On the isometric minimal immersion into a Euclidean space (Q1819116)

Let \(M ^n\) be a complete \(n\)-dimensional Riemannian manifold and \(p \in M ^n\). Denote by \(B _p (t)\) the geodesic ball of \(M ^n\) with center \(p\) and radius \(t\). The author gives a necessary condition for the existence of a minimal isometric immersion of \(M ^n\) into a Euclidean space. More precisely, the author proves: if there exists a minimal isometric immersion of \(M ^n\) into a Euclidean space, then for each \(p \in M ^n\) the function \(\text{vol} (B _p (t)) / t ^n \) is monotone non-decreasing with respect to \(t\). A classification of non-positively curved minimal hypersurfaces of Euclidean spaces is given in the third paragraph of the paper.