Rigidity of Non-Steady Gradient Ricci Solitons

Abstract

Let $(M,g)$ be a connected, compact Riemannian manifold of dimensionan n. We demonstrate that, after a suitable normalization, a shrinking gradient Ricci soliton $(M,g,f,\lambda )$ is trivial exactly when the mean value of f is less than or equal to $\frac{n}{2}$. Moreover, we prove that a normalized non-steady gradient Ricci soliton $(M,g,f,\lambda )$ is trivial if and only if its scalar curvature S satisfies the relation $S=\lambda \left(f+\frac{n}{2}\right).$ In addition, we establish that if $(M,g,f,\lambda )$ admits an isometric immersion as a hypersurface in the Euclidean space, then the soliton must necessarily be of a shrinking type. In such a case, the constant $\lambda$ and the mean curvature of M satisfy a certain inequality, with equality occurring precisely when M is isometric to a round sphere.

1. Introduction

Let $n\ge 2$, and consider an n-dimensional Riemannian manifold $(M,g)$, which we assume to be connected throughout this paper. Denote by $\mathfrak{X}\left(M\right)$ the set of all smooth vector fields defined on M.

In the present paper, we adopt the curvature sign convention following [1], which is the reverse of the one appearing in [2]. Specifically, the Riemann curvature tensor is defined as the $(1,3)$-type tensor field

$$R(X,Y)Z=[{\nabla }_X,{\nabla }_Y]Z-{\nabla }_{[X,Y]}Z,$$

for all vector fields $X,Y,Z\in \mathfrak{X}\left(M\right)$.

At any point $p\in M$, let $\{e_1,\dots ,e_n\}$ be a local orthonormal frame of $T_{p}M$. The Ricci tensor $\mathrm{Ric}$ and the scalar curvature S are then defined, respectively, by

$$\begin{array}{cc}\hfill \mathrm{Ric}(X,Y)& =\mathrm{tr}\left(Z\mapsto R(Z,X)Y\right)\hfill \\ \hfill & =\sum _{i=1}^{n}g(R(X,e_i)e_i,Y),\hfill \end{array}$$

for all $X,Y\in T_{p}M$, and

$$\begin{array}{cc}\hfill S\left(p\right)& =\mathrm{tr}\left(\mathrm{Ric}\right)\hfill \\ \hfill & =\sum _{i=1}^n\mathrm{Ric}(e_i,e_i),\hfill \end{array}$$

where $\mathrm{tr}$ denotes the trace operator.

Consider a smooth manifold M endowed with a one-parameter family of Riemannian metrics $g\left(s\right)$. This family is called a solution of the Ricci flow whenever

$$-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial g}{\partial s}}\left(s\right)={\mathrm{Ric}}_{g\left(s\right)},$$

(1)

where ${\mathrm{Ric}}_{g\left(s\right)}$ is the Ricci tensor of the metric $g\left(s\right)$.

A solution $g\left(s\right)$ of the Ricci flow is said to be self-similar if there exists a positive function $\omega \left(s\right)$ and a one-parameter family of diffeomorphisms $\varphi \left(s\right)$ on M such that

$$g\left(s\right)=\omega \left(s\right)\varphi {\left(s\right)}^{*}g\left(0\right)$$

satisfies (1). Here, $\varphi {\left(s\right)}^{*}$ denotes the pullback of $g\left(0\right)$ under $\varphi \left(s\right)$.

For a given Riemannian manifold $(M,g)$, any self-similar solution with $g\left(0\right)=g$ is associated with a vector field X and a constant $\lambda$ satisfying

$$\mathrm{Ric}+\frac{1}{2}{L}_{X}g=\lambda g,$$

(2)

where $\mathrm{Ric}$ is the Ricci tensor of the metric g and $L_{X}g$ stands for the Lie derivative of g in the direction of X.

Conversely, if X and $\lambda$ solve (2), then X generates a one-parameter family of diffeomorphisms $\varphi \left(s\right)$ such that

$$g\left(s\right)=(1-2\lambda s)\varphi {\left(s\right)}^{*}g$$

is a self-similar solution to the Ricci flow (1). For details, see [3,4].

