# Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Jia-Yong Wu  Department of Mathematics, Shanghai Maritime University, Haigang
Avenue 1550, Shanghai 201306, P. R. China<EMAIL_ADDRESS>,  Peng Wu
Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.
<EMAIL_ADDRESS>and  William Wylie  Department of Mathematics,
Syracuse University, Syracuse, NY 13244, U.S.A<EMAIL_ADDRESS>
(Date: September 5, 2014)

###### Abstract.

We prove that a four-dimensional gradient shrinking Ricci soliton with $\delta
W^{\pm}=0$ is either Einstein, or a finite quotient of
$S^{3}\times\mathbb{R}$, $S^{2}\times\mathbb{R}^{2}$ or $\mathbb{R}^{4}$. We
also prove that a four-dimensional cscK gradient Ricci soliton is either
Kähler-Einstein, or a finite quotient of $M\times\mathbb{C}$, where $M$ is a
Riemann surface. The main arguments are curvature decompositions, the
Weitzenböck formula for half Weyl curvature, and the maximum principle.

###### Key words and phrases:

Gradient shrinking Ricci soliton, harmonic half Weyl curvature, cscK gradient
Kähler-Ricci soliton, curvature decomposition, Weitzenböck formula, maximum
principle.

###### 2010 Mathematics Subject Classification:

Primary 53C24, 53C25.

##  1\. Introduction

In this paper we investigate four-dimensional gradient shrinking Ricci
solitons with half harmonic Weyl curvature. A Riemannian metric $g$ on a
smooth manifold $M^{n}$ is called a gradient Ricci soliton, if there exists an
$f\in C^{\infty}(M)$ and a $\lambda\in\mathbb{R}$, such that

(1) $\begin{split}\mathrm{Ric}+\nabla^{2}f=\lambda g.\end{split}$

The function $f$ is called a potential function for the gradient Ricci
soliton. By convention we denote it by the triple $(M^{n},g,f)$. The gradient
Ricci soliton is called shrinking, steady, or expanding, if $\lambda>0$,
$\lambda=0$, or $\lambda<0$, respectively. Gradient Ricci solitons play an
important role in the Ricci flow as they are self-similar solutions to the
flow, making them possible singularity models and critical points of
Perelman’s entropies. See [5] for an excellent survey of Ricci solitons.

Any Einstein metric is also a gradient Ricci soliton with $f$ taken to be a
constant function. The only space which is both Einstein and a gradient Ricci
soliton with a non-constant potential function is a _Gaussian_ which is flat
Euclidean space. If $\lambda\neq 0$, $f$ can be chosen as
$f(x)=\frac{\lambda}{2}d(p,x)^{2}$ and $d(p,x)$ is the Euclidean distance to
some point $p\in\mathbb{R}^{n}$. A simplest way to construct a non-Einstein
gradient Ricci soliton is to take the product of an Einstein metric and a
Gaussian.

In dimensions $2$ and $3$ these are the only examples of shrinking gradient
Ricci solitons as follows from the work of Hamilton [24], Ivey [26], Perelman
[36], L. Ni, Wallach [33], and H.-D. Cao, B.-L. Chen, and X.-P. Zhu [8]. In
higher dimensions many authors have studied the classification of gradient
Ricci solitons. For the purposes of this introduction we focus only on results
for shrinking gradient Ricci solitons. Any locally conformal flat gradient
shrinking Ricci soliton is a finite quotient of $S^{n}$,
$S^{n-1}\times\mathbb{R}$, or $\mathbb{R}^{n}$ as follows from the works of
various authors, see [17, 10, 33, 46, 38, 30]. Recall that a Riemannian
manifold is locally conformally flat if the Weyl tensor vanishes. Fernández-
López, García-Río [19], and Munteanu, Sesum [30] generalized the
classification of locally conformally flat gradient shrinking Ricci solitons
by showing that a gradient shrinking Ricci solitons with harmonic Weyl
curvature is either Einstein, or a finite quotient of
$N^{k}\times\mathbb{R}^{n-k}$ for $0\leq k\leq n$, where $N^{k}$ is a
$k$-dimensional Einstein manifold of positive scalar curvature. H.-D. Cao, Q.
Chen [7] also proved that a Bach-flat gradient shrinking Ricci soliton is
either Einstein, or a finite quotient of $N^{n-1}\times\mathbb{R}$ or
$\mathbb{R}^{n}$, where $N^{n-1}$ is an $(n-1)$-dimensional Einstein manifold.
Naber [32], Munteanu, M.-T. Wang [31], Catino [12], and M. Cai [2] proved
various rigidity results under appropriate curvature pinching assumptions.

Dimension four is the lowest dimension where there are interesting examples of
shrinking gradient Ricci solitons. The first examples where constructed by
Koiso [28] and Cao [4], also see [18, 43]. Note that all of the known
interesting examples are Kähler. In dimension 4, the Hodge star splits the
space of $2$-forms into the self-dual and anti-self dual parts and
consequently the curvature tensor and Weyl tensor respect this splitting (see
section 2). It is thus natural to consider self dual or anti-self dual part of
Weyl curvature $W^{\pm}$ commonly called the half Weyl curvature. X. Chen, Y.
Wang [15] and H.-D. Cao, Q. Chen [7] proved that a half conformally flat
($W^{\pm}=0$) four-dimensional gradient shrinking Ricci soliton is a finite
quotient of $S^{4}$, $\mathbb{C}P^{2}$, $S^{3}\times\mathbb{R}$, or
$\mathbb{R}^{4}$. In [45], the second author proved that a compact four-
dimensional gradient shrinking Ricci soliton with $\delta W^{\pm}=0$ and half
two-nonnegative curvature operator (which is equivalent to half nonnegative
isotropic curvature) is a finite quotient of $S^{4}$ or Kähler-Einstein.

In this paper we complete the classification of four-dimensional gradient
shrinking Ricci solitons with harmonic half Weyl curvature,

######  Theorem 1.1.

A four-dimensional gradient shrinking Ricci soliton with $\delta W^{\pm}=0$ is
either Einstein, or a finite quotient of $S^{3}\times\mathbb{R}$,
$S^{2}\times\mathbb{R}^{2}$, or $\mathbb{R}^{4}$.

As opposed to the locally conformally flat or half-locally conformally flat
conditions we note that all Einstein metrics satisfy $\delta W=0$ and are Bach
flat and all Einstein $4$-manifolds satisfy $\delta W^{\pm}=0$. Thus, for
example Theorem 1.1 shows that one can differentiate whether a gradient Ricci
soliton is Einstein or non-Einstein simply by checking $\delta W^{\pm}$ (or
the Bach tensor). There are many interesting results about the tensor
$W^{\pm}$ in the geometry of $4$-manifolds see for example Gursky [22] for an
interesting gap theorem for $\|W^{\pm}\|_{L^{2}}$.

We also note that any $4$-dimensional Kähler metric with constant scalar
curvature satisfies $\delta W^{+}=0$ and
$\frac{|W^{+}|^{2}}{R^{2}}=\frac{1}{24}$ (see [16]). In the process of
detecting $S^{2}\times\mathbb{R}^{2}$ in the proof of Theorem 1.1, we prove,

######  Theorem 1.2.

A four-dimensional gradient Kähler-Ricci soliton with constant scalar
curvature is either Kähler-Einstein, or a finite quotient of
$M\times\mathbb{C}$, where $M$ is a Riemann surface.

Theorem 1.2 has also been obtained by Fernandez-Lopez and Garcia-Rio [20]
recently using a different method, and they were also able to classify six-
dimensional gradient Kähler-Ricci solitons with constant scalar curvature.

