On Gradient Shrinking Ricci Solitons with Radial Conditions

Abstract

In this paper, we prove an n-dimensional radially flat gradient shrinking Ricci solitons with $div^2W(\nabla f,\nabla f)=0$ is rigid. Moreover, we show that a four-dimensional radially flat gradient shrinking Ricci soliton with $\text {div}^2W^\pm (\nabla f,\nabla f)=0$ is either Einstein or a finite quotient of ${\mathbb {R}}^4$, ${\mathbb {S}}^2\times {\mathbb {R}}^2$ or ${\mathbb {S}}^3\times {\mathbb {R}}$.

1 Introduction

A complete Riemannian manifold $(M^n,g,f)$ is called a gradient shrinking Ricci soliton if there exists a smooth function f on $M^n$ such that the Ricci tensor Ric of the metric g satisfies the equation

$$\begin{aligned} Ric+\nabla ^2f=\lambda g. \end{aligned}$$

(1.1)

for some constant $\lambda >0$. By scaling the metric g, one customarily assumes $\lambda =\frac{1}{2}$. In this paper, we focus on the geometric structures of gradient shrinking Ricci solitons.

A gradient soliton is rigid if it is a flat bundle $N\times _{\Gamma }{\mathbb {R}}^k$, where $\Gamma$ acts freely on an Einstein manifold N and by orthogonal transformations on ${\mathbb {R}}^k$ (see Peterson–Wylie [11] for details).

Some classification theorems for gradient shrinking Ricci solitons have been proved in the recent years by various authors. An n-dimensional locally conformally flat gradient shrinking Ricci soliton was classified (cf. [6, 12, 17]). Moreover, an n-dimensional gradient shrinking Ricci soliton with harmonic Weyl tensor is rigid (see [7, 9]). More generally, Cao and Chen [1] proved that an n-dimensional $(n\ge 5)$ complete non-compact Bach-flat gradient shrinking Ricci soliton is a finite quotient of ${\mathbb {R}}^n$ or ${\mathbb {R}}\times N^{n-1}$, where N is an $(n-1)$-dimensional Einstein manifold. In 2016, Catino et al. [3] showed that a gradient shrinking Ricci soliton with fourth-order divergence-free Weyl tensor (i.e. $\text {div}^4W=0$) is rigid.

In the case of dimension 4, Naber [10] classified four-dimensional non-compact shrinking Ricci solitons with bounded nonnegative Riemannian curvature. Cao and Chen [1] proved that a four-dimensional complete non-compact Bach-flat gradient shrinking Ricci soliton is a finite quotient of ${\mathbb {R}}^4$ or ${\mathbb {R}}\times {\mathbb {S}}^3$. Under the condition of $W^+=0$, Chen and Wang [5] showed that a four-dimensional complete gradient shrinking Ricci soliton is isometric to finite quotients of ${\mathbb {R}}^4$, ${\mathbb {S}}^3\times {\mathbb {R}}$, ${\mathbb {S}}^4$, or ${\mathbb {C}}P^2$. More generally, Wu et al. [13] proved that a four-dimensional gradient shrinking Ricci soliton with half harmonic Weyl tensor (i.e. $\text {div}W^\pm =0$) is either Einstein or a finite quotient of ${\mathbb {R}}^4$, ${\mathbb {R}}^2\times {\mathbb {S}}^2$ or ${\mathbb {R}}\times {\mathbb {S}}^3$.

Under radial conditions, Peterson and Wylie [11] proved that a gradient soliton is rigid if and only if it has constant scalar curvature and is radially flat, that is, $sec(E,\nabla f)=0$, where sec stands for the sectional curvature operator and E is an arbitrary vector field. Moreover, they [12] showed that a complete gradient Ricci soliton with constant scalar curvature and $W(\nabla f,\cdot ,\cdot ,\nabla f)=o(|\nabla f|^2)$ is a flat bundle of rank 0, 1, or n over an Einstein manifold. Moreover, the authors [14] proved that a gradient shrinking Ricci soliton is rigid if $Rm(\cdot ,\nabla f)\nabla f=0$ and $\text {div}^2B\le 0$. They [14] also showed that a complete non-compact gradient expanding Ricci soliton with nonnegative Ricci curvature is rigid if $Rm(\cdot ,\nabla f)\nabla f=0$ and $\text {div}^2B\ge 0$.

In order to state our results precisely, we introduce the following definitions for the Riemannian curvature:

$$\begin{aligned}&(\text {div}Rm)_{ijk}:=\nabla _lR_{ijkl},\\&(\text {div}^2Rm)_{ik}:=\nabla _j\nabla _lR_{ijkl}. \end{aligned}$$

For the Weyl tensor, we define:

$$\begin{aligned}&(\text {div}W)_{ijk}:=\nabla _lW_{ijkl},\\&(\text {div}^2W)_{ik}:=\nabla _j\nabla _lW_{ijkl},\\&(\text {div}W^\pm )_{ijk}:=\nabla _lW_{ijkl}^\pm ,\\&(\text {div}^2W^\pm )_{ik}:=\nabla _j\nabla _lW_{ijkl}^\pm . \end{aligned}$$

We explore the geometric structures of gradient shrinking Ricci solitons with some radially conditions. The main results of this paper are the following theorems.