A quadruple $(M,g,X,\lambda )$ satisfying (2) is called a Ricci soliton. Depending on the sign of $\lambda$, the soliton is called shrinking $(\lambda >0)$, steady $(\lambda =0)$, or expanding $(\lambda <0)$. If X is a Killing vector field (so that $L_{X}g=0$), then the soliton is said to be trivial, and (2) reduces to

$$\mathrm{Ric}=\lambda g,$$

which means that $(M,g)$ is Einstein.

When the vector field X is the gradient of a smooth function f, denoted by $\nabla f$, it is defined through the relation

$$g(\nabla f,Y)=Y\left(f\right),$$

for every $Y\in \mathfrak{X}\left(M\right)$.

Under this assumption, Equation (2) can be rewritten as

$$Ric+Hessf=\lambda g,$$

(3)

with $\mathrm{Hess}f$ representing the Hessian of f, i.e., the $(0,2)$-tensor defined by the formula

$$\mathrm{Hess}f=\frac{1}{2}{L}_{\nabla f}g.$$

(4)

A quadruple $(M,g,f,\lambda )$ satisfying Equation (3) is called a gradient Ricci soliton.

In the special case where the potential function f is constant, the soliton is said to be trivial. Then, the metric g is Einstein, that is,

$$Ric=\lambda g.$$

For any vector field $X\in \mathfrak{X}\left(M\right)$, the divergence of X is defined as

$$div\left(X\right)=\sum _{i=1}^{n}g({\nabla }_{e_i}X,e_i),$$

(5)

where $\{e_1,\dots ,e_n\}$ is a local orthonormal frame on M.

It is worth noting that

$$div\left(X\right)=\frac{1}{2}\mathrm{tr}\left(L_{X}g\right).$$

(6)

For a smooth function f on M, the Laplacian is defined by

$$\Delta f=div(\nabla f).$$

Using (4) and (6), we obtain

$$\begin{array}{cc}\hfill \Delta f& =div(\nabla f)\hfill \\ \hfill & =\frac{1}{2}\mathrm{tr}\left(L_{\nabla f}g\right)\hfill \\ \hfill & =\mathrm{tr}\left(\mathrm{Hess}f\right).\hfill \end{array}$$

Recall that for a $(1,1)$-tensor field T, its covariant derivative is defined as

$$\left({\nabla }_{X}T\right)\left(Y\right)={\nabla }_X\left(T\left(Y\right)\right)-T\left({\nabla }_{X}Y\right),$$

for all $X,Y\in \mathfrak{X}\left(M\right)$.

The divergence of T is the vector field

$$\begin{array}{cc}\hfill div\left(T\right)& =trace(\nabla T)\hfill \\ \hfill & =\sum _{i=1}^n\left({\nabla }_{e_i}T\right)\left(e_i\right),\hfill \end{array}$$

and if $\left\{e_i\right\}$ is chosen to be parallel at a point, this expression simplifies to

$$div\left(T\right)=\sum _{i=1}^n{\nabla }_{e_i}\left(T\left(e_i\right)\right).$$

The squared norm of a $(1,1)$-tensor field T on $(M,g)$ is defined by

$${\left|T\right|}^2=\sum _{i=1}^n{\left|T\left(e_i\right)\right|}^2,$$

where $\{e_1,\dots ,e_n\}$ is a local orthonormal frame on M, and $|T\left(e_i\right){|}^2=g(T\left(e_i\right),T\left(e_i\right))$.

In the case where T is self-adjoint (i.e., symmetric), this expression can equivalently be written as

$$\begin{array}{c}\hfill {\left|T\right|}^2=\sum _{i=1}^n{\lambda }_i^2,\end{array}$$

(7)

with ${\lambda }_1,\dots ,{\lambda }_n$ denoting the eigenvalues of T.

It is well known that every compact Ricci soliton, as well as every noncompact shrinking soliton, must be of a gradient type (see [5,6]). Over the past years, gradient Ricci solitons have been intensively studied due to their central role in the Ricci flow, their rigidity properties, and their appearance as models for singularity formation. For further accounts, we refer to [7,8,9,10,11,12,13,14].