Recently there have been some other interesting classification results for
Kähler-Ricci solitons. Y. Su and K. Zhang [41] have proved that a complete
noncompact gradient Kähler-Ricci soliton with vanishing Bochner tensor is
Kähler-Einstein. Q. Chen, M. Zhu [14] have proved that a gradient steady
Kähler-Ricci soliton with harmonic Bochner tensor is Calabi-Yau, and a
gradient shrinking (expanding) Kähler-Ricci soliton with harmonic Bochner
tensor is either Kähler-Einstein, or a finite quotient of
$N^{k}\times\mathbb{C}^{n-k}$, where $N^{k}$ is a Kähler-Einstein manifold of
positive (negative) scalar curvature. Calamai, Petrecca [3] also have proved
that an extremal Kähler-Ricci soliton with positive holomorphic sectional
curvature is Kähler-Einstein.

In the process of proving Theorem 1.1 we also observe that in a result of
Catino [12], the nonnegative Ricci curvature assumption is not required (by
replacing $|\mathrm{Ric}|^{2}$ by $|\mathrm{Ric}|$ in his argumnet),

######  Theorem 1.3 (Catino [12]).

A gradient shrinking Ricci soliton with

$|W|R\leq\sqrt{\frac{2(n-1)}{n-2}}\left(\overset{\circ}{\mathrm{Ric}}-\frac{R}{\sqrt{n(n-1)}}\right)^{2}$

is a finite quotient of $S^{n}$, $S^{n-1}\times\mathbb{R}$, or
$\mathbb{R}^{n}$.

######  Remark 1.1.

We also point out that our proof is different from the $\delta W=0$ case. For
gradient shrinking Ricci solitons with $\delta W=0$, the proofs of Fernández-
López, García-Río [19], and Munateanu, Sesum [30] rely on the following
identity. If the Ricci curvature is bounded below and $|\mathrm{Rm}|\leq
e^{a(r+1)}$ for some $a\in\mathbb{R}$ (X. Cao [9]), or if
$\int_{M}|\mathrm{Rm}|^{2}e^{-\delta f}<\infty$ for some $\delta<1$ (Munteanu,
Sesum [30]), then

$\int_{M}|\delta\mathrm{Rm}|^{2}e^{-f}dv=\int_{M}|\nabla\mathrm{Ric}|^{2}e^{-f}dv.$

Unfortunately, it is not clear whether there is an analogous identity for half
curvature.

The main arguments to prove Theorem 1.1 are curvature decompostions, the
Weitzenböck formula for half Weyl curvature [11, 45], and an analogous
argument of Catino [12]. As with much of the work mentioned above, another
important component is the $D$-tensor introduced by Cao and Chen in [6, 7]. We
observe that the $D$-tensor arises naturally from the standard curvature
decomposition. First we show, using the decomposition of the curvature tensor,
that if $\delta W^{\pm}=0$ then $\nabla f$ is an eigenvector of the Ricci
tensor, and observe that $W^{\pm}$ can be expressed explicitly in terms of the
traceless Ricci curvature. Next applying the Weitzenböck formula and the
maximum principle, we prove an identity involving the (anti-)self-dual Weyl
curvature, the traceless Ricci curvature and the scalar curvature, which
further imply that either $\nabla f\equiv 0$, hence $(M,g)$ is an Einstein
manifold; or $W^{\pm}\equiv 0$, hence $(M,g)$ is a finite quotient of
$S^{3}\times\mathbb{R}$ or $\mathbb{R}^{4}$; or $(M,g)$ is a cscK gradient
shrinking Kähler-Ricci soliton, hence is a finite quotient of
$S^{2}\times\mathbb{R}^{2}$.

The rest of the paper is organized as follows. In section 2, we discuss
curvature decompositions and the relationship between $W^{\pm}$ and the
traceless Ricci curvature when $\delta W^{\pm}=0$. In section 3, we prove
Theorem 1.1, and Theorem 1.2 by applying the Weitzenböck formula and the
maximum principle.

Acknowledgement. Part of the work was done when the first author was visiting
the Department of Mathematics at Cornell University, he greatly thanks
Professor Xiaodong Cao for his help and Department of Mathematics for their
hospitality. The second author thanks Professors Jeffrey Case, Yuanqi Wang and
Yuan Yuan for helpful discussions. The first author is partially supported by
NSFC (11101267, 11271132) and the China Scholarship Council (201208310431).
The second author is partially supported by an AMS-Simons postdoc travel
grant.

##  2\. Curvature decompositions on four-dimensional gradient Ricci solitons

In this section we will discuss curvature decompositions on four-dimensional
gradient Ricci solitons. First we fix some notation. Our sign conventions for
the curvature tensor will be so that

$\begin{split}&R_{ijkl}=g_{hk}R^{h}_{ijl},\ \ K(e_{i},e_{j})=R_{ijij},\ \
R_{ik}=g^{jl}R_{ijkl},\ \ R=g^{ij}R_{ij}.\\\ \end{split}$

And our convention for the inner product of two $(0,4)$-tensors $S,T$ will be

$\langle S,T\rangle=\frac{1}{4}S_{ijkl}T^{ijkl}\ $

so that our convention agrees with the one in Derdzinski’s Weitzenböck formula
[16].

On an oriented $4$-manifold, the Hodge star
$\star:\wedge^{2}M\rightarrow\wedge^{2}M$ has eigenvalues $1$ and $-1$. Thus
we can break $\wedge^{2}M=\wedge^{+}M+\wedge^{-}M$ according to the
eigenspaces of $\star$. Given a basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$ of
$T_{p}M$, for any pair $(ij)$, $1\leq i\neq j\leq 4$, denote
$(i^{\prime}j^{\prime})$ to be the dual of $(ij)$, i.e., the pair such that
$e_{i}\wedge e_{j}\pm e_{i^{\prime}}\wedge e_{j^{\prime}}\in\wedge^{\pm}M$. In
other words, $(iji^{\prime}j^{\prime})=\sigma(1234)$ for some even permutation
$\sigma\in S_{4}$. So for any $(0,4)$-tensor $T$, its (anti-)self-dual part is

$T^{\pm}_{ijkl}=\frac{1}{4}(T_{ijkl}\pm T_{ijk^{\prime}l^{\prime}}\pm
T_{i^{\prime}j^{\prime}kl}+T_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}}),$

It is well known that for four-manifolds, Weyl curvature has a very
interesting symmetry,

######  Lemma 2.1.

Let $(M,g)$ be a four-dimensional Riemannian manifold. Then

$\begin{split}W_{ijkl}=&W_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}},\end{split}$

therefore,

$\begin{split}W^{\pm}_{ijkl}=&\pm W^{\pm}_{ijk^{\prime}l^{\prime}}=\pm
W^{\pm}_{i^{\prime}j^{\prime}kl}=W^{\pm}_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}}=\frac{1}{2}(W_{ijkl}{\pm}W_{ijk^{\prime}l^{\prime}}).\end{split}$

In particular, for any $u\in C^{\infty}(M)$,

$\begin{split}|\iota_{\nabla u}W^{\pm}|^{2}=&\frac{1}{4}|W^{\pm}|^{2}|\nabla
u|^{2}.\end{split}$

Now we discuss curvature decompositions on four-dimensional gradient Ricci
solitons. First recall that, similar to Einstein metrics, for gradient Ricci
solitons, we have

$\begin{split}\delta(e^{-f}\mathrm{Rm})=0,\end{split}$

following from which we have the following basic identities,

######  Lemma 2.2.

Let $(M,g,f)$ be a gradient Ricci soliton. Then

$\begin{split}\nabla_{k}R_{jl}-\nabla_{l}R_{jk}=&R_{ijkl}\nabla^{i}f,\\\
(\delta\mathrm{Rm})_{jkl}=\nabla^{i}R_{ijkl}=&R_{ijkl}\nabla^{i}f,\\\
\nabla_{i}R=2\nabla^{j}R_{ij}=&2R_{ij}\nabla^{j}f.\end{split}$

In [6, 7], H.-D. Cao, Q. Chen introduced a $(0,3)$-tensor
$D=\frac{n-2}{n-3}\delta W-\iota_{\nabla f}W$, which plays an important role
in their classification of locally conformally flat gradient steady Ricci
solitons and Bach-flat gradient shrinking Ricci solitons. We observe that $D$
and its “self-dual” and “anti-self-dual” parts $D^{\pm}$, arise naturally from
the standard curvature decomposition. For our purpose we only calculate the
four-dimensional case, for general dimensions the argument is the same.

