On gradient $ρ$-Einstein solitons with Bach tensor radially nonnegative
Maria Andrade, Valter Borges, Hiuri Reis
TL;DR
The paper investigates gradient $\rho$-Einstein solitons under the condition that the Bach tensor is radially nonnegative. By analyzing the Bach tensor in the radial direction and exploiting rectifiability when $\rho\neq0$, it proves that around any regular point, the soliton metric is locally a warped product with an Einstein fiber, and that $\nabla f$ is an eigenvector of the Bach tensor with a computable eigenvalue. Consequently, these solitons have harmonic Weyl curvature and are Bach-flat; this enables a local-to-global perspective that yields classifications in the locally conformally flat steady case, and a low-dimensional ( $n\in\{3,4\}$ ) complete classification under the Bach-nonnegativity hypothesis. The results further show that, under the nonnegativity condition, the solitons decompose locally as warped products, leading to rotationally symmetric models in low dimensions and a clear dichotomy between flat Gaussian-type and Bryant-type solitons in the locally conformally flat setting. Overall, the work extends the understanding of rigidity and symmetry for gradient $\rho$-Einstein solitons constrained by the Bach tensor and connects local curvature structure to global geometric classifications.
Abstract
In this paper, we study $n$-dimensional gradient $ρ$-Einstein solitons whose Bach tensor is radially nonnegative. Under this assumption, we show that such $ρ$-Einstein solitons are locally warped products of an interval and an Einstein manifold, provided either $ρ\neq0$ or $ρ=0$ and the soliton is rectifiable. We obtain as a consequence that these solitons must have harmonic Weyl tensor and vanishing Bach tensor. We also finish the classification of complete locally conformally flat steady $ρ$-Einstein solitons and classify these manifolds when their Bach tensor is radially nonnegative and $n\in\{3,4\}$.