Classification of Gradient Ricci solitons with harmonic Weyl curvature

TL;DR

This paper classifies gradient Ricci solitons with harmonic Weyl curvature, identifying their local geometric structures and providing a complete classification for the complete case, using novel methods to handle higher dimensions.

Contribution

It introduces a new refined adapted frame method for classifying such solitons in dimensions five and above, overcoming previous technical challenges.

Findings

Local classification into four geometric types

Complete classification for complete solitons

Development of a novel method for higher dimensions

Abstract

We make classifications of gradient Ricci solitons $(M, g, f)$ with harmonic Weyl curvature. As a local classification, we prove that the soliton metric $g$ is locally isometric to one of the following four types: an Einstein manifold, the Riemannian product of a Ricci flat manifold and an Einstein manifold, a warped product of $\mathbb{R}$ and an Einstein manifold, and a singular warped product of $\mathbb{R}^2$ and a Ricci flat manifold. Compared with the previous four-dimensional study in \cite{Ki}, we have developed a novel method of {\it refined adapted frame fields} and overcome the main difficulty arising from a large number of Riemmannian connection components in dimension$ \geq 5$. Next we have obtained a classification of {\it complete} gradient Ricci solitons with harmonic Weyl curvature. For the proof, using the real analytic nature of $g$ and $f$, we elaborate geometric arguments to fit together local regions.