Initial value problem for cohomogeneity one gradient Ricci solitons

TL;DR

This paper investigates the existence and uniqueness of gradient Ricci solitons near singular orbits in manifolds with cohomogeneity one group actions, generalizing previous Einstein metric results to solitons.

Contribution

It establishes existence results for gradient Ricci solitons near singular orbits under specific technical conditions and discusses non-uniqueness of solutions.

Findings

Existence of solutions near singular orbits under certain conditions.

Non-uniqueness of initial data leading to multiple solutions.

Generalization of Einstein metric initial value problem to 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.