In this paper, we first use the method of Colding and Minicozzi II  to show that K. Smoczyk's classification theorem  for complete self-shrinkers in higher codimension also holds under a weaker condition. Then as an application, we give some rigidity results for self-shrinkers in arbitrary codimension.
"Classification and rigidity of self-shrinkers in the mean curvature flow." J. Math. Soc. Japan 66 (3) 709 - 734, July, 2014. https://doi.org/10.2969/jmsj/06630709