Gradient Ricci Solitons with Half Harmonic Weyl Curvature and two Ricci Eigenvalues

Commun. Korean Math. Soc. Published online February 25, 2021

Jongsu Kim and Yutae Kang
Sogang University; Sogang University

Abstract : In this article we classify four dimensional gradient Ricci solitons $(M, g, f)$ with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point.
Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$,
$(V, g)$ is isometric to one of the following:

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{\rm (i)} an Einstein manifold.

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{\rm (ii)} a domain in the Riemannian product $ (\mathbb{R}^2, g_0) \times (N, \tilde{g})$, where $g_0$ is the flat metric on $\mathbb{R}^2$ and $(N, \tilde{g})$ is a two dimensional Riemannian manifold of constant curvature $\lambda \neq 0$.

% And $f = \frac{\lambda}{2} s^2+C_1$, for a constant $C_1$.

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{\rm (iii)} a domain in $\mathbb{R} \times W$ with the warped product metric $ ds^2 + h(s)^2 \tilde{g},$
where $\tilde{g}$ is a constant curved metric on a three dimensional manifold $W$.