Let $(M^n, g)$ be a complete smooth Riemannian manifold, the metric $g$ is called a Ricci-harmonic soliton if there exists a vector field $X$ and a constant $\lambda$, such that

$\begin{equation} \left\{ \begin{aligned} &R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\frac{1}{2}\mathcal{L}_Xg=\lambda g_{ij}, \\ &\tau_g\phi=\nabla_X\phi, \end{aligned} \right. \end{equation}$

(1.1)

where $\phi:(M^n, g)\rightarrow (N^m, h)$ is a map between the Remannian manifolds $(M^n, g)$ and $(N^m, h)$, $Rc$ is the Ricci curvature of $(M, g)$, $\tau_g\phi={\rm trace}\nabla d\phi$ and $\alpha$ is a nonnegative constant.

We call the Ricci-harmonic soliton (1.1) a shrinking, steady, expanding Ricci-harmonic soliton if $\lambda>0$, $\lambda=0$, or $\lambda<0$. If $X$ is a gradient of some function $f$, then $\mathcal{L}_Xg=\nabla^2f$, we call the Ricci-harmonic soliton a gradient Ricci-harmonic soliton with potential function $f$.

Similar to the Ricci soliton, the Ricci-harmonic soliton is a self-similar solution to the Ricci-harmonic flow,

$ \begin{equation} \left\{ \begin{aligned} &\frac{\partial}{\partial t}g=-2Rc+2\alpha(t)\nabla\phi\otimes\nabla\phi, \\ &\frac{\partial}{\partial t}\phi=\tau_g\phi, \end{aligned} \right. \end{equation}$

(1.2)

Perelman [9] proved the result that every compact Ricci soliton is a gradient one. Manola, Gabriele and Carlo [4] gave another proof by Perelman's work [10] and previous others, see Hamilton [6] (dimension 2) and Ivey [7] (dimension 3). Aquino, Barros and Ribeiro [1] showed that the potential function in the compact Ricci soliton equals to the Hodge-de Rham potential. Müller [8] proved the result that a shrinking Ricci-harmonic breather is a gradient soliton. Yang and Shen [13] obtained a monotone volume formula for a general geometric flow by utilizing Perelman's method under the Ricci flow.

Cao and Zhou [2] proved that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth by estimating the bounds for the potential function. Zhang [14] proved a more precise estimate, $Vol(B(o, r))\leq C(R+1)^{n-2\delta}$, when the scalar curvature is bounded below by a positive constant $\delta$. Yang and Shen [12] found that the complete noncompact gradient shrinking Ricci-harmonic soliton still has at most Euclidean volume growth by proving $R-\alpha|\nabla\phi|^2$ is nonnegative and estimating the bounds of the potential function $f$.

In Section 2, we prove the result that every compact shrinking Ricci-harmonic soliton is a gradient one. The method is inspired by Manola, Gabriele and Carlo's work [4] and different from that in [8]. In Section 3, we extend Zhang's work [14] to the case of complete noncompact Ricci-harmonic soliton.

Our main theorems in this paper are below.

Theorem 1.1  Every compact shrinking Ricci-harmonic soliton is a gradient one.

Theorem 1.2  Let $(M^n, g)$ be the complete noncompact shrinking gradient Ricci-harmonic soliton structure (3.1), if there exists a nonnegative constant $\delta$ such that $R-\alpha|\nabla\phi|^2\geq\delta$, then there is a constant $C<+\infty$ depending only on $g$ and $x_0$ such that

$\begin{equation} Vol(B_{x_0}(r))\leq C(r+1)^{n-2\delta} \end{equation}$

(1.3)

for all $r>r_0$, where $B_{x_0}(r)$ is a geodesic ball with radius $r$ and $r_0$ is a positive constant.

Remark 1.1  The condition $R-\alpha|\nabla\phi|^2\geq\delta$ added in Theorem 1.3 is reasonable for Yang and Shen [12] proved $R-\alpha|\nabla\phi|^2\geq0$.