For n-dimensional gradient shrinking Ricci solitons, we have the following rigid result.

Theorem 1.1

Let $(M^n,g,f)$ be a radially flat gradient shrinking Ricci soliton with $\text {div}^2W(\nabla f,\nabla f)=0$. Then $(M^n,g,f)$ is rigid.

For 4-dimensional gradient shrinking Ricci solitons, we have the following classification theorem.

Theorem 1.2

Let $(M^4,g,f)$ be a four-dimensional radially flat gradient shrinking Ricci soliton. If $\text {div}^2W^\pm (\nabla f,\nabla f)=0$. Then $(M^4,g,f)$ is either Einstein or a finite quotient of ${\mathbb {R}}^4$, ${\mathbb {S}}^2\times {\mathbb {R}}^2$ or ${\mathbb {S}}^3\times {\mathbb {R}}$.

Remark 1.3

It is clear from the proof that the assumptions on the vanishing of $\text {div}^2W(\nabla f,\nabla f)$ and $\text {div}^2W^\pm (\nabla f,\nabla f)$ in the above theorems can be trivially relaxed to a (suitable) inequality. To be precise, the condition of $\text {div}^2W(\nabla f,\nabla f)=0$ in Theorem 1.1 can be relaxed to $\int \text {div}^2W(\nabla f,\nabla f)e^{-f}\ge 0$, while the the condition of $\text {div}^2W(\nabla f,\nabla f)=0$ in Theorem 1.2 can be relaxed to $\int \text {div}^2W^\pm (\nabla f,\nabla f)e^{-f}\ge 0$.

We arrange this paper as follows. In Sect. 2, we fix our notations and recall some known results about gradient Ricci solitons that we shall need in the proof of Theorems 1.1 and 1.2. We prove the rigid result for n-dimensional gradient shrinking Ricci solitons (Theorem 1.1) in Sect. 4. Before we classify the four-dimensional gradient shrinking Ricci soliton, we present some useful formulas of the curvature in Sect. 5. Finally, we finish the proof of Theorem 1.2 in Sect. 6.

2 Preliminaries

First of all, we recall that on any n-dimensional $(n\ge 3)$ Riemannian manifold, the Weyl tensor is given by

$$\begin{aligned} W_{ijkl}:= & {} R_{ijkl}-\frac{1}{n-2}(g_{ik}R_{jl}-g_{il}R_{jk}-g_{jk}R_{il}+g_{jl}R_{ik})\\&+\frac{R}{(n-1)(n-2)}(g_{ik}g_{jl}-g_{il}g_{jk}), \end{aligned}$$

The Cotton tensor is defined as

$$\begin{aligned} C_{ijk}:=\nabla _iR_{jk}-\nabla _jR_{ik}-\frac{1}{2(n-1)}(g_{jk}\nabla _iR-g_{ik}\nabla _jR). \end{aligned}$$

The relation between the Cotton tensor and the divergence of the Weyl tensor is

$$\begin{aligned} C_{ijk}=-\frac{n-2}{n-3}\nabla _lW_{ijkl}. \end{aligned}$$

In the case of dimension 4, denote $(i'j')$ to be the dual of any pair (ij) with $1\le i\ne j\le 4$. That is the pair such that $e_i\wedge e_j\pm e_{i'}\wedge e_{j'}\in \wedge ^{\pm }M^4$. In other words, $(i'j'ij)=\sigma (1234)$ for some even permutation $\sigma \in S_4$, i.e.

$$\begin{aligned}&(iji'j')\in \{(1234),(1342),(1423),(2143),(2314),(2431),\\&\quad (3124),(3241),(3412),(4132),(4213),(4321)\}. \end{aligned}$$

For any (0, 4)-tensor T of an 4-dimensional manifold, its (anti-)self-dual part is

$$\begin{aligned} T^\pm _{ijkl}=\frac{1}{4}(T_{ijkl}\pm T_{ijk'l'}\pm T_{i'j'kl}+T_{i'j'k'l'}). \end{aligned}$$

On four-manifolds, the Weyl tensor has sufficiently exotic symmetries.

Proposition 2.1

(Wu et al. [13]) Let (M, g) be a four-dimensional Riemannian manifold. Then

$$\begin{aligned} W_{ijkl}=W_{i'j'k'l'}, \end{aligned}$$

therefore,

$$\begin{aligned} W_{ijkl}^\pm =\pm W_{ijk'l'}^\pm =\pm W_{i'j'kl}^\pm =W_{i'j'k'l}^\pm =\frac{1}{2}(W_{ijkl}\pm W_{ijk'l'}). \end{aligned}$$

It follows that

$$\begin{aligned} W^\pm _{ijkl}=\frac{1}{2}(W_{ijkl}\pm W_{i'j'kl}). \end{aligned}$$

Therefore, we have

$$\begin{aligned} (\text {div}W^\pm )_{ijk}:=\nabla _lW^\pm _{ijkl}=\frac{1}{2}(\nabla _lW_{ijkl}\pm \nabla _lW_{i'j'kl}), \end{aligned}$$

and

$$\begin{aligned} (\text {div}^2W^\pm )_{ik}:=\nabla _j\nabla _lW^\pm _{ijkl}=\frac{1}{2}(\nabla _j\nabla _lW_{ijkl}\pm \nabla _j\nabla _lW_{i'j'kl}). \end{aligned}$$

Next, we recall some basic facts about complete gradient shrinking Ricci solitons.

