On the Triviality of a Certain Kind of Shrinking Solitons (Journal of Mathematical Study, 2019)
In this paper, we study shrinking gradient Ricci solitons whose Ricci tensorhas one eigenvalue of multiplicity at least $n−2.$ Firstly, we show that if the minimaleigenvalue of Ricci tensor has multiplicity at least $n−1$ at each point, then the solitonare Einstein. While on the shrinking gradient Ricci solitons whose maximal eigenvaluehas multiplicity at least $n−1,$ the triviality are also true if we naturally require thepositivity of Ricci tensor.We further prove that if the maximal (or minimal) eigenvalue of Ricci tensor has multiplicity at least $n−2$ at each point , and in addition the sectional curvature is boundedfrom above, then the soliton are Einstein.
Keywords: Einstein manifold; shrinking gradient Ricci soliton; positive Ricci curvature; pinched sectional curvature.