Differential Equations on Complex Projective Hypersurfaces of Low Dimension

Let $n=2,3,4,5$ and let $X$ be a smooth complex projective hypersurface of ${\mathbb{P}}^{n+1}$. In this paper we find an effective lower bound for the degree of $X$, such that every holomorphic entire curve in $X$ must satisfy an algebraic differential equation of order $k=n=\dim X$, and also similar bounds for order $k>n$. Moreover, for every integer $n\ge 2$, we show that there are no such algebraic differential equations of order $k<n$ for a smooth hypersurface in ${\mathbb{P}}^{n+1}$.