A Bernstein Theorem for Minimal Maps with Small Second Fundamental Form

We consider minimal maps $f:M\to N$ between Riemannian manifolds $(M,{\mathrm{g}}_M)$ and $(N,{\mathrm{g}}_N)$, where $M$ is compact and where the sectional curvatures satisfy ${\sec }_N\le \sigma \le {\sec }_M$ for some $\sigma >0$. Under certain assumptions on the differential of the map and the second fundamental form of the graph $\mathrm{\Gamma }(f)$ of $f$, we show that $f$ is either the constant map or a totally geodesic isometric immersion.