Abstract
Let M n , n ≥ 3, be a complete oriented immersed minimal hypersurface in Euclidean space R n+1. We show that if the total scalar curvature on M is less than the n/2 power of 1/C s , where C s is the Sobolev constant for M, then there are no L 2 harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.
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