Proposition 2.2

(Yang and Zhang [15]) Let $(M^n,f,g)$ be a gradient Ricci soliton satisfying (1.1). Then we have

$$\begin{aligned} (\text {div}^2Rm)_{ik}=2\lambda R_{ik}+\nabla _lR_{ik}\nabla _lf-\frac{1}{2}\nabla _i\nabla _kR-R_{ik}^2-R_{ijkl}R_{jl}, \end{aligned}$$

(2.1)

and

$$\begin{aligned} (\text {div}^2W)_{ik}=\frac{n-3}{n-2}(\text {div}^2Rm)_{ik}-\frac{n-3}{2(n-1)(n-2)}(g_{ik}\Delta R-\nabla _k\nabla _iR). \end{aligned}$$

(2.2)

Lemma 2.3

(Cao and Zhou [2]) Let $(M^n,g)$ be a complete gradient shrinking soliton with (1.1). Then,

• 1. the potential function f satisfies the estimates

  $$\begin{aligned} \frac{1}{4}(r(x)-c_1)^2\le f(x)\le \frac{1}{4}(r(x)+c_2)^2, \end{aligned}$$

  (2.3)

  where $r(x)=d(x_0,x)$ is the distance function from some fixed point $x_0\in M$, $c_1$ and $c_2$ are positive constants depending only on n and the geometry of g on the unit ball $B(x_0,1)$;

• 2. there exists some constant $C>0$ such that

  $$\begin{aligned} Vol(B(x_0,s))\le Cs^n \end{aligned}$$

  (2.4)

  for $s>0$ sufficiently large.

Lemma 2.4

(Peterson and Wylie [11]) A gradient soliton is rigid if and only if it has constant scalar curvature and is radially flat, that is, $\text {sec}(E,\nabla f)=0$.

3 Two Formulas for Gradient Ricci Solitons

In this section, we prove two identities for gradient Ricci solitons. First of all, we give a identity of $\text {div}Rm^2(\nabla f,\nabla f)$.

Proposition 3.1

Let $(M^n,g,f)$ be a gradient Ricci soliton. Then, we have

$$\begin{aligned} \text {div}^2Rm(\nabla f,\nabla f)=\lambda R_{ik}\nabla _i f\nabla _k f-R_{ijkl}R_{jl}\nabla _if\nabla _kf, \end{aligned}$$

(3.1)

Proof

It follows from (2.1) that

$$\begin{aligned}&(\text {div}^2Rm)_{ik}\nabla _if\nabla _kf\\&\quad = 2\lambda R_{ik}\nabla _i f\nabla _k f+\nabla _lR_{ik}\nabla _lf\nabla _if\nabla _kf-\frac{1}{2}\nabla _i\nabla _kR\nabla _if\nabla _kf\\&\quad \quad -R_{ik}^2\nabla _if\nabla _kf-R_{ijkl}R_{jl}\nabla _if\nabla _kf\\&\quad = 2\lambda R_{ik}\nabla _i f\nabla _k f+\nabla _l(R_{ik}\nabla _if)\nabla _lf\nabla _kf-R_{ik}\nabla _i\nabla _lf\nabla _lf\nabla _kf\\&\quad \quad -\frac{1}{2}\nabla _i\nabla _kR\nabla _if\nabla _kf-R_{ip}R_{kp}\nabla _if\nabla _kf-R_{ijkl}R_{jl}\nabla _if\nabla _kf\\&\quad = 2\lambda R_{ik}\nabla _i f\nabla _k f+\frac{1}{2}\nabla _l\nabla _kR\nabla _lf\nabla _kf-\lambda R_{ik}\nabla _i f\nabla _k f+R_{ik}R_{il}\nabla _lf\nabla _kf\\&\quad \quad -\frac{1}{2}\nabla _i\nabla _kR\nabla _if\nabla _kf-R_{ip}R_{kp}\nabla _if\nabla _kf-R_{ijkl}R_{jl}\nabla _if\nabla _kf\\&\quad =\lambda R_{ik}\nabla _i f\nabla _k f-R_{ijkl}R_{jl}\nabla _if\nabla _kf, \end{aligned}$$

where we used $Ric(\nabla f,\cdot )=\frac{1}{2}\nabla R$ and (1.1) in the third equality. $\square$

Lemma 3.2

Let $(M^n,g,f)$ be a gradient radially flat Ricci soliton. Then, we have

$$\begin{aligned} (\text {div}^2W)_{ik}\nabla _if\nabla _kf=\frac{n-3}{2(n-1)(n-2)}(\nabla _k\nabla _iR\nabla _if\nabla _kf-|\nabla f|^2\Delta R). \end{aligned}$$

(3.2)