The outline of the paper is as follows. In Section 2, we recall the basic notions and formulas that will be needed in the sequel. Section 3 is concerned with non-steady gradient Ricci solitons satisfying the identity $S=\lambda f+c$, where S denotes the scalar curvature, $\lambda$ the soliton constant, and f the potential function. We prove that such a soliton is trivial exactly when

$$S=\lambda \left(f+\frac{n}{2}\right).$$

Further consequences are obtained in the situation where S satisfies a Poisson-type equation.

In Section 5 we turn to the study of compact solitons. We show that compact expanding or steady Ricci solitons are always trivial, whereas a compact normalized shrinking gradient Ricci soliton is trivial precisely when the average value of its potential function is at most $\frac{n}{2}$. Finally, we investigate compact Ricci solitons that can be realized as hypersurfaces in Euclidean space, proving in particular that such solitons must be shrinking, and that $\lambda$ and the mean curvature of M satisfy an inequality, with equality if and only if M is a sphere.

2. Preliminaries

Consider a Ricci soliton $(M,g,X,\lambda )$. Taking the trace of Equation (2) yields

$$S+divX=n\lambda ,$$

(8)

where S denotes the scalar curvature of the metric g. In the special case of a gradient Ricci soliton $(M,g,f,\lambda )$, this reduces to

$$S+\Delta f=n\lambda ,$$

(9)

with $\Delta f$ the Laplacian of the potential f.

A well-known relation derived from the contracted second Bianchi identity (see [14]) states that

$$\nabla S=2div\left(\mathrm{Ric}\right).$$

(10)

Using Equations (2) and (10) in the case of a gradient soliton $(M,g,f,\lambda )$, one obtains

$$S+{|\nabla f|}^2=2\lambda f+c,$$

(11)

for some constant c (cf. [15]).

When the soliton is non-steady, the potential f can be shifted to eliminate c, and Equation (11) simplifies to

$$S+{|\nabla f|}^2=2\lambda f.$$

(12)

In this case, the soliton is said to be normalized.

The squared norm of the Ricci tensor, equivalently of the associated Ricci operator, is defined according to (7) and can be written as

$${\left|\mathrm{Ric}\right|}^2=\sum _{i=1}^n{\lambda }_i^2,$$

where ${\lambda }_i$ denote the eigenvalues of the Ricci operator.

In a similar manner, the squared norm of the Hessian of a smooth function f, written as $|{\nabla }^2{f|}^2$, is defined as the squared norm of the corresponding self-adjoint endomorphism ${\nabla }^{2}f$ (a $(1,1)$-tensor) associated with the $(0,2)$-tensor $Hessf$ through the relation

$$Hessf(X,Y)=g({\nabla }^{2}f\left(X\right),Y),$$

(13)

for all $X,Y\in \mathfrak{X}\left(M\right)$.

By Cauchy–Schwarz, one has

$${\left|\mathrm{Ric}\right|}^2\ge {\displaystyle \frac{S^2}{n}}.$$

(14)

The following identities from [14] will be useful later:

$$\Delta S-g(\nabla S,\nabla f)+{2\left|\mathrm{Ric}\right|}^2=2\lambda S,$$

(15)

and

$${\displaystyle \frac{1}{2}}{\Delta |\nabla f|}^2={\left|{\nabla }^{2}f\right|}^2-\mathrm{Ric}(\nabla f,\nabla f).$$

(16)

The latter is simply Bochner’s formula adapted to the setting of gradient Ricci solitons.

3. Non-Steady Gradient Ricci Solitons with $\mathit{S}=\lambda \mathit{f}+\mathit{c}$

We next compute explicit formulas for the norms of the Ricci and Hessian operators in the case of non-steady solitons for which $S=\lambda f+c$.

Theorem 1.

Let $(M,g,f,\lambda )$ be a normalized non-steady gradient Ricci soliton of dimension n satisfying $S=\lambda f+c$ for some constant c. Then

$$\begin{array}{ccc}\hfill {\left|\mathrm{Ric}\right|}^2& =& {\lambda }^2\left(2f-\frac{n}{2}\right)+\lambda c,\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill |{\nabla }^2{f|}^2& =& \lambda \left(\frac{n}{2}\lambda -c\right).\hfill \end{array}$$

(18)

Proof. 

