Initial value problem for cohomogeneity one gradient Ricci solitons

Consider a smooth manifold MM. Let GG be a compact Lie group which acts on MM with cohomogeneity one. Let QQ be a singular orbit for this action. We study the gradient Ricci soliton equation Hess(u)+Ric(g)+ϵ2g=0 around QQ. We show that there always exists a solution on a tubular neighbourhood of QQ for any prescribed GG-invariant metric gQgQ and shape operator LQLQ, provided that the following technical assumption is satisfied: if P=G/KP=G/K is the principal orbit for this action, the KK-representations on the normal and tangent spaces to QQ have no common sub-representations. We also show that the initial data are not enough to ensure uniqueness of the solution, providing examples to explain this indeterminacy. This work generalises the paper “The initial value problem for cohomogeneity one Einstein metrics” of 2000 by J.-H. Eschenburg and McKenzie Y. Wang to the gradient Ricci solitons case.