Proof

From (2.2) and Proposition 3.1, we have

$$\begin{aligned}&(\text {div}^2W)_{ik}\nabla _if\nabla _kf \nonumber \\&\quad = \frac{n-3}{n-2}(\text {div}^2Rm)_{ik}\nabla _if\nabla _kf+\frac{n-3}{2(n-1)(n-2)}(\nabla _k\nabla _iR-g_{ik}\Delta R)\nabla _if\nabla _kf \nonumber \\&\quad =\frac{n-3}{n-2}(\lambda R_{ik}\nabla _i f\nabla _k f-R_{ijkl}R_{jl}\nabla _if\nabla _kf) \nonumber \\&\quad \quad +\frac{n-3}{2(n-1)(n-2)}(\nabla _k\nabla _iR\nabla _if\nabla _kf-|\nabla f|^2\Delta R). \end{aligned}$$

(3.3)

Note that $sec(E,\nabla f)=0$ implies that $Ric(\nabla f,\nabla f)=0$. In the following, we show that $R_{ijkl}R_{jl}\nabla _if\nabla _kf=0$ on $M^n$.

$\forall x\in M^n$, we divide the arguments into two cases:

• Case 1: $\nabla f(x)=0$. It is clear that (3.2) holds at x.

• Case 2: $\nabla f(x)\ne 0$. In this case, we denote by $\{e_i\}_{i=1}^n$ a local orthonormal frame of $M^n$ with $e_1=\frac{\nabla f}{|\nabla f|}$. We use $\{\alpha _i\}_{i=1}^n$ to represent eigenvalues of the Ricci tensor with corresponding orthonormal eigenvectors $\{e_i\}_{i=1}^n$, respectively. Since $sec(E,\nabla f)=0$, at x we have

  $$\begin{aligned} R_{ijkl}R_{jl}\nabla _if\nabla _kf=\sum _{s=2}^nR_{1s1s}\alpha _s|\nabla f|^2=0. \end{aligned}$$

  (3.4)

  At x, (3.2) follows from (3.3) and (3.4).

This completes the proof of Lemma 3.2. $\square$

4 Proof of Theorem 1.1

For gradient shrinking Ricci soliton satisfying the equation

$$\begin{aligned} Ric+\nabla ^2f=\frac{1}{2}g, \end{aligned}$$

(4.1)

integrating the right side of (3.2), we have the following lemma.

Lemma 4.1

Let $(M^n,g,f)$ be a gradient shrinking Ricci soliton. Then, we have

$$\begin{aligned}&\int \nabla _k\nabla _iR\nabla _if\nabla _kfe^{-f}-\int |\nabla f|^2\Delta Re^{-f} \nonumber \\&\quad = -\frac{1}{2}\int |\nabla R|^2e^{-f}+2\int RRic(\nabla f,\nabla f)e^{-f}-(n-1)\int Ric(\nabla f,\nabla f)e^{-f}.\nonumber \\ \end{aligned}$$

(4.2)

Proof

We divide the arguments into the compact case and the complete non-compact case:

• The compact case:

Integrating by parts,we have

$$\begin{aligned}&\int \nabla _k\nabla _iR\nabla _if\nabla _kfe^{-f}\nonumber \\&\quad =-\int \nabla _iR\nabla _k\nabla _if\nabla _kfe^{-f}-\int \nabla _iR\nabla _if\Delta fe^{-f}+\int \langle \nabla R,\nabla f\rangle |\nabla f|^2e^{-f}\nonumber \\&\quad =-\frac{1}{2}\int \langle \nabla R,\nabla f\rangle e^{-f}+\int \nabla _iRR_{ik}\nabla _kfe^{-f}+\int R\langle \nabla R,\nabla f\rangle e^{-f}\nonumber \\&\quad \quad -\frac{n}{2}\int \langle \nabla R,\nabla f\rangle e^{-f}\nonumber \\&\quad \quad +\int \langle \nabla R,\nabla f\rangle |\nabla f|^2e^{-f}\nonumber \\&\quad =\frac{1}{2}\int |\nabla R|^2e^{-f}+\int R\langle \nabla R,\nabla f\rangle e^{-f} \nonumber \\&\quad \quad -\frac{n+1}{2}\int \langle \nabla R,\nabla f\rangle e^{-f}\nonumber \\&\quad \quad +\int \langle \nabla R,\nabla f\rangle |\nabla f|^2e^{-f} \end{aligned}$$

(4.3)

where we used (4.1) and $\Delta f=\frac{n}{2}-R$ in the second equality. Moreover, we used $Ric(\nabla f,\cdot )=\frac{1}{2}\nabla R$ in the third equality.

Integrating by parts, we obtain

$$\begin{aligned}&\int |\nabla f|^2\Delta Re^{-f}\nonumber \\&\quad = -2\int \nabla _iR\nabla _i\nabla _j f\nabla _jfe^{-f}+\int \langle \nabla R,\nabla f\rangle |\nabla f|^2e^{-f}\nonumber \\&\quad =-\int \langle \nabla R,\nabla f\rangle e^{-f}+2\int \nabla _iRR_{ij}\nabla _jfe^{-f}+\int \langle \nabla R,\nabla f\rangle |\nabla f|^2e^{-f}\nonumber \\&\quad =-\int \langle \nabla R,\nabla f\rangle e^{-f}+\int |\nabla R|^2e^{-f}+\int \langle \nabla R,\nabla f\rangle |\nabla f|^2e^{-f}, \end{aligned}$$

(4.4)

where we used (4.1) in the second equality and $Ric(\nabla f,\cdot )=\frac{1}{2}\nabla R$ in the third.

