  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.

Differential Geometry