######  Lemma 2.3.

Let $(M,g,f)$ be a four-dimensional gradient Ricci soliton. Then

$\begin{split}D_{jkl}=&2\nabla^{i}W_{ijkl}-W_{ijkl}\nabla^{i}f\\\
=&\frac{1}{2}(R_{jl}\nabla_{k}f-R_{jk}\nabla_{l}f)+\frac{1}{12}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})-\frac{R}{6}(g_{jl}\nabla_{k}f-g_{jk}\nabla_{l}f),\\\
D^{\pm}_{jkl}\overset{\triangle}{=}&2\nabla^{i}W^{\pm}_{ijkl}-W^{\pm}_{ijkl}\nabla^{i}f\\\
=&\frac{1}{4}(R_{jl}\nabla_{k}f-R_{jk}\nabla_{l}f)+\frac{1}{24}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})\\\
-&\frac{R}{12}(g_{jl}\nabla_{k}f-g_{jk}\nabla_{l}f)\\\
\pm&\frac{1}{4}(R_{jl^{\prime}}\nabla_{k^{\prime}}f-R_{jk^{\prime}}\nabla_{l^{\prime}}f)\pm\frac{1}{24}(\nabla_{k^{\prime}}Rg_{jl^{\prime}}-\nabla_{l^{\prime}}Rg_{jk^{\prime}})\\\
\mp&\frac{R}{12}(g_{jl^{\prime}}\nabla_{k^{\prime}}f-g_{jk^{\prime}}\nabla_{l^{\prime}}f).\end{split}$

In particular,

(2)
$\begin{split}|D^{+}|^{2}=|D^{-}|^{2}=\frac{1}{2}|D|^{2}=&\frac{1}{4}|\overset{\circ}{\mathrm{Rc}}|^{2}|\nabla
f|^{2}-\frac{1}{48}|R\nabla f-2\nabla R|^{2}.\end{split}$

Proof. We apply the standard curvature decomposition to both sides of the
identity $\nabla^{i}R_{ijkl}=R_{ijkl}\nabla^{i}f$,

For the left hand side, we have

$\begin{split}\nabla^{i}R_{ijkl}=&\nabla^{i}W_{ijkl}+\frac{1}{2}\nabla^{i}(R_{ik}g_{jl}+R_{jl}g_{ik}-R_{il}g_{jk}-R_{jk}g_{il})\\\
-&\frac{\nabla^{i}R}{6}(g_{ik}g_{jl}-g_{il}g_{jk})\\\
=&\nabla^{i}W_{ijkl}+\frac{1}{2}(\nabla_{k}R_{jl}-\nabla_{l}R_{jk}+\frac{1}{2}\nabla_{k}Rg_{jl}-\frac{1}{2}\nabla_{l}Rg_{jk})\\\
-&\frac{1}{6}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})\\\
=&\nabla^{i}W_{ijkl}+\frac{1}{2}\nabla^{i}R_{ijkl}+\frac{1}{4}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})-\frac{1}{6}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk}),\end{split}$

Therefore we get

$\begin{split}\nabla^{i}R_{ijkl}=&2\nabla^{i}W_{ijkl}+\frac{1}{6}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk}),\end{split}$

On the other hand, observe that by the second Bianchi identity, we have
$\nabla^{i}R_{i^{\prime}j^{\prime}kl}=0$, so we get

$\begin{split}4\nabla^{i}R^{\pm}_{ijkl}=&\nabla^{i}(R_{ijkl}+R_{ijk^{\prime}l^{\prime}}+R_{i^{\prime}j^{\prime}kl}+R_{i^{\prime}j^{\prime}k^{\prime}l^{\prime}})\\\
=&\nabla^{i}(R_{ijkl}+R_{ijk^{\prime}l^{\prime}})\\\
=&2\nabla^{i}(W_{ijkl}+W_{ijk^{\prime}l^{\prime}})+\frac{1}{6}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})+\frac{1}{6}(\nabla_{k^{\prime}}Rg_{jl^{\prime}}-\nabla_{l^{\prime}}Rg_{jk^{\prime}})\\\
=&4\nabla^{i}W^{\pm}_{ijkl}+\frac{1}{6}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})+\frac{1}{6}(\nabla_{k^{\prime}}Rg_{jl^{\prime}}-\nabla_{l^{\prime}}Rg_{jk^{\prime}}),\end{split}$

therefore we obtain

(3)
$\begin{split}R_{ijkl}\nabla^{i}f+R_{ijk^{\prime}l^{\prime}}\nabla^{i}f=&4\nabla^{i}W^{\pm}_{ijkl}+\frac{1}{6}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})\\\
+&\frac{1}{6}(\nabla_{k^{\prime}}Rg_{jl^{\prime}}-\nabla_{l^{\prime}}Rg_{jk^{\prime}})\end{split}$

For the right hand side, we have

$\begin{split}R_{ijkl}\nabla^{i}f=&W_{ijkl}\nabla^{i}f+\frac{1}{2}(R_{ik}g_{jl}+R_{jl}g_{ik}-R_{il}g_{jk}-R_{jk}g_{il})\nabla^{i}f\\\
-&\frac{R}{6}(g_{ik}g_{jl}-g_{il}g_{jk})\nabla^{i}f\\\
=&W_{ijkl}\nabla^{i}f+\frac{1}{2}(R_{jl}\nabla_{k}f-R_{jk}\nabla_{l}f+\frac{1}{2}\nabla_{k}Rg_{jl}-\frac{1}{2}\nabla_{l}Rg_{jk})\\\
-&\frac{R}{6}(g_{jl}\nabla_{k}f-g_{jk}\nabla_{l}f).\end{split}$

Taking the difference, we get

$\begin{split}D_{jkl}=&2\nabla^{i}W_{ijkl}-W_{ijkl}\nabla^{i}f\\\
=&\frac{1}{2}(R_{jl}\nabla_{k}f-R_{jk}\nabla_{l}f)+\frac{1}{12}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})-\frac{R}{6}(g_{jl}\nabla_{k}f-g_{jk}\nabla_{l}f),\end{split}$

Therefore

$\begin{split}D^{\pm}_{jkl}=&2\nabla^{i}W^{\pm}_{ijkl}-W^{\pm}_{ijkl}\nabla^{i}f\\\
=&\frac{1}{2}[(2\nabla^{i}W_{ijkl}-W_{ijkl}\nabla^{i}f)\pm(2\nabla^{i}W_{ijk^{\prime}l^{\prime}}-W_{ijk^{\prime}l^{\prime}}\nabla^{i}f)]\\\
=&\frac{1}{4}(R_{jl}\nabla_{k}f-R_{jk}\nabla_{l}f)+\frac{1}{24}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})-\frac{R}{12}(g_{jl}\nabla_{k}f-g_{jk}\nabla_{l}f)\\\
\pm&\frac{1}{4}(R_{jl^{\prime}}\nabla_{k^{\prime}}f-R_{jk^{\prime}}\nabla_{l^{\prime}}f)\pm\frac{1}{24}(\nabla_{k^{\prime}}Rg_{jl^{\prime}}-\nabla_{l^{\prime}}Rg_{jk^{\prime}})\\\
\mp&\frac{R}{12}(g_{jl^{\prime}}\nabla_{k^{\prime}}f-g_{jk^{\prime}}\nabla_{l^{\prime}}f),\end{split}$

∎

Fernández-López and García-Río [19] proved that if a gradient Ricci soliton
satisfies $\delta W=0$, then $\nabla f$ is an eigenvector of the Ricci tensor.
Following from Lemme 2.3, it is easy to see that in dimensions four, $\delta
W^{\pm}=0$ provides the same information, see also X. Cao, Tran [11],

######  Lemma 2.4.

Let $(M,g,f)$ be a four-dimensional gradient Ricci soliton. If $\delta
W^{\pm}=0$, then $\nabla f$, whenever nonzero, is an eigenvector of the Ricci
tensor.