Combining (4.3) and (4.4), we obtain (4.2).

• The complete non-compact case:

Let $\phi (t)=\frac{s-t}{s}$ on (0, s), $\phi =0$ on $[s,\infty )$ for any fixed constant $s>0$. From Lemma 2.3, we know that f is of quadratic growth and the volume of a geodesic ball is at most Euclidean growth. Moreover, the support of $\phi (f)$ is a closed subset of the compact set $\{x|\frac{1}{4}(r(x)-c_1)^2\le s\}$ for any fixed $s>0$. Therefore, $\phi (f)$ has compact support on $M^n$.

Integrating by parts, we have

$$\begin{aligned}&\int \nabla _k\nabla _iR\nabla _if\nabla _kf\phi (f)e^{-f}\nonumber \\&\quad =-\int \nabla _iR\nabla _k\nabla _if\nabla _kf\phi (f)e^{-f}-\int \nabla _iR\nabla _if\Delta f\phi (f)e^{-f}\nonumber \\&\quad \quad +\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi (f)e^{-f}\nonumber \\&\quad \quad -\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi '(f)e^{-f}\nonumber \\&\quad =-\frac{1}{2}\int \langle \nabla R,\nabla f\rangle \phi (f)e^{-f}+\int \nabla _iRR_{ik}\nabla _kf\phi (f)e^{-f}+\int R\langle \nabla R,\nabla f\rangle \phi (f)e^{-f}\nonumber \\&\quad \quad -\frac{n}{2}\int \langle \nabla R,\nabla f\rangle \phi (f)e^{-f}+\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi (f)e^{-f}-\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi '(f)e^{-f}\nonumber \\&\quad =\frac{1}{2}\int |\nabla R|^2\phi (f)e^{-f}+\int R\langle \nabla R,\nabla f\rangle \phi (f)e^{-f}-\frac{n+1}{2}\int \langle \nabla R,\nabla f\rangle \phi (f)e^{-f}\nonumber \\&\quad \quad +\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi (f)e^{-f}-\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi '(f)e^{-f}, \end{aligned}$$

(4.5)

where we used (4.1) and $\Delta f=\frac{n}{2}-R$ in the second equality. Moreover, we used $Ric(\nabla f,\cdot )=\frac{1}{2}\nabla R$ in the third equality.

Integrating by parts, we obtain

$$\begin{aligned}&\int |\nabla f|^2\Delta R\phi (f)e^{-f}\nonumber \\&\quad =-2\int \nabla _iR\nabla _i\nabla _j f\nabla _jf\phi (f)e^{-f}+\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi (f)e^{-f}\nonumber \\&\qquad -\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi '(f)e^{-f}\nonumber \\&\quad =-\int \langle \nabla R,\nabla f\rangle \phi (f)e^{-f}+2\int \nabla _iRR_{ij}\nabla _jf\phi (f)e^{-f}\nonumber \\&\quad \quad +\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi (f)e^{-f}-\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi '(f)e^{-f}\nonumber \\&\quad =-\int \langle \nabla R,\nabla f\rangle \phi (f)e^{-f}+\int |\nabla R|^2\phi (f)e^{-f}\nonumber \\&\quad \quad +\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi (f)e^{-f}-\int \langle \nabla R,\nabla f\rangle |\nabla f|^2\phi '(f)e^{-f}, \end{aligned}$$

(4.6)

where we used (4.1) in the second equality and $Ric(\nabla f,\cdot )=\frac{1}{2}\nabla R$ in the third.

Combining (4.5) and (4.6), we obtain

$$\begin{aligned}&\int \nabla _k\nabla _iR\nabla _if\nabla _kf\phi (f)e^{-f}-\int |\nabla f|^2\Delta R\phi (f)e^{-f}\nonumber \\&\quad =-\frac{1}{2}\int |\nabla R|^2\phi (f)e^{-f}+\int R\langle \nabla R,\nabla f\rangle \phi (f)e^{-f}\nonumber \\&\qquad -\frac{n-1}{2}\int \langle \nabla R,\nabla f\rangle \phi (f)e^{-f},\nonumber \\&\quad =-\frac{1}{2}\int |\nabla R|^2\phi (f)e^{-f}+2\int RRic(\nabla f,\nabla f)\phi (f)e^{-f}\nonumber \\&\qquad -(n-1)\int Ric(\nabla f,\nabla f)\phi (f)e^{-f}, \end{aligned}$$

(4.7)

where we used $\nabla R=2Ric(\nabla f,\cdot )$.

