L^2-harmonic 1-forms on Complete Manifolds

We study the global behavior of complete minimal δ-stable hypersurfaces in Rby using L~2-harmonic 1-forms.We show that a complete minimal δ-stable(δ >(n-1)~2/n~2)hypersurface in Rhas only one end.We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincar′e inequality.These results are improvements of the first author’s theorems on hypersurfaces and quaternionic K¨ahler manifolds.

We study the global behavior of complete minimal δ-stable hypersurfaces in R^(n+1) by using L^2-harmonic 1-forms.We show that a complete minimal δ-stable(δ >(n-1)~2/n^2)hypersurface in R^(n+1) has only one end.We also obtain two vanishing theorems of complete noncompact quaternionic manifolds satisfying the weighted Poincar′e inequality.These results are improvements of the first author's theorems on hypersurfaces and quaternionic K¨ahler manifolds.