Lemma 2.1  (Log Sobolev inequality, see [3]) Let $(M^n, g)$ be a compact Riemannian manifold. For any $a>0$, there exists a constant $C(a, g)$ such that if $\varphi>0$ satisfies $\displaystyle\int_M\varphi^2dVol=1$, then

$\begin{equation} \int_M\varphi^2\log\varphi dVol\leq a\int_M|\nabla\varphi|^2dVol+C(a, g). \end{equation}$

(2.1)

Lemma 2.2  Let $(M^n, g)$ be a compact Riemannian manifold, $F:M\rightarrow\mathbb{R}$ be a smooth function and $\lambda$ be a positive constant, then there exists a smooth function $f:M\rightarrow\mathbb{R}$ satisfies the equation

$\begin{equation} F+2\triangle f-|\nabla f|^2+2\lambda f={\rm Const}.. \end{equation}$

(2.2)

Proof  Define a functional $W$

$W(g, f)=\int_M(F+2\triangle f-|\nabla f|^2+2\lambda f)e^{-f}dVol$

and

$\mu(g)=\inf\{W(g, f):f\in C^\infty(M)\ {\rm with}\ \int_Me^{-f}dVol=1\}.$

Let $\omega=e^{-\frac{f}{2}}$, we have

$\int_M\omega^2dVol=1$

and

$\begin{eqnarray*} W(g, f)&=&\int_M(F+|\nabla f|^2+2\lambda f)e^{-f}dVol\\ &=&\int_M[(F-4\lambda\log\omega)\omega^2+4|\nabla\omega|^2]dVol\\ &:=&H(g, \omega). \end{eqnarray*}$

Since $F$ is bounded below on $M$ and from Log Sobolev inequality (Lemma 2.1), there exist a constant $C<+\infty$ such that

$\int_M\omega^2\log\omega dVol\leq\frac{1}{\lambda}\int_M|\nabla\omega|^2+C.$

Then the positive minimizer $\omega_1$ realizing $\mu(g)$ is the lowest positive eigenvalue of the nonlinear operator

$\Theta(\omega):=-4\Delta\omega+(F-4\lambda\log\omega)\omega=\mu(g)\omega.$

Choose $\omega$ such that $H(g, \omega)\leq C_1$, then $\displaystyle\int_M\omega^2dVol=1, $ and there exists a positive constant $C_2$ with

$C_1\geq H(g, \omega)\geq2\int_M|\nabla\omega|^2dVol-C_2.$

Hence any minimizing sequence for $H(g, \cdot)$ is bounded in $W^{1, 2}(M)$. We get a minimizer $\omega_1\in W^{1, 2}(M)$ and $\omega_1$ is a weak solution to

$-4\Delta\omega+(F-4\lambda\log\omega)\omega=\mu(g)\omega.$

By elliptic regularity theory (see Gilbarg and Trudinger [5]), we have $\omega_1\in C^\infty$. It's easy to verify that $\omega_1>0$. Then there exists a smooth function $f_1=-2\log\omega_1$ realizing $\mu(g)$, i.e.,

$F+2\triangle f_1-|\nabla f_1|^2+2\lambda f_1=\mu(g)$

for $\lambda>0$.

Proof of Theorem 1.1  Considering the compact shrinking Ricci-harmonic soliton

$ \begin{equation} \left\{ \begin{aligned} &R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\frac{1}{2}\mathcal{L}_Xg=\lambda g_{ij}, \\ &\tau_g\phi=\nabla_X\phi. \end{aligned} \right. \end{equation}$

(2.3)

From Lemma 2.2, there exists a smooth function $f:M\rightarrow \mathbb{R}$ satisfying

$R-\alpha|\nabla\phi|^2+2\triangle f-|\nabla f|^2+2\lambda f={\rm Const.}, $

we have

$\begin{aligned} \nabla_j[2(R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij})e^{-f}]\\ =(\nabla_iR-2\alpha\nabla_j\nabla_i\phi\nabla_j\phi\\ -2\alpha\nabla_i\phi\tau_g\phi+2\triangle\nabla_if)e^{-f}\\ -2\nabla_jf(R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij})e^{-f}\\ =(\nabla_iR-\alpha\nabla_i|\nabla\phi|^2-2\alpha\nabla_i\phi\nabla_j\phi X^j+2\nabla_i\triangle f\\ +2R_{ij}\nabla_jf)e^{-f}\\ -(2R_{ij}\nabla_jf-2\alpha\nabla_i\phi\nabla_j\phi\nabla_jf+\nabla_i|\nabla f|^2-2\lambda\nabla_if)e^{-f}\\ =\nabla_i(R-\alpha|\nabla\phi|^2+2\triangle f-|\nabla f|^2+2\lambda f)e^{-f}\\ +2\alpha\nabla_i\phi\nabla_j\phi(\nabla_jf-X^j)e^{-f}\\ =2\alpha\nabla_i\phi\nabla_j\phi(\nabla_jf-X^j)e^{-f}. \end{aligned}$