Note that $R\ge 0$ (see Chen [4]), $R+|\nabla f|^2- f=Const.$ and f is at most quadratic growth, we know that $|\nabla f|$ is at most linear growth and R is at most quadratic growth. It follows from Lemma 2.2 that

$$\begin{aligned} \int |\nabla R|^2 e^{-f}\le 2\int |Ric|^2|\nabla f|^2e^{-f}\le \int |Ric|^2e^{-cf}<+\infty , \end{aligned}$$

(4.8)

and

$$\begin{aligned} \int |RRic(\nabla f,\nabla f)|e^{-f}\le \int R|Ric||\nabla f|^2e^{-f}\le \sqrt{n}\int |Ric|^2e^{-cf}<+\infty ,\nonumber \\ \end{aligned}$$

(4.9)

where c is a constant in (0, 1]. Moreover, we have

$$\begin{aligned}&\int |Ric(\nabla f,\nabla f)|e^{-f}\le \int |Ric||\nabla f|^2e^{-f}\nonumber \\&\quad \le \frac{1}{4}\int |Ric|^2e^{-f}+\int |\nabla f|^4e^{-f}<+\infty . \end{aligned}$$

(4.10)

(4.2) follows by taking $s\rightarrow +\infty$ in (4.7).

This completes the proof of Lemma 4.1. $\square$

Now we are ready to prove Theorem 1.1.

Theorem 4.2

Let $(M^n,g,f)$ be a radially flat gradient shrinking Ricci soliton with $\text {div}^2W(\nabla f,\nabla f)=0$. Then $(M^n,g,f)$ is rigid.

Proof

Integrating (3.2) and using the condition of $\text {div}^2W(\nabla f,\nabla f)=0$, we obtain

$$\begin{aligned} 0= & {} \int (\text {div}^2W)_{ik}\nabla _if\nabla _kfe^{-f}\nonumber \\= & {} \frac{n-3}{2(n-1)(n-2)}\int (\nabla _k\nabla _iR\nabla _if\nabla _kf-|\nabla f|^2\Delta R)e^{-f}. \end{aligned}$$

(4.11)

Note that $sec(E,\nabla f)=0$ implies $Ric(\nabla f,\nabla f)=0$. Applying Lemma 4.1 to (4.11), we have

$$\begin{aligned} \int |\nabla R|^2e^{-f}=0. \end{aligned}$$

It follows that $|\nabla R|=0$ a.e. on $M^n$. Since any gradient shrinking Ricci soliton is analytic in harmonic coordinates, we have $|\nabla R|\equiv 0$ on $M^n$, i.e. $M^n$ has constant scalar curvature.

From Lemma 2.4, we know that $(M^n,g,f)$ is rigid. $\square$

5 Curvatures of Four-Dimensional Gradient Ricci Solitons

In this section, we present a formula (Proposition 5.3) on curvatures that are essential in the proof of Theorem 1.2. First of all, we recall two propositions that are proved by the authors [16]

Proposition 5.1

(Yang and Zhang [16]) On a four-dimensional Riemannian manifold, we have

$$\begin{aligned} R_{i'j'ik}=R_{j'ii'k}=R_{ii'j'k}=0, \end{aligned}$$

(5.1)

and

$$\begin{aligned} \nabla _iR_{i'j'kl}=\nabla _{i'}R_{j'ikl}=\nabla _{j'}R_{ii'kl}=0. \end{aligned}$$

(5.2)

Proposition 5.2

(Yang and Zhang [16]) Let $(M^4,f,g)$ be a four-dimensional gradient Ricci soliton. Then we have

$$\begin{aligned} \nabla _{j}\nabla _{l}R_{i'j'kl}=-R_{i'j'kl}R_{jl}. \end{aligned}$$

(5.3)

Proof

The second Bianchi identity implies that

$$\begin{aligned} \nabla _j\nabla _{l}R_{i'j'kl}=\nabla _j\nabla _{j'}R_{i'k}-\nabla _j\nabla _{i'}R_{j'k}. \end{aligned}$$

(5.4)

Using (1.1), we have

$$\begin{aligned} \nabla _jR_{ik}-\nabla _iR_{jk}= & {} \nabla _i\nabla _{j}\nabla _{k}f-\nabla _j\nabla _{i}\nabla _{k}f \nonumber \\= & {} R_{ijkl}\nabla _{l}f, \end{aligned}$$

(5.5)

By direct computation, we obtain

$$\begin{aligned} \nabla _j\nabla _{j'}R_{i'k}= & {} \nabla _j(\nabla _kR_{j'i'}+R_{kj'i'l}\nabla _lf)\nonumber \\= & {} \nabla _j\nabla _kR_{j'i'}+\nabla _jR_{kj'i'l}\nabla _lf+R_{kj'i'l}\nabla _j\nabla _lf\nonumber \\= & {} \nabla _k\nabla _jR_{j'i'}+R_{jkj'l}R_{li'}+R_{jki'l}R_{j'l}\nonumber \\&+\nabla _jR_{kj'i'l}\nabla _lf+R_{kj'i'l}(\lambda g_{jl}-R_{jl})\nonumber \\= & {} \nabla _k\nabla _jR_{j'i'}+R_{jkj'l}R_{li'}+R_{jki'l}R_{j'l}-R_{kj'i'l}R_{jl}, \end{aligned}$$

(5.6)

where we used (5.5) in the first identity, (1.1) in the third and Proposition 5.1 in the last.