Proof. In equation (3), if $\delta W^{\pm}=0$, then

$\begin{split}R_{ijkl}\nabla^{i}f+R_{ijk^{\prime}l^{\prime}}\nabla^{i}f=&\frac{1}{6}(\nabla_{k}Rg_{jl}-\nabla_{l}Rg_{jk})+\frac{1}{6}(\nabla_{k^{\prime}}Rg_{jl^{\prime}}-\nabla_{l^{\prime}}Rg_{jk^{\prime}})\end{split}$

Let $e_{1}=\frac{\nabla f}{|\nabla f|}$, and extend it to an orthonormal basis
$\\{e_{1},e_{2},e_{3},e_{4}\\}$ of $T_{p}M$. Let $j=k=1$, $l\neq 1$, then
since $(klk^{\prime}l^{\prime})=\sigma(1234)$, we have
$g_{jl}=g_{jk^{\prime}}=g_{jl^{\prime}}=0$. Therefore we get

$\begin{split}0=&\nabla_{l}R=2R_{lj}\nabla^{j}f=2|\nabla f|R_{1l}.\end{split}$

∎

Combining Lemma 2.3 and Lemma 2.4, we make a key observation that if $\delta
W^{\pm}=0$, then $W^{\pm}$ has a nice expression in terms of Ricci curvature
and scalar curvature,

######  Proposition 2.1.

Let $(M,g,f)$ be a four-dimensional gradient Ricci soliton with $\delta
W^{\pm}=0$. Denote $a_{1},a_{2},a_{3},a_{4}$ be the eigenvalues of the
traceless Ricci tensor with corresponding eigenvectors $e_{1}=\frac{\nabla
f}{|\nabla f|},e_{2},e_{3},e_{4}$. Then whenever $\nabla f\neq 0$,

$\begin{split}b_{1}\overset{\triangle}{=}W^{\pm}_{1212}=&-\frac{1}{12}(a_{1}+3a_{2})=\frac{1}{12}(a_{3}+a_{4}-2a_{2}),\\\
b_{2}\overset{\triangle}{=}W^{\pm}_{1313}=&-\frac{1}{12}(a_{1}+3a_{3})=\frac{1}{12}(a_{2}+a_{4}-2a_{3}),\\\
b_{3}\overset{\triangle}{=}W^{\pm}_{1414}=&-\frac{1}{12}(a_{1}+3a_{4})=\frac{1}{12}(a_{2}+a_{3}-2a_{4}),\\\
W^{\pm}_{1j1l}=&0,\quad\mathrm{if}\ j\neq l.\end{split}$

Proof. By Lemma 2.4 we have $R_{1j}=0$ for $j\neq 1$, which gives us

$\nabla_{1}R=2R_{1j}\nabla^{j}f=2R_{11}|\nabla f|.$

If $\delta W^{\pm}=0$, then

$\begin{split}-W^{\pm}_{1j1j}|\nabla f|=&-W^{\pm}_{ij1j}\nabla^{i}f\\\
=&\frac{1}{4}(R_{jj}\nabla_{1}f-R_{1j}\nabla_{j}f)+\frac{1}{24}(\nabla_{1}Rg_{jj}-\nabla_{j}Rg_{1j})\\\
-&\frac{R}{12}(g_{jj}\nabla_{1}f-g_{1j}\nabla_{j}f)\\\
\pm&\frac{1}{4}(R_{jj^{\prime}}\nabla_{1^{\prime}}f-R_{j1^{\prime}}\nabla_{j^{\prime}}f)\pm\frac{1}{24}(\nabla_{1^{\prime}}Rg_{jj^{\prime}}-\nabla_{j^{\prime}}Rg_{j1^{\prime}})\\\
\mp&\frac{R}{12}(g_{jj^{\prime}}\nabla_{1^{\prime}}f-g_{j1^{\prime}}\nabla_{j^{\prime}}f)\\\
=&\frac{1}{4}R_{jj}|\nabla f|+\frac{1}{12}R_{11}|\nabla f|-\frac{R}{12}|\nabla
f|.\end{split}$

If $\nabla f\neq 0$, then we get

$\begin{split}-W^{\pm}_{1212}=&\frac{1}{4}R_{22}+\frac{1}{12}R_{11}-\frac{R}{12}\\\
=&\frac{1}{12}[3(R_{22}-\frac{R}{4})+(R_{11}-\frac{R}{4})]\\\
=&\frac{1}{12}(a_{1}+3a_{2}),\end{split}$

similarly we get $W^{\pm}_{1313}$ and $W^{\pm}_{1414}$.

If $j\neq l$, then it is easy to compute that $W^{\pm}_{1j1l}=0$.

∎

##  3\. Proof of Theorem 1.1

First recall the Weitzenböck formula for $W^{\pm}$ (see [11] or [45]),

######  Proposition 3.1.

Let $(M,g,f)$ be a four-dimensional gradient Ricci soliton. Then

$\begin{split}\Delta_{f}|W^{\pm}|^{2}=&2|\nabla
W^{\pm}|^{2}+4\lambda|W^{\pm}|^{2}-36\det
W^{\pm}-\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle.\end{split}$

######  Remark 3.1.

The second author [45] derived a Weitzenböck formula for generalized
$m$-quasi-Einstein metrics (or “Einstein metrics” on smooth metric measure
spaces), which are defined by

$\begin{split}\mathrm{Ric}+\nabla^{2}f-\frac{1}{m}df\otimes df=\lambda
g,\end{split}$

for some $f,\lambda\in C^{\infty}(M)$ and an
$m\in\mathbb{R}\cup\\{\pm\infty\\}$, from which we expect a similar rigidity
result for quasi-Einstein four-manifolds with $\delta W^{\pm}=0$.

