non-increasing map between
certain positively curved manifolds is either homotopy trivial, Riemannian
submersion, local isometry or isometric immersion. This implies that an area
non-increasing self map of $\mathbb{CP}^n$, $n\ge 2$ is either homotopically
trivial or is an isometry. This confirms a speculation of Tsai-Tsui-Wang. We
also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci
flow to establish related results on manifolds with positive $1$-isotropic
curvature.