By the same arguments, we have

$$\begin{aligned} \nabla _j\nabla _{i'}R_{j'k}= & {} \nabla _k\nabla _jR_{i'j'}+R_{jki'l}R_{lj'}+R_{jkj'l}R_{i'l}-R_{ki'j'l}R_{jl}. \end{aligned}$$

(5.7)

Applying (5.6) and (5.7) to (5.4), we have

$$\begin{aligned} \nabla _j\nabla _{l}R_{ijkl}= & {} \nabla _j\nabla _{j'}R_{i'k}-\nabla _j\nabla _{i'}R_{j'k}\\= & {} (R_{ki'j'l}-R_{kj'i'l})R_{jl}\\= & {} -R_{i'j'kl}R_{jl}, \end{aligned}$$

where we used (5.2).

This completes the proof of proposition 5.2. $\square$

Proposition 5.3

Let $(M^4,f,g)$ be a four-dimensional gradient Ricci soliton. Then we have

$$\begin{aligned} \nabla _j\nabla _lW_{i'j'kl}\nabla _if\nabla _kf=\frac{1}{2}R_{i'j'kl}\nabla _j\nabla _lf\nabla _i f\nabla _k f. \end{aligned}$$

(5.8)

Proof

Both sides of (5.8) are zero at the critical point of f. In the following, we consider regular points of f. Let x be a regular point of f, we denote by $\{e_i\}_{i=1}^4$ a local orthonormal frame with $e_1=\frac{\nabla f}{|\nabla f|}$. We use $\{\alpha _i\}_{i=1}^4$ to represent eigenvalues of the Ricci tensor with corresponding orthonormal eigenvectors $\{e_i\}_{i=1}^4$, respectively.

Note that

$$\begin{aligned} R_{ijkl}\nabla _{l}f= & {} \nabla _i\nabla _{j}\nabla _{k}f-\nabla _j\nabla _{i}\nabla _{k}f \nonumber \\= & {} \nabla _jR_{ik}-\nabla _iR_{jk}, \end{aligned}$$

(5.9)

where we used (1.1).

Moreover, we have

$$\begin{aligned} -2\nabla _lW_{i'j'kl}= & {} C_{i'j'k}\nonumber \\= & {} \nabla _{i'}R_{j'k}-\nabla _{j'}R_{i'k}-\frac{1}{6}(g_{j'k}\nabla _{i'}R-g_{i'k}\nabla _{j'}R)\nonumber \\= & {} -\nabla _lR_{i'j'kl}-\frac{1}{6}(g_{j'k}\nabla _{i'}R-g_{i'k}\nabla _{j'}R), \end{aligned}$$

where we used the second Bianchi identity. It follows that

$$\begin{aligned}&\nabla _j\nabla _lW_{i'j'kl}\nabla _if\nabla _kf\\&\quad = \frac{1}{2}\nabla _j\nabla _lR_{i'j'kl}\nabla _if\nabla _kf+\frac{1}{12}(g_{j'k}\nabla _j\nabla _{i'}R-g_{i'k}\nabla _j\nabla _{j'}R)\nabla _if\nabla _kf\\&\quad =-\frac{1}{2}R_{i'j'kl}R_{jl}\nabla _if\nabla _kf+\frac{1}{12}(g_{j'1}\nabla _j\nabla _{1'}R-g_{1'1}\nabla _j\nabla _{j'}R)|\nabla f|^2\\&\quad =-\frac{\lambda }{2}R_{i'j'kj}\nabla _if\nabla _kf+\frac{1}{2}R_{i'j'kl}\nabla _j\nabla _lf\nabla _i f\nabla _k f\\&\quad =\frac{1}{2}R_{i'j'kl}\nabla _j\nabla _lf\nabla _i f\nabla _k f, \end{aligned}$$

where we used Proposition 5.2 in the second equality and (1.1) in the third. Moreover, we used (5.1) in the last equality. $\square$

6 Proof of Theorem 1.2

We finish the proof of Theorem 1.2 in this section. First of all, we prove a integral identity for four-dimensional gradient shrinking Ricci solitons with (4.1).

Lemma 6.1

Let $(M^4,f,g)$ be a four-dimensional gradient shrinking Ricci soliton, then we have

$$\begin{aligned} \int \text {div}^2W^\pm (\nabla f,\nabla f)e^{-f}=\frac{1}{2}\int \text {div}^2W(\nabla f,\nabla f)e^{-f}. \end{aligned}$$

(6.1)

Proof

Since $W^\pm _{ijkl}=\frac{1}{2}(W_{ijkl}\pm W_{i'j'kl})$, we only need to show that

$$\begin{aligned} \int \nabla _j\nabla _lW_{i'j'kl}\nabla _if\nabla _kfe^{-f}=0. \end{aligned}$$

(6.2)

For the complete non-compact case, let $\phi (t)=\frac{s-t}{s}$ on (0, s), $\phi =0$ on $[s,\infty )$ for any fixed constant $s>0$. Moreover, the support of $\phi (f)$ is a closed subset of the compact set $\{x|\frac{1}{4}(r(x)-c_1)^2\le s\}$ for any fixed $s>0$. Then we have $\phi (f)$ having compact support in $M^4$.