Next we compute,

######  Proposition 3.2.

Let $(M,g,f)$ be a four-dimensional gradient Ricci soliton. Let $h=f-\ln
R^{2}$, then

(4)
$\begin{split}\Delta_{h}\left(\frac{|W^{\pm}|}{R}\right)\geq&\frac{1}{2|W^{\pm}|R^{2}}\Big{(}R^{2}|W^{\pm}|^{2}-36R\det
W^{\pm}+4|W^{\pm}|^{2}|\overset{\circ}{\mathrm{Ric}}|^{2}\\\
&-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\Big{)}.\end{split}$

Proof. Recall the Kato inequality $|\nabla T|^{2}\geq|\nabla|T||^{2}$ for any
tensor $T$. From Proposition 3.1, we get

$\begin{split}\Delta_{f}|W^{\pm}|=&\frac{1}{2|W^{\pm}|}\Big{[}2|\nabla
W^{\pm}|^{2}-2|\nabla|W^{\pm}||^{2}+4\lambda|W^{\pm}|^{2}-36\det W^{\pm}\\\
&-\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\Big{]}\\\
\geq&\frac{1}{2|W^{\pm}|}\Big{[}4\lambda|W^{\pm}|^{2}-36\det
W^{\pm}-\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\Big{]}\end{split}$

and recall that

(5) $\begin{split}\Delta_{f}R=&2\lambda R-2|\mathrm{Ric}|^{2}.\end{split}$

Therefore we compute

$\begin{split}\Delta_{f}\left(\frac{|W^{\pm}|}{R}\right)=&\frac{\Delta_{f}|W^{\pm}|}{R}-\frac{|W^{\pm}|\Delta_{f}R}{R^{2}}-2\frac{\nabla|W^{\pm}|\nabla
R}{R^{2}}+2\frac{|W^{\pm}||\nabla R|^{2}}{R^{3}}\\\
\geq&\frac{1}{2|W^{\pm}|}(4\lambda|W^{\pm}|^{2}-36\det
W^{\pm}-\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle)\\\
-&\frac{|W^{\pm}|}{R^{2}}(2\lambda R-2|\mathrm{Ric}|^{2})\\\
-&2\frac{\nabla|W^{\pm}|\nabla R}{R^{2}}+2\frac{|W^{\pm}||\nabla
R|^{2}}{R^{3}}\\\
=&-2\frac{1}{R}\Big{\langle}\nabla\Big{(}\frac{|W^{\pm}|}{R}\Big{)},\nabla
R\Big{\rangle}+\frac{1}{2|W^{\pm}|R^{2}}\Big{(}4|W^{\pm}|^{2}|\mathrm{Ric}|^{2}\\\
-&36R\det
W^{\pm}-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\Big{)}\\\
=&-\Big{\langle}\nabla\Big{(}\frac{|W^{\pm}|}{R}\Big{)},\nabla\ln
R^{2}\Big{\rangle}+\frac{1}{2|W^{\pm}|R^{2}}\Big{(}R^{2}|W^{\pm}|^{2}\\\
-&36R\det
W^{\pm}+4|W^{\pm}|^{2}|\overset{\circ}{\mathrm{Ric}}|^{2}-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\Big{)}.\end{split}$

∎

We have,

######  Lemma 3.1.

Let $(M,g,f)$ be a four-dimensional gradient shrinking Ricci soliton with
$\delta W^{\pm}=0$, then whenever $\nabla f\neq 0$,

$\begin{split}R^{2}|W^{\pm}|^{2}-36R\det
W^{\pm}+4|W^{\pm}|^{2}|\overset{\circ}{\mathrm{Ric}}|^{2}-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\geq
0,\end{split}$

with equality if and only if, either

$\mathrm{(1)}$, $a_{2}=a_{3}=a_{4}$, i.e., $W^{\pm}=0$; or

$\mathrm{(2)}$, $a_{1}=a_{i}=-a,a_{j}=a_{k}=a$, $2\leq i,j,k\leq 4$, and
$R=4a$ for some $a>0$, where $a_{1},a_{2},a_{3},a_{4}$ are eigenvalues of
$\overset{\circ}{\mathrm{Ric}}$ with corresponding orthonormal eigenvectors
$e_{1}=\frac{\nabla f}{|\nabla f|},e_{2},e_{3},e_{4}$.

######  Remark 3.2.

Lemma 3.1 also works for gradient steady and expanding solitons, and the sign
of $a$ in the second equality case changes correspondingly.

The proof of the Lemma will be presented at the end of this section. We first
prove Theorem 1.1.

Proof of Theorem 1.1. B.-L. Chen [13] proved that any gradient shrinking Ricci
soliton has $R\geq 0$. Moreover, either $R>0$ on $M$, or $R\equiv 0$ on $M$,
and if $R\equiv 0$ then $(M,g)$ is a finite quotient of $\mathbb{R}^{4}$, see
[39, 40]. From now on we assume $R>0$.

If $M$ is compact, then from Proposition 3.2,

$\begin{split}0=&\int_{M}\Delta_{h}\left(\frac{|W^{\pm}|}{R}\right)e^{-h}dv\\\
\geq&\int_{M}\frac{1}{2|W^{\pm}|}\Big{(}R^{2}|W^{\pm}|^{2}-36R\det
W^{\pm}+4|W^{\pm}|^{2}|\overset{\circ}{\mathrm{Ric}}|^{2}\\\
&-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\Big{)}e^{-f}dv\end{split}$

so by Lemma 3.1 we get

(6) $\begin{split}&R^{2}|W^{\pm}|^{2}-36R\det
W^{\pm}+4|W^{\pm}|^{2}|\overset{\circ}{\mathrm{Ric}}|^{2}-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle\equiv
0.\end{split}$

If $M$ is noncompact, by equation (2) in Lemma 2.3, if $\delta W^{\pm}=0$,
then

$\begin{split}|W^{\pm}|\leq|\overset{\circ}{\mathrm{Ric}}|<|\mathrm{Ric}|,\end{split}$

Munteanu, Sesum [30] proved that for a gradient shrinking Ricci soliton,

$\begin{split}\int_{M}|\mathrm{Ric}|^{2}e^{-\delta f}dv<\infty,\end{split}$

for any $\delta>0$. Therefore if $\delta W^{\pm}=0$, then

$\Big{\|}\frac{|W^{\pm}|}{R}\Big{\|}_{L^{2}_{h}(M)}=\int_{M}|W^{\pm}|^{2}e^{-f}dv<\infty.$

By a maximum principle of Naber [32] and Petersen and Wylie [38], if
$\int_{M}e^{-h}dv<\infty$, then any $L_{h}^{2}$-integrable $h$-subharmonic
function is constant, therefore we conclude that $\frac{|W^{\pm}|}{R}$ is
constant, which also implies equation (6).

Recall that any gradient Ricci soliton is a real-analytic manifold (see [27]
or [29]), hence all $|\nabla f|^{2}$, $|W^{\pm}|^{2}$, and $R$ are analytic
functions on $M$, therefore either $\nabla f\equiv 0$ or $W^{\pm}\equiv 0$, or
the second equality case in Lemma 3.1 holds on $M$.

Case 1. If $\nabla f\equiv 0$ on $M$,then $(M,g)$ is Einstein.

Case 2. If $W^{\pm}\equiv 0$ on $M$, then $(M,g)$ is a finite quotient of
$S^{3}\times\mathbb{R}$ or $\mathbb{R}^{4}$ by [15, 7].

Case 3. If $\nabla f\not\equiv 0$ and $W^{\pm}\not\equiv 0$, then the second
case in Lemma 3.1 holds in an open dense set $S$ of $M$. Without loss of
generality, assume $\delta W^{+}=0$ (if $\delta W^{-}=0$ changing the
orientation we get $\delta W^{+}=0$). First it is easy to see that

$R_{11}=\overset{\circ}{R}_{11}+\frac{R}{4}=0,\quad R_{ii}=0,\quad
R_{jj}=R_{kk}=2a,$

$2\leq i,j,k\leq 4$, therefore by Lemma 2.4, $\nabla
R=\nabla_{1}Re_{1}=2R_{11}|\nabla f|=0$, that is $R\equiv$const on $S$. By the
continuity of $R$, we have $R\equiv$const on $M$, and
$|W^{\pm}|^{2}=\frac{R^{2}}{24}\equiv$const. Moreover in this case, $W^{\pm}$
has only two distinct eigenvalues:

$\pm\frac{R}{6},\mp\frac{R}{12},\mp\frac{R}{12},$

(in fact $\mathfrak{R}^{\pm}$ is two-nonnegative), hence by a Theorem of
Derdzinski (Proposition 5 in [16]), $g$ is a Kähler metric.

