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We establish a dichotomy on the curvature decay for four dimensional completenoncompact non Ricci flat steady gradient Ricci soliton with linear curvaturedecay and proper potential function. A similar dichotomy is also shown inhigher dimensions under the additional assumption that the Ricci curvature isnonnegative outside a compact subset.

On a dichotomy of the curvature decay of steady Ricci soliton