Using Proposition 5.3, we have

$$\begin{aligned}&\int \nabla _j\nabla _lW_{i'j'kl}\nabla _if\nabla _kf\phi (f)e^{-f}\nonumber \\&\quad =\frac{1}{2}\int R_{i'j'kl}\nabla _j\nabla _lf\nabla _i f\nabla _k f\phi (f)e^{-f}\nonumber \\&\quad =-\frac{1}{2}\int \nabla _j(R_{i'j'kl}\nabla _i f\nabla _k f\phi (f)e^{-f})\nabla _lf. \end{aligned}$$

(6.3)

Next, we prove that $\nabla _j(R_{i'j'kl}\nabla _i f\nabla _k f\phi (f)e^{-f})\nabla _lf\equiv 0$ on $M^4$. We divide the arguments into two cases:

• Case 1: $|\nabla f|^2=0$ on some nonempty open set. In this case, since any gradient shrinking Ricci soliton is analytic in harmonic coordinates, it follows that $|\nabla f|^2=0$ on $M^4$. It is clear that $\nabla _j(R_{i'j'kl}\nabla _i f\nabla _k f\phi (f)e^{-f})\nabla _lf\equiv 0$ on $M^4$.

• Case 2: The set $\Theta :=\{x\in M^4|\nabla f(x)\ne 0\}$ is dense in $M^4$. $\forall x\in \Theta$, we denote by $\{e_i\}_{i=1}^4$ a local orthonormal frame of with $e_1=\frac{\nabla f}{|\nabla f|}$. Note that

  $$\begin{aligned} \nabla _j(R_{i'j'kl}\nabla _i f\nabla _k f\phi (f)e^{-f})\nabla _lf=\nabla _j(R_{1'j'11}|\nabla f|^2\phi (f)e^{-f})|\nabla f|=0 \end{aligned}$$

  on $\Theta$, the continuity implies that $\nabla _j(R_{i'j'kl}\nabla _i f\nabla _k f\phi (f)e^{-f})\nabla _lf\equiv 0$ on $M^4$.

  It follows from (6.3) that

  $$\begin{aligned} \int \nabla _j\nabla _lW_{i'j'kl}\nabla _if\nabla _kf\phi (f)e^{-f}=0 \end{aligned}$$

  (6.4)

  (6.2) follows by taking $s\rightarrow +\infty$ in (6.4).

From the proof of the complete non-compact case, we know that (6.2) also holds for the compact case.

This completes the proof of Lemma 6.1. $\square$

Next, we present a result that will be needed in the proof of Theorem 1.2.

Lemma 6.2

(Fernández-López et al. [8]) No complete gradient shrinking Ricci soliton may exist with $R=\lambda$.

Now we are ready to finish the proof of Theorem 1.2.

Theorem 6.3

Let $(M^4,g,f)$ be a four-dimensional radially flat gradient shrinking Ricci soliton. If $\text {div}^2W^\pm (\nabla f,\nabla f)=0$. Then $(M^4,g,f)$ is either Einstein or a finite quotient of ${\mathbb {R}}^4$, ${\mathbb {S}}^2\times {\mathbb {R}}^2$ or ${\mathbb {S}}^3\times {\mathbb {R}}$.

Proof

It follows from Lemma 6.1 that $\int \text {div}^2W(\nabla f,\nabla f)e^{-f}=0$. From the proof of Theorem 1.1, we know that $(M^4,g,f)$ is rigid, i.e. it is Einstein, or a finite quotient of ${\mathbb {R}}^4$ or of the product ${\mathbb {R}}^k\times N^{4-k}$ $(k=1,2,3)$, where N is Einstein with Einstein constant $\lambda$. We divide the arguments into three cases:

• Case 1: $(M^4,g,f)$ is a finite quotient of ${\mathbb {R}}\times N^3$. In this case, $N^3$ is a three-dimensional Einstein manifold with positive Einstein constant $\lambda$. Note that the Weyl tensor vanishes identically in dimension 3, we can derive that $N^3$ has positive constant sectional curvature $\frac{3}{2}\lambda$. Therefore, $(M^4,g,f)$ is a finite quotient of ${\mathbb {R}}\times {\mathbb {S}}^3$.

• Case 2: $(M^4,g,f)$ is a finite quotient of ${\mathbb {R}}^2\times N^2$. In this case, two-dimensional Einstein manifold $N^2$ with positive Einstein constant $\lambda$ must be ${\mathbb {S}}^2$.

• Case 3: $(M^4,g,f)$ is a finite quotient of ${\mathbb {R}}^3\times N^1$. In this case, the scalar curvature of $M^4$ is $\lambda$, which contradicts Lemma 6.2.

To conclude, $(M^4,g,f)$ is either Einstein or a finite quotient of ${\mathbb {R}}^4$, ${\mathbb {S}}^2\times {\mathbb {R}}^2$ or ${\mathbb {S}}^3\times {\mathbb {R}}$. $\square$