Since $S=\lambda f+c$, it follows that $\nabla S=\lambda \nabla f$ and $\Delta S=\lambda \Delta f$. Substituting these into (15) gives

$$\lambda \Delta f-{\lambda |\nabla f|}^2+{\left|\mathrm{Ric}\right|}^2=2\lambda (\lambda f+c).$$

(19)

On the other hand, inserting $S=\lambda f+c$ into (9) and (12) yields

$$\Delta f=n\lambda -\lambda f-c,$$

(20)

and

$${|\nabla f|}^2=\lambda f-c.$$

(21)

Combining (19)–(21) immediately gives formula (17). Likewise, replacing these values into Bochner’s formula (16) and recalling that

$$\mathrm{Ric}(\nabla f,\nabla f)=\frac{1}{2}g(\nabla f,\nabla S)$$

(which is just (10)) yields (18). □

Remark 1.

Adding Equations (17) and (18) leads to the useful identity

$${\left|\mathrm{Ric}\right|}^2+{\left|{\nabla }^{2}f\right|}^2=2{\lambda }^{2}f.$$

(22)

4. Criteria for Triviality of Non-Steady Gradient Ricci Solitons

We are now in a position to prove one of our principal results. The argument relies essentially on relations (17), (18), and (22).

Theorem 2.

Let $(M,g,f,\lambda )$ be a normalized non-steady gradient Ricci soliton of dimension n, and let S denote its scalar curvature. Then $(M,g,f,\lambda )$ is trivial if and only if

$$S=\lambda \left(f+\frac{n}{2}\right).$$

Proof. 

Suppose first that $(M,g,f,\lambda )$ is trivial. From (9) and (12) we deduce that $S=n\lambda$ and $f=\frac{n}{2}$, hence the claimed identity holds.

Conversely, assume that $S=\lambda \left(f+\frac{n}{2}\right)$. Setting $c=\frac{n}{2}\lambda$ in (18), we obtain $\left|{\nabla }^{2}f\right|=0$. Plugging this into (22) yields

$${\left|\mathrm{Ric}\right|}^2=2{\lambda }^{2}f.$$

Now, by inequality (14), one has

$$\begin{array}{cc}\hfill 2{\lambda }^{2}f& ={\left|\mathrm{Ric}\right|}^2\hfill \\ \hfill & \ge {\displaystyle \frac{S^2}{n}}\hfill \\ \hfill & ={\displaystyle \frac{{\lambda }^2{\left(f+\frac{n}{2}\right)}^2}{n}}.\hfill \end{array}$$

This inequality simplifies to

$$0\ge {\left(f-\frac{n}{2}\right)}^2,$$

forcing $f=\frac{n}{2}$. Therefore the soliton is trivial. □

Next we recall a related criterion, originally proved in [16], for compact shrinking gradient Ricci solitons. We provide here a concise proof.

Theorem 3.

Let $(M,g,f,\lambda )$ be a compact shrinking gradient Ricci soliton of dimension n. If the scalar curvature S satisfies the Poisson equation

$$▵S=\sigma ,\sigma =\lambda (n\lambda -S),$$

then either $(M,g,f,\lambda )$ is trivial, or $\lambda \ge {\lambda }_1$, where ${\lambda }_1$ denotes the first eigenvalue of the Laplacian$▵$.

Proof. 

From the hypothesis we obtain

$$-▵(n\lambda -S)=\lambda (n\lambda -S).$$

(23)

If S is constant, (23) forces $S=n\lambda$. Substituting this into (9), we obtain $▵f=0$. Since M is compact, f must be constant, and the soliton is trivial.

If S is not constant, then (23) shows that $\lambda$ is an eigenvalue of $▵$. Hence $\lambda \ge {\lambda }_1$. □

We now present an application which ensures the existence of non-trivial examples.

Theorem 4.

Let $(M,g,f,\lambda )$ be a compact normalized shrinking non-trivial gradient Ricci soliton of dimension n. If the scalar curvature S satisfies

$$▵S=\sigma ,\sigma =\lambda (n\lambda -S),$$

then λ is an eigenvalue of $▵$, and moreover

$$S=\lambda f+c,$$

for some constant $c\ne \frac{n}{2}\lambda$.

