A Bernstein Theorem for Minimal Maps with Small Second Fundamental Form

Abstract

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 $\Gamma(f)$ of $f$, we show that $f$ is either the constant map or a totally geodesic isometric immersion.