As $\nabla_iX^j+\nabla_jX^i=-2R_{ij}+2\alpha\nabla_i\phi\nabla_j\phi+2\lambda g_{ij}$ and $\alpha\geq0$, we have

$\begin{aligned} \nabla_i[(\nabla_jf-X^j)(R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij})e^{-f}]\\ =(\nabla_i\nabla_jf-\nabla_iX^j)(R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij})e^{-f}\\ +2\alpha|\nabla_{\nabla f-X}\phi|^2e^{-f}\\ =\frac{1}{2}(2\nabla_i\nabla_jf-\nabla_iX^j-\nabla_jX^i)(R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij})e^{-f}\\ +2\alpha|\nabla_{\nabla f-X}\phi|^2e^{-f}\\ =|R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij}|^2e^{-f}\\ +2\alpha|\nabla_{\nabla f-X}\phi|^2e^{-f}\geq&0.\\ \end{aligned}$

Denote $|R_{ij}-\alpha\nabla_i\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij}|^2e^{-f}+2\alpha|\nabla_{\nabla f-X}\phi|^2e^{-f}$ by $Q$, we conclude that

$ \begin{equation} 0\leq Q={\rm div}[(\nabla_jf-X^j)(R_{ij}-\alpha\nabla_i\nabla_j\phi+\nabla_i\nabla_jf-\lambda g_{ij})e^{-f}]. \end{equation}$

(2.4)

Integrating $Q$, we have $Q\equiv0$ by Stokes's theorem and the compactness of $M^n$. This implies compact shrinking Ricci-harmonic soliton (2.3) is a gradient Ricci-harmonic soliton with $X=\nabla f$.

Similar to the proof of Theorem 1.1, we have the direct corollary.

Corollary 2.1  Every compact steady Ricci-harmonic soliton is a gradient one.

Proposition 2.1  For the compact shrinking Ricci-harmonic soliton (2.3), the potential function $f$ equals a Hodge-de Rham potential up to a constant.

Proof  By the Hodge-de Rham decomposition theorem, there exists a divergence-free vector field $Y$ and a function $b$ on $M^n$, such that

$\begin{equation} X=Y+\nabla b, \end{equation}$

(2.5)

we deduce ${\rm div}X=\triangle b$. By Theorem 1.1, we can find a potential function $f$ to $(M, g, X)$ satisfying $X=\nabla f$, then ${\rm div}X=\triangle f$.

We conclude that $f=b+{\rm Const.}$ for $\triangle(f-b)=0$ and $M$ is compact.

Remark 2.1  Proposition 2.1 provides another way to find the potential function $f$ to the generic compact shrinking Ricci-harmonic soliton structure $(M^n, g, X)$. Normalizing $f$, we can replace $f$ by the Hodge-de Rham potential to vector field $X$.

In this section, we consider the complete noncompact gradient shrinking Ricci-harmonic soliton

$\begin{equation} \left\{ \begin{aligned} &R_{ij}-\alpha\nabla_i\phi\nabla_j\phi+\nabla_i\nabla_jf=\frac{1}{2} g_{ij}, \\ &\tau_g\phi=\langle\nabla\phi, \nabla f\rangle. \end{aligned} \right. \end{equation}$

(3.1)

Lemma 3.1  Let $(M^n, g)$ be the complete noncompact gradient shrinking Ricci-harmonic soliton structure (3.1), we have the following four equalities

$\begin{eqnarray} &&R-\alpha|\nabla\phi|^2+\triangle f=\frac{n}{2}, \end{eqnarray}$

(3.2)

$\begin{eqnarray} &&R-\alpha|\nabla\phi|^2+|\nabla f|^2-f=0.\end{eqnarray}$

(3.3)

Proof  Taking trace of the first equation of $(3.