Assume $M$ is compact, since $R\equiv$const, by the soliton equation $R+\Delta
f=4\lambda$, so $f\equiv$const (and $g$ is Kähler-Einstein), contradiction!
Therefore $(M,g,f)$ is a complete noncompact cscK gradient shrinking Kähler-
Ricci soliton.

Furthermore, since the eigenvalues of Ricci curvature are $0,0,2a,2a$, so we
have $|\mathrm{Ric}|^{2}=8a^{2}$, plugging into equation (5), we get

$\begin{split}0=\Delta_{f}R=&2\lambda R-2|\mathrm{Ric}|^{2}\\\ =&8\lambda
a-16a^{2},\end{split}$

therefore $a=\frac{\lambda}{2}$, which in particular implies that
$0\leq\mathrm{Ric}\leq\lambda g$. By Proposition 1.3 in [39], $(M,g,f)$ is
rigid, i.e., it is a finite quotient of $N^{k}\times\mathbb{R}^{4-k}$, where
$N^{k}$ is an Einstein manifold. Since $g$ is Kähler, so $(M,g)$ is a finite
quotient of $N^{2}\times\mathbb{R}^{2}$. Since $R$ is positive, $N^{2}$ has to
be $S^{2}$, therefore $(M,g)$ is a finite quotient of
$S^{2}\times\mathbb{R}^{2}$.

∎

Proof of Theorem 1.2. On a four-manifold, a Kähler metric with constant scalar
curvature satisfies

$\delta W^{+}=0,\quad\frac{|W^{+}|^{2}}{R^{2}}=\frac{1}{24}.$

If $(M,g,f)$ is a gradient shrinking Ricci soliton, it follows directly from
Theorem 1.1.

If $(M,g,f)$ is a gradient steady Ricci soliton, then $R\equiv$const implies
that $R\equiv 0$ (see [21] or [44]), and by Proposition 4.3 in [39], $(M,g,f)$
is a finite quotient of $M\times\mathbb{C}$, where $M$ is a flat Riemann
surface.

If $(M,g,f)$ is a gradient expanding Ricci soliton, then by Proposition 3.2
and Lemma 3.1 we have

$\begin{split}R^{2}|W^{+}|^{2}-36R\det
W^{+}+4|W^{+}|^{2}|\overset{\circ}{\mathrm{Ric}}|^{2}-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{+},W^{+}\rangle\equiv
0.\end{split}$

Similar to the proof of Theorem 1.1, there are three cases,

Case 1. If $\nabla f\equiv 0$ on $M$, then $(M,g)$ is Kähler-Einstein.

Case 2. If $W^{+}\equiv 0$ on $M$, then $D^{+}\equiv 0$, hence by Lemma 2.3,
$D\equiv 0$, therefore $W\equiv 0$ by Theorem 5.1 in H.-D. Cao and Q. Chen
[7], and $(M,g,f)$ is a finite quotient of Gaussian expanding soliton by Y. Su
and K. Zhang [41].

Case 3. If $\nabla f\not\equiv 0$ and $W^{+}\not\equiv 0$, then it follows
from Case 3 in the proof of Theorem 1.1 that it is rigid, hence a finite
quotient of $M\times\mathbb{C}$, where $M$ is a Riemann surface of constant
negative curvature.

∎

Proof of Lemma 3.1. By proposition 2.1, we express each term in terms of
eigenvalues of traceless Ricci tensor,

$\begin{split}|\overset{\circ}{\mathrm{Rc}}|^{2}=&a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}\\\
=&2(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4}),\\\
|W^{\pm}|^{2}=&4(b_{1}^{2}+b_{2}^{2}+b_{3}^{2})\\\
=&\frac{1}{6}(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}-a_{2}a_{3}-a_{2}a_{4}-a_{3}a_{4})\\\
36\det
W^{\pm}=&\frac{1}{6}(a_{3}+a_{4}-2a_{2})(a_{2}+a_{4}-2a_{3})(a_{2}+a_{3}-2a_{4})\\\
=&\frac{1}{6}(-2a_{2}^{3}-2a_{3}^{3}-2a_{4}^{3}+3a_{2}^{2}a_{3}+3a_{3}^{2}a_{2}+3a_{2}^{2}a_{4}\\\
&\hskip
42.67912pt+3a_{4}^{2}a_{2}+3a_{3}^{2}a_{4}+3a_{4}^{2}a_{3}-12a_{2}a_{3}a_{4}),\\\
\langle(\overset{\circ}{\mathrm{Rc}}\circ\overset{\circ}{\mathrm{Rc}})^{\pm},W^{\pm}\rangle=&W^{\pm}_{ijij}\overset{\circ}{R}_{ii}\overset{\circ}{R}_{jj}\\\
=&2[b_{1}(a_{1}a_{2}+a_{3}a_{4})+b_{2}(a_{1}a_{3}+a_{2}a_{4})+b_{3}(a_{1}a_{4}+a_{2}a_{3})]\\\
=&\frac{1}{6}(2a_{2}^{3}+2a_{3}^{3}+2a_{4}^{3}+a_{2}^{2}a_{3}+a_{3}^{2}a_{2}+a_{2}^{2}a_{4}\\\
&\hskip
42.67912pt+a_{4}^{2}a_{2}+a_{3}^{2}a_{4}+a_{4}^{2}a_{3}-12a_{2}a_{3}a_{4}).\end{split}$

Therefore we get

(7) $\begin{split}6\phi=&6(R^{2}|W^{\pm}|^{2}-36R\det
W^{\pm}+4|W^{\pm}|^{2}|\overset{\circ}{\mathrm{Ric}}|^{2}-R\langle(\overset{\circ}{\mathrm{Ric}}\circ\overset{\circ}{\mathrm{Ric}})^{\pm},W^{\pm}\rangle)\\\
=&R^{2}(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}-a_{2}a_{3}-a_{2}a_{4}-a_{3}a_{4})\\\
-&4R(a_{2}^{2}a_{3}+a_{3}^{2}a_{2}+a_{2}^{2}a_{4}+a_{4}^{2}a_{2}+a_{3}^{2}a_{4}+a_{4}^{2}a_{3}-6a_{2}a_{3}a_{4})\\\
+&8(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4})\times\\\
&\ \
(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}-a_{2}a_{3}-a_{2}a_{4}-a_{3}a_{4}).\end{split}$

By abusing the noataion, we identify $\phi$ and $6\phi$. Observe that $\phi$
is a fourth-order homogeneous symmetric polynomial if we assume
$R=k(a_{2}+a_{3}+a_{4})$ for some $k\in\mathbb{R}$.

First we show $\phi\geq 0$ using Timofte’s criterion for positivity of
homogeneous symmetric polynomials (see Corollary 5.6 in [42]),

######  Proposition 3.3 (Timofte [42]).

Let $p$ be a fourth-order homogeneous symmetric polynomial on
$\mathbb{R}^{n}$, then

$p\geq 0\ \mathrm{on}\ \mathbb{R}^{n}\quad\Longleftrightarrow\quad
p(t\cdot\vec{1}_{\mathbb{R}^{i}},\vec{1}_{\mathbb{R}^{n-i}})\geq 0,\ \forall\
t\in[-1,1],\ i=1,2,...,n-1,$

where $\vec{1}=(1,1,..,1)$.

