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We present some results on the boundedness of the mean curvature of properbiharmonic submanifolds in spheres. A partial classification result for properbiharmonic submanifolds with parallel mean curvature vector field in spheres isobtained. Then, we completely classify the proper biharmonic submanifolds inspheres with parallel mean curvature vector field and parallel Weingartenoperator associated to the mean curvature vector field.

Biharmonic submanifolds with parallel mean curvature vector field in spheres