To appear in J.ReineAng.Math. (Crelles) STABLE MINIMAL HYPERSURFACES IN A NONNEGATIVELY CURVED MANIFOLD

The classical Bernstein Theorem asserts that an entire minimal graph in R must be planar. This theorem was subsequently generalized to higher dimensions by the works of Fleming [10], Almgren [1], DeGiorgi [6], and Simons [23]. The final result states that an entire n-dimensional minimal graph in R must be given by a linear function over R providing that n ≤ 7. On the other hand, nonlinear entire minimal graphs in R for n ≥ 8 were found by Bombieri, DeGiorgi, and Guisti in [2]. Due to the fact that minimal graphs are area minimizing, they are contained in a larger class of submanifolds given by the stable minimal hypersurfaces in R. In 1979, do Carmo and Peng [7] proved that a complete, stable, minimally immersed hypersurface M in R must be planar. At the same time, Fischer-Colbrie and Schoen [9] independently showed that a complete, stable, minimally immersed hypersurface M in a complete 3-dimensional manifold N with nonnegative scalar curvature must be either conformally a plane R or conformally a cylinder R× S. For the special case when N is R, they also proved that M must be planar. In 1984, Gulliver [11] studied a yet larger class of submanifolds in R. He proved that a complete, oriented, minimally immersed hypersurface with finite index in R must have finite total curvature. In particular, applying a theorem of Huber [13], one concludes that the hypersurface must be conformally equivalent to a compact Riemann surface with finitely many punctures. The same result was also independently proved by Fischer-Colbrie in [8]. In addition, she also proved that a complete, oriented, minimally immersed hypersurface with finite index in a complete 3-dimensional manifold with nonnegative scalar curvature must be conformally equivalent to a compact Riemann surface with finite punctures. Shortly