If $a_{1}\neq 0$, without of loss of generality, assume
$R=-ka_{1}=k(a_{2}+a_{3}+a_{4})$. In our case $n=3$, so we need to show that

$\phi(t,1,1)\geq 0,\quad\phi(t,t,1)\geq 0,\quad\forall\ t\in[-1,1].$

For $\phi(t,1,1)$, plugging into equation (7), recall that $R=k(t+2)$, we get

$\begin{split}\phi(t,1,1)=&(t-1)^{2}\Big{[}k^{2}(t+2)^{2}-8k(t+2)+8(t^{2}+2t+3)\Big{]}.\end{split}$

Consider $\phi(t,1,1)$ as a quadratic function of $k$. When $-1\leq t\leq 1$,
the discriminant

$\begin{split}\mathfrak{D}=&-32(t+2)^{2}(t-1)^{4}(t+1)^{2}\leq 0,\end{split}$

and $\mathfrak{D}<0$ when $-1<t<1$. Therefore for all $-1\leq t\leq 1$,

$\begin{split}\phi(t,1,1)\geq 0,\end{split}$

and $\phi(t,1,1)=0$ if and only if $t=1$, or $t=-1$ and $R=4$.

For $\phi(t,t,1)$, since we assume $a_{1}\neq 0$, so $t\neq-\frac{1}{2}$.
Plugging into equation (7), recall that $R=k(2t+1)$, we get

$\begin{split}\phi(t,t,1)=&(t-1)^{2}\Big{[}k^{2}(2t+1)^{2}-8kt(2t+1)+8(3t^{2}+2t+1)\Big{]}.\end{split}$

Consider $\phi(t,t,1)$ as a quadratic function of $k$, we see that when
$-1\leq t\leq 1$ and $t\neq-\frac{1}{2}$, the discriminant

$\begin{split}\mathfrak{D}=&-32(2t+1)^{2}(t-1)^{4}(t+1)^{2}\leq 0,\end{split}$

and $\mathfrak{D}<0$ when $-1<t<1$ and $t\neq-\frac{1}{2}$. Therefore for all
$-1\leq t\leq 1$ and $t\neq-\frac{1}{2}$,

$\begin{split}\phi(t,t,1)\geq 0,\end{split}$

and $\phi(t,t,1)=0$ if and only if $t=1$, or $t=-1$ and $R=-4$.

If $a_{1}=0$ (which corresponds to $t=-\frac{1}{2}$ in $\phi(t,t,1)$), i.e.,
$a_{2}+a_{3}+a_{4}=0$, then $\phi$ can be simplified as

$\begin{split}\phi=&3R^{2}(a_{2}^{2}+a_{3}^{2}+a_{2}a_{3})-36Ra_{2}a_{3}(a_{2}+a_{3})+24(a_{2}^{2}+a_{3}^{2}+a_{2}a_{3})^{2}.\end{split}$

Consider $\phi$ as a quadratic function of $R$, then its discriminant

$\begin{split}\mathfrak{D}=&36^{2}a_{2}^{2}a_{3}^{2}(a_{2}+a_{3})^{2}-36[a_{2}^{2}+a_{3}^{2}+(a_{2}+a_{3})^{2}]^{3}\leq
0.\end{split}$

Recall a well-known inequality: if $a+b+c=0$, then
$3\sqrt{6}|abc|\leq(a^{2}+b^{2}+c^{2})^{\frac{3}{2}}$, with equality if and
only if $a=-2b$, or $b=-2c$, or $c=-2a$. Since $-a_{2}-a_{3}+(a_{2}+a_{3})=0$,
we have

$\begin{split}\mathfrak{D}=&36^{2}a_{2}^{2}a_{3}^{2}(a_{2}+a_{3})^{2}-36[a_{2}^{2}+a_{3}^{2}+(a_{2}+a_{3})^{2}]^{3}\\\
\leq&24[a_{2}^{2}+a_{3}^{2}+(a_{2}+a_{3})^{2}]^{3}-36[a_{2}^{2}+a_{3}^{2}+(a_{2}+a_{3})^{2}]^{3}\\\
\leq&0,\end{split}$

with equality if and only if $a_{2}=a_{3}=0$. So $\phi\geq 0$, and $\phi=0$ if
and only if $a_{2}=a_{3}=a_{4}=0$.

Therefore we proved that $\phi(R,a_{2},a_{3},a_{4})\geq 0$ on
$\mathbb{R}^{4}$.

Next we show that when $a_{2}\neq a_{3}\neq a_{4}$, then $\phi>0$.

Assume that $a_{2}\neq a_{3}\neq a_{4}$ and $\phi(R,a_{2},a_{3},a_{4})=0$.
Taking the first derivatives we get,

$\begin{split}\phi_{a_{2}}=&R^{2}(2a_{2}-a_{3}-a_{4})-4R(2a_{2}a_{3}+2a_{2}a_{4}+a_{3}^{2}+a_{4}^{2}-6a_{3}a_{4})\\\
&\hskip
79.6678pt+16(2a_{2}^{3}+a_{2}a_{3}^{2}+a_{2}a_{4}^{2}-a_{3}^{2}a_{4}-a_{3}a_{4}^{2}-2a_{2}a_{3}a_{4})=0,\\\
\phi_{a_{3}}=&R^{2}(2a_{3}-a_{2}-a_{4})-4R(2a_{2}a_{3}+2a_{3}a_{4}+a_{2}^{2}+a_{4}^{2}-6a_{2}a_{4})\\\
&\hskip
79.6678pt+16(2a_{3}^{3}+a_{2}^{2}a_{3}+a_{3}a_{4}^{2}-a_{2}^{2}a_{4}-a_{2}a_{4}^{2}-2a_{2}a_{3}a_{4})=0,\\\
\phi_{a_{4}}=&R^{2}(2a_{4}-a_{2}-a_{2})-4R(2a_{2}a_{4}+2a_{3}a_{4}+a_{2}^{2}+a_{3}^{2}-6a_{2}a_{3})\\\
&\hskip
79.6678pt+16(2a_{4}^{3}+a_{2}^{2}a_{4}+a_{3}^{2}a_{4}-a_{2}^{2}a_{3}-a_{2}a_{3}^{2}-2a_{2}a_{3}a_{4})=0.\end{split}$

Taking the difference, since $a_{2}\neq a_{3}\neq a_{4}$, we get

$\begin{split}0=\frac{\phi_{a_{2}}-\phi_{a_{3}}}{a_{2}-a_{3}}=&3R^{2}-R(4(a_{2}+a_{3})-32a_{4})\\\
&\hskip
17.07182pt+16(2a_{2}^{2}+2a_{3}^{2}+2a_{4}^{2}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4}),\\\
0=\frac{\phi_{a_{2}}-\phi_{a_{4}}}{a_{2}-a_{4}}=&3R^{2}-R(4(a_{2}+a_{4})-32a_{3})\\\
&\hskip
17.07182pt+16(2a_{2}^{2}+2a_{3}^{2}+2a_{4}^{2}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4}),\\\
0=\frac{\phi_{a_{3}}-\phi_{a_{4}}}{a_{3}-a_{4}}=&3R^{2}-R(4(a_{3}+a_{4})-32a_{2})\\\
&\hskip
17.07182pt+16(2a_{2}^{2}+2a_{3}^{2}+2a_{4}^{2}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4}).\end{split}$

Taking the difference again, we get $a_{2}=a_{3}=a_{4}$, contradiction! So
$\phi>0$ when $a_{2}\neq a_{3}\neq a_{4}$.

Therefore $\phi\geq 0$, and by Timofte’s criterion and above argument,
$\phi=0$ if and only if,

either $a_{2}=a_{3}=a_{4}$, i.e. $W^{\pm}=0$;

or $a_{1}=a_{i}=-a,a_{j}=a_{k}=a$, $2\leq i,j,k\leq 4$, and $R=4a$, for some
$a>0$.

