Classification of Gradient Ricci Solitons with Harmonic Weyl Curvature

We make classifications of gradient Ricci solitons (M, g, f) with harmonic Weyl curvature. As a local classification, we show that the associated Riemannian metric g is locally among the types (i)-(iv) in Theorem 1. Compared with the previous four-dimensional study in [25], we have developed a novel method of refined adapted frame fields and overcome the main difficulty arising from a large number of Riemmannian connection components in dimension >= 5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \ge 5$$\end{document}. Next we have obtained a classification of complete ones 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.