Initial value problem for cohomogeneity one gradient Ricci solitons

Abstract

Consider a smooth manifold $M$. Let $G$ be a compact Lie group which acts on $M$ with cohomogeneity one. Let $Q$ be a singular orbit for this action. We study the gradient Ricci soliton equation $\Hess(u)+\Ric(g)+\frac{\epsilon}{2}g=0$ around $Q$. We show that there always exists a solution on a tubular neighbourhood of $Q$ for any prescribed $G$-invariant metric $g_Q$ and shape operator $L_Q$, provided that the following technical assumption is satisfied: if $P=G/K$ is the principal orbit for this action, the $K$-representations on the normal and tangent spaces to $Q$ 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 papaer "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.