∎

## References

  * [1] Besse, A., _Einstein manifolds_ , Berlin-Heidelberg, Springer-Verlag, 1987.
  * [2] Cai, M., _On shrinking gradient Ricci soliton with nonnegative sectional curvature_ , arXiv:math.DG/1303.2728.
  * [3] Calamai, S., Petrecca D., _On Calabi extremal Kähler-Ricci solitons_ , arXiv:math.DG/1401.6293, 2014.
  * [4] Cao, H.-D., _Existence of gradient Kähler-Ricci solitons_ , Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), A. K. Peters (ed.), Wellesley, MA, 1996, 1 16.
  * [5] Cao, H.-D., _Recent progress on Ricci solitons_ , Recent advances in geometric analysis, 1 38, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010.
  * [6] Cao, H.-D., Chen, Q., _On locally conformally flat gradient steady Ricci solitons._ , Trans. Amer. Math. Soc. 364(2012), 2377-2391.
  * [7] Cao, H.-D., Chen, Q., _On Bach-flat gradient shrinking Ricci solitons_ , Duke Math. J. 162 (2013), 1003–1204.
  * [8] Cao, H.-D., Chen, B.-L., Zhu, X.-P., _Recent developments on Hamilton’s Ricci flow_ , Surveys in differential geometry. Vol. XII. Geometric flows, 47–112, Surv. Differ. Geom. 12, Int. Press, Somerville, MA, 2008.
  * [9] Cao, X., _Closed gradient shrinking ricci soli tons with positive curvature operator_ , J. Geom. Anal. 17 (2007), 451–459.
  * [10] Cao, X., Wang, B., Zhang, Z., _On locally conformally flat gradient Ricci solitons_ , Comm. Contemp. Math. 13 (2010), 269–282.
  * [11] Cao, X., Tran, H., _The Weyl curvature of gradient Ricci solitons_ , arXiv:math.DG/1311.0846.
  * [12] Catino, G., _Complete gradient shrinking Ricci solitons with pinched curvature_ , Math. Ann. 35 (2013), 629–635.
  * [13] Chen, B.-L., _Strong uniqueness of the Ricci flow_ , J. Differential Geom. 82 (2009), 363–382.
  * [14] Chen, Q., Zhu, M., _On rigidity of gradient Kähler-Ricci solitons with harmonic Bochner tensor_ , Proc. Amer. Math. Soc. 140 (2012), 4017–4025.
  * [15] Chen, X., Wang, Y., _On four-dimensional anti-self-dual gradient Ricci solitons_ , J. Geom. Anal. DOI 10.1007/s12220-014-9471-8.
  * [16] Derdziński, A., _Self-dual Kähler manifolds and Einstein manifolds of dimension four_ , Com. Math. 49 (1983), 405–433.
  * [17] Eminenti, M., La Nave, G., Mantegazza. C., _Ricci solitons: the equation point of view_ , Manuscripta Math. 127 (2008), 345–367.
  * [18] Feldman, M., Ilmanen, T. and Knopf, D., _Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons_ , J. Diff. Geom., 65, 2003, 169 209.
  * [19] Fernández-López, M., García-Río, E., _Rigidity of shrinking Ricci solitons_ , Math. Z. 269 (2011), 461–466.
  * [20] Fernández-López, M., García-Río, E., _On gradient Ricci solitons with constant scalar curvature_ , arXiv:math.DG/1409.3359.
  * [21] Fernández-López, M., García-Río, E., _Maximum principles and gradient Ricci solitons_ , J. Differ. Equ. 251 (2011), 73–81.
  * [22] Grusky, M., _Four-manifolds with $\delta W^{+}=0$ and Einstein constants of the sphere_, Math. Ann. 318 (2000), 417–431.
  * [23] Hamilton, R. S., _Three manifolds with positive Ricci curvature_ , J. Differential Geom. 17 (1982), 255–306.
  * [24] Hamilton, R. S., _The Ricci flow on surfaces_ , Contemporary Mathematics 71 (1988), 237–261.
  * [25] Hamilton, R. S., _The formation of singularities in the Ricci flow_ , Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Int. Press, Cambridge, MA, 1995.
  * [26] Ivey, T., _Ricci solitons on compact three-manifolds_ , Differential Geom. Appl. 3 (1993), 301–307.
  * [27] Ivey, T., _Local existence of Ricci solitons_ , Manuscripta Math. 91 (1996), 151 162.
  * [28] Koiso, N., _On rotationally symmmetric Hamilton’s equation for Kähler-Einstein metrics_ , Recent Topics in Diff. Anal. Geom., Adv. Studies Pure Math., 18-I, Academic Press, Boston, MA, 1990, 327 337.
  * [29] Kotschwar, B., _A local version of Bando’s theorem on the real-analyticity of solutions to the Ricci flow_. Bull. London Math. Soc. 45 (2013), 153–158.
  * [30] Munteanu, O., Sesum, N., _On gradient Ricci solitons_ , J. Geom. Anal. 23 (2013), 539–561.
  * [31] Munteanu, O., Wang, M.-T., _The curvature of gradient Ricci solitons_ , Math. Res. Lett. 18 (2011), 1051–1069.
  * [32] Naber, A., _Noncompact shrinking four solitons with nonnegative curvature_ , J. Reine Angew. Math. 645 (2010), 125–153.
  * [33] Ni, L., Wallach, N., _On a classification of the gradient shrinking Ricci solitons_ , Math. Res. Lett. 15 (2008), 941–955.
  * [34] Ni, L., Wallach, N., _On 4-dimensional gradient shrinking solitons_ , Int. Math. Res. Notices 4 (2008), rnm152.
  * [35] Perelman, G., _The entropy formula for the Ricci slow and its geometric applications_ , arXiv:math.DG/0211159, 2002.
  * [36] Perelman, G., _Ricci flow with surgery on three-manifolds_ , arXiv:math.DG/0303109, 2003.
  * [37] Perelman, G., _Finite extinction time for the solutions to the Ricci flow on certain three-manifolds_ , arXiv:math.DG/0307245, 2003.
  * [38] Petersen, P., Wylie, W., _On the classification of gradient Ricci solitons_ , Geom. Topol. 14 (2010), 2277–2300.
  * [39] Petersen, P., Wylie, W., _Rigidity of gradient Ricci solitons_ , Pacific J. Math. 241 (2009), 329–345.
  * [40] Pigola, S., Rimoldi, M., Setti, A., _Remarks on non-compact gradient Ricci solitons_ , Math. Z. 268 (2011), 777–790.
  * [41] Su, Y., Zhang, K., _On the Kähler-Ricci solitons with vanishing Bochner-Weyl tensor_ , Acta Math. Sci. Ser. B Engl. Ed. 32B (2012), 1239–1244.
  * [42] Timofte, V., _On the positivity of symmetric polynomial functions. Part I: general results_ , J. Math. Anal. Appl. 284 (2003), 174–190.
  * [43] Wang, X. J. and Zhu, X. H., _Kähler-Ricci solitons on toric manifolds with positive first Chern class_ , Adv. Math., 188(1), 2004, 87 103.
  * [44] Wu, P., _On the potential function of gradient steady Ricci solitons_ , J. Geom. Anal. 23 (2011), 221–228.
  * [45] Wu, P., _Berger curvature decomposition, Weitzenböck formula, and canonial metrics on four-manifolds_ , preprint.
  * [46] Zhang, Z.-H., _Gradient shrinking solitons with vanishing Weyl tensor_ , Pacific J. Math. 242 (2009), 2755–2759.