Proof. 

Since the soliton is assumed to be non-trivial, S cannot be constant. As in the previous theorem, this implies that $\lambda$ is an eigenvalue of $▵$.

On the other hand, using (9) we compute

$$\begin{array}{ccc}\hfill ▵S& =& \lambda (n\lambda -S)\hfill \\ & =& \lambda ▵f,\hfill \end{array}$$

so that

$$▵(S-\lambda f)=0.$$

Compactness of M then implies $S-\lambda f=c$ for some constant c. By Theorem 2, c cannot equal $\frac{n}{2}\lambda$, which completes the proof. □

5. Compact Shrinking Gradient Ricci Solitons

We begin with a basic observation.

Proposition 1.

A compact Ricci soliton is trivial precisely when its scalar curvature is constant.

Proof. 

Recall that every compact Ricci soliton is automatically a gradient Ricci soliton.

Let $(M,g,f,\lambda )$ be such a soliton with constant scalar curvature S.

Integrating Equation (9) over M, and observing that

$$\begin{array}{cc}\hfill {\int }_M\Delta fdV& ={\int }_{M}div(\nabla f)dV\hfill \\ \hfill & =0,\hfill \end{array}$$

we obtain

$${\int }_{M}SdV={\int }_{M}n\lambda dV.$$

(24)

Here, $dV$ denotes the Riemannian volume element on M.

Since both $\lambda$ and the scalar curvature S are constants, Equation (24) simplifies to

$$(n\lambda -S){\int }_{M}dV=0,$$

or equivalently,

$$(n\lambda -S)\mathrm{Vol}\left(M\right)=0,$$

where $\mathrm{Vol}\left(M\right)$ stands for the total volume of M.

Consequently, we deduce that $S=n\lambda$. Substituting this back into Equation (9) gives $\Delta f=0$. Because M is compact, it follows that f must be constant.

The converse direction is immediate according to (9). □

It was shown in [17] that for compact Ricci solitons of dimension $n\ge 3$, the case of non-constant scalar curvature forces the soliton to be shrinking. In contrast, steady and expanding compact Ricci solitons always have constant scalar curvature. From this we deduce the following, for which we give a direct argument.

Proposition 2.

Every compact steady or expanding Ricci soliton is trivial.

Proof. 

Let $(M,g,f,\lambda )$ be a compact gradient Ricci soliton. By combining (9) and (11), we obtain

$$\Delta f-{|\nabla f|}^2+2\lambda f=n\lambda -c.$$

(25)

Case 1. $\lambda =0$. At a maximum point $x_0$ of f, one has

$$\nabla f\left(x_0\right)=0,$$

and

$$\Delta f\left(x_0\right)\le 0.$$

It follows by Equation (25) that $c\ge 0$.

At a minimum point $x_1$, similarly

$$\nabla f\left(x_1\right)=0,$$

and

$$\Delta f\left(x_1\right)\ge 0,$$

giving $c\le 0.$ Thus $c=0$, so (25) reduces to

$$\Delta f={|\nabla f|}^2\ge 0,$$

showing that f is subharmonic. By the maximum principle, f is constant.

• Case 2. $\lambda <0$. Define

$$\tilde{f}=f-{\displaystyle \frac{n}{2}}+{\displaystyle \frac{c}{2\lambda }}.$$

Then (25) rewrites as

$$\Delta \tilde{f}-{|\nabla \tilde{f}|}^2+2\lambda \tilde{f}=0.$$

(26)

At a maximum point $x_0$ of $\tilde{f}$ we have

$$\nabla \tilde{f}\left(x_0\right)=0,$$

and

$$\Delta \tilde{f}\left(x_0\right)\le 0,$$

which together with $\lambda <0$ imply $\tilde{f}\left(x_0\right)\le 0$.

