Gap theorems for certain submanifolds of Euclidean spaces and hyperbolic space forms (Q1101705)

In the present paper, the authors prove Green-Wu type gap theorems about open submanifolds of noncompact space forms. For example, they show: (I) Let M be a connected minimal submanifold of dimension m properly immersed into Euclidean space \(R^ n\). Let \(\rho\) denote the distance in \(R^ n\) to a fixed point of \(R^ n\). Then M is totally geodesic if one of the following conditions holds: (a) \(m\geq 3\), M has one end and the second fundamental form \(\alpha_ M\) of the immersion \(M\to R^ n\) satisfies \(\limsup \rho (x)| \alpha_ M| (x)<\kappa_ 0<1,\) where \(\kappa_ 0\) is defined by \(\kappa_ 0\{(1-\kappa^ 2_ 0)^{- 1}+1\}=\sqrt{2},\) (b) \(m=2\), M has one end and \(\sup \rho^ 2(x)| \alpha_ M| (x)<+\infty,\) (c) \(2m>n\), M is imbedded and \(\sup \rho^{\epsilon}(x)| \alpha_ M| (x)<+\infty\) for some constant \(\epsilon >m.\) (II) Let M be a connected, noncompact Riemannian submanifold of dimension m properly immersed into \(R^ n\). Suppose that M has one end and the second fundamental form \(\alpha_ M\) of M satisfies \(\sup \rho^{\epsilon}(x)| \alpha_ M| (x)<+\infty\) for a constant \(\epsilon >2\). Then M is totally geodesic if \(2m>n\) and the sectional curvature is nonpositive everywhere on M, or if \(m=n-1\) and the scalar curvature is non-positive everywhere on M. The authors get similar results for submanifolds of hyperbolic space forms or complex hyperbolic space forms, too.