Minimal submanifolds with small total scalar curvature in Euclidean space

Abstract

Let M be an n-dimensional complete minimal submanifold in R^n+p. Lei Ni proved that if M has sufficiently small total scalar curvature, then M has only one end. We improve the upper bound of total scalar curvature. We also prove that if M has the same upper bound of total scalar curvature, there is no nontrivial L² harmonic 1-form on M.