At a minimum point $x_1$, the analogous reasoning yields

$$\tilde{f}\left(x_1\right)\ge 0.$$

Therefore, for an arbitrary point $x,$ we have

$$0\le \tilde{f}\left(x_1\right)\le \tilde{f}\left(x\right)\le \tilde{f}\left(x_0\right)\le 0,$$

from which we deduce that $\tilde{f}\left(x\right)=0$, that is $\tilde{f}$ vanishes identically, showing that f is constant and equal to ${\displaystyle \frac{n}{2}}-{\displaystyle \frac{c}{2\lambda }}$. Thus the soliton $\left(M,g,f,\lambda \right)$ is trivial. □

For a function f on a compact Riemannian manifold $(M,g)$, its average value is defined by

$$f_{av}={\displaystyle \frac{1}{Vol\left(M\right)}}{\int }_{M}fdV.$$

In particular, if $(M,g,f,\lambda )$ is a compact gradient Ricci soliton, integrating (9) gives

$$S_{av}=n\lambda .$$

(27)

Moreover, when $(M,g,f,\lambda )$ is a non-steady compact gradient Ricci soliton, one can normalize by replacing f with $f-\frac{c}{2\lambda }$ so that (11) is equivalent to (12).

We shall always assume this normalization. From (27) and (12) it follows that

$${|\nabla f|}_{av}^2=2\lambda \left(f_{av}-\frac{n}{2}\right).$$

(28)

Thus, in the shrinking case ($\lambda >0$) we have $f_{av}\ge \frac{n}{2}$, while in the expanding case ($\lambda <0$) we obtain $f_{av}\le \frac{n}{2}$. This motivates seeking conditions ensuring triviality in the shrinking case.

Theorem 5.

Let $(M,g,f,\lambda )$ be a compact normalized shrinking gradient Ricci soliton. Then it is trivial if and only if $f_{av}\le \frac{n}{2}$.

Proof. 

From (9) and (12), we obtain

$$n\lambda -\Delta f+{|\nabla f|}^2=2\lambda f.$$

Integrating over M gives

$$n\lambda Vol\left(M\right)+{\int }_M{|\nabla f|}^{2}dV=2\lambda {\int }_{M}fdV,$$

which can be rewritten as

$$f_{av}={\displaystyle \frac{n}{2}}+{\displaystyle \frac{1}{2\lambda Vol\left(M\right)}}{\int }_M{|\nabla f|}^{2}dV.$$

(29)

If $f_{av}\le \frac{n}{2}$, then (29) forces $\nabla f=0$, so f is constant and the soliton is trivial.

Conversely, if the soliton is trivial, then $\nabla f=0$ and (9), (12) yield $S=n\lambda =2\lambda f$, hence $f=\frac{n}{2}$ and $f_{av}=\frac{n}{2}$. □

Remark 2.

The assumption $f_{av}\le \frac{n}{2}$ in Theorem 5 is equivalent to the condition ${\left(fS\right)}_{av}\le \frac{n^2\lambda }{2}$ appearing in Theorem 1.1 of [16].

However, phrasing the condition directly in terms of $f_{av}$ is often more convenient.

Next we study compact Ricci solitons that can be realized as hypersurfaces in Euclidean space. The following result shows that such solitons must be shrinking.

Theorem 6.

If a compact Ricci soliton admits an isometric immersion as a hypersurface in Euclidean space, then it must be shrinking. In other words, no compact steady or expanding Ricci soliton can be immersed in codimension one into ${\mathbb{R}}^{n+1}$.

Proof. 

Let $(M,g,f,\lambda )$ be a compact gradient Ricci soliton immersed as a hypersurface in ${\mathbb{R}}^{n+1}$ with $\lambda \le 0$.

By Proposition 2, the soliton is trivial, hence $S=n\lambda \le 0$.

On the other hand, every compact hypersurface in Euclidean space contains a point where all principal curvatures are positive (see [18]), which forces $S>0$.

This contradiction shows that $\lambda$ cannot be nonpositive, so the soliton must be shrinking. □

Remark 3.

Theorem 6 extends a result of [19], where it was proved that if $(M,g,X,\lambda )$ is a compact Ricci soliton realized as a hypersurface in Euclidean space, with X the tangential part of the position vector field, then $\lambda \ne 0$.

We now establish a geometric estimate relating the soliton constant $\lambda$ to the mean curvature of the hypersurface. We first recall a standard inequality.

Let $(M,g)$ be a hypersurface of a Riemannian manifold $(\overline{M},\overline{g})$ with unit normal vector field $\nu$, shape operator A, and mean curvature H. Then, the scalar curvature $S_M$ of $(M,g)$ is given by

$$S_M=S_{\overline{M}}+n^2{H}^2-{\left|A\right|}^2-2\mathrm{Ric}(\nu ,\nu ),$$

(30)

where $S_{\overline{M}}$ denotes the scalar curvature of $(\overline{M},\overline{g})$, and ${\left|A\right|}^2$ is the squared norm of the shape operator. Recall that the mean curvature H of $(M,g)$ is defined as

$$H={\displaystyle \frac{1}{n}}trA.$$

Theorem 7.

Let $(M,g,f,\lambda )$ be a compact gradient Ricci soliton of dimension n, isometrically immersed in ${\mathbb{R}}^{n+1}$. Then the soliton constant λ and the mean curvature H satisfy

$$0<\lambda \le (n-1){\left(H^2\right)}_{av},$$

with equality if and only if M is isometric to a round sphere.

Proof. 

By Theorem 6, the soliton is shrinking, hence $\lambda >0$.

Since the ambient space is Euclidean, Formula (30) reduces to

$$S_M=n^2{H}^2-{\parallel A\parallel }^2.$$

The Cauchy–Schwarz inequality implies ${\parallel A\parallel }^2\ge n{H}^2$, hence

$$S_M\le n(n-1)H^2.$$

(31)

From (9) and (31), one obtains

$$n\lambda -\Delta f\le n(n-1)H^2.$$

Integrating over M yields

$$n\lambda Vol\left(M\right)\le n(n-1){\int }_M{H}^{2}dV,$$

that is,

$$0<\lambda \le (n-1){\left(H^2\right)}_{av}.$$

If equality holds, then integrating (9) shows that

$${\left(S_M\right)}_{av}=n(n-1){\left(H^2\right)}_{av}.$$

Thus

$${\int }_M(S_M-n(n-1)H^2)dV=0.$$

With (30), this forces $S_M=n(n-1)H^2$ pointwise, which means M is totally umbilical. A complete totally umbilical hypersurface in Euclidean space with $H\ne 0$ is a round sphere, as claimed. □

Corollary 1.

Let $(M,g,f,\lambda )$ be a compact gradient Ricci soliton immersed as a hypersurface in ${\mathbb{R}}^{n+1}$. If its mean curvature H satisfies

$$\lambda \ge (n-1){\left(H^2\right)}_{av},$$

then M is isometric to a round sphere.

6. Conclusions and Future Directions

In this work, we obtained new characterization criteria for compact gradient Ricci solitons. We showed that a normalized shrinking gradient Ricci soliton is trivial if and only if the average of the potential function does not exceed $\frac{n}{2}$. For non-steady normalized gradient Ricci solitons, triviality is guaranteed precisely when the scalar curvature satisfies the relation

$$S=\lambda \left(f+\frac{n}{2}\right).$$

Moreover, in the context of isometric immersions, we proved that a compact gradient Ricci soliton realized as a hypersurface in Euclidean space necessarily corresponds to the shrinking case, with an explicit inequality linking $\lambda$ to the mean curvature, attaining equality uniquely on spheres.

These results emphasize the rigidity of compact gradient Ricci solitons and further illuminate the deep interaction between curvature conditions and potential functions.

Several directions remain open for further investigation. It would be natural to explore whether analogous characterizations hold for noncompact settings, especially for complete shrinking or steady gradient Ricci solitons.

Another avenue is to examine the behavior under different ambient spaces: for instance, characterizing Ricci solitons realized as hypersurfaces in space forms or more general ambient manifolds such as Einstein spaces.

It is also worthwhile to investigate gradient Ricci solitons arising on Lorentzian manifolds, particularly when considered as timelike hypersurfaces within a Lorentzian ambient space.

Finally, it would be interesting to analyze stability aspects, such as whether the inequalities obtained persist under perturbations of the metric or immersion, and how these results connect to the broader study of geometric flows and their singularity models.