Rigidity results on gradient Schouten solitons
Romildo Pina, Ilton Menezes
TL;DR
This work analyzes gradient Schouten solitons on product manifolds $M=(B^n,g^*)\times (F^m,g_F)$ where $g^*=\psi^{-2}g$ is conformal to a pseudo-Euclidean metric and $F^m$ is Einstein. By exploiting pseudo-orthogonal invariance, the authors derive a system of PDEs for the potential $h$ and warp function $\psi$, which reduces to ODEs in radial coordinates; in the gradient Schouten case $\rho=\frac{1}{2(n-1)}$ they obtain all solutions explicitly. They classify gradient Schouten solitons on the form $M=(\mathbb{R}^n,g^*)\times F^m$ with $\psi(r)=k_2 r^s$, showing that only $s=1$ or $s=\frac{1}{2}$ occur and providing precise expressions for $h$, $\lambda_F$, and $\tilde{\lambda}$. In the complete Riemannian setting, the results yield a rigidity: $(\mathbb{R}^n,g^*)$ becomes $\mathbb{S}^{n-1}\times \mathbb{R}$ with $F^m$ compact (and $\lambda_F=(n-2)$), and the paper furnishes explicit complete examples of solitons with varying $\tilde{\lambda}$.
Abstract
In this paper we consider $ρ$-Einstein solitons of type $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$, where $\left(B^n,g^{*}\right)$ is conformal to a pseudo-Euclidean space and invariant under the action of the pseudo-orthogonal group, and $\left(F^m,g_{F}\right)$ is an Einstein manifold. We provide all the solutions for the gradient Schouten soliton case. Moreover, in the Riemannian case, we prove that if $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$ is a complete gradient Schouten soliton then $\left(B^{n},g^{*}\right)$ is isometric to $\mathbb{S}^{n-1}\times \mathbb{R}$ and $F^m$ is a compact Einstein manifold.