A Bernstein Theorem for Minimal Maps with Small Second Fundamental Form

TL;DR

This paper proves a Bernstein-type theorem for minimal maps between Riemannian manifolds with bounded sectional curvature, showing such maps are either constant or totally geodesic under certain conditions.

Contribution

It establishes a new Bernstein theorem for minimal maps with small second fundamental form, extending classical results to a broader geometric setting.

Findings

Minimal maps are either constant or totally geodesic under given curvature bounds.

Conditions on the differential and second fundamental form determine the map's rigidity.

Results apply to compact domain manifolds with sectional curvature constraints.

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.