Vanishing theorems for L2L2 harmonic forms on complete submanifolds in Euclidean space

The aim of this paper is to prove vanishing theorems for L2L2 harmonic forms of higher order on complete non-compact submanifolds in Euclidean space. Firstly, by assuming that the submanifold has flat normal bundle, we can explicitly express the Weitzenböck formulae for harmonic p  -forms. Using this formulae, we can obtain some L2L2 vanishing theorems by adding a relation of the square length of the second fundamental form with the squared mean curvature, or by assuming that the total curvature of the submanifold is bounded from above by an explicit positive constant. Secondly, by using a monotonicity formulae for general harmonic forms, we obtain a vanishing theorem under an appropriate decay of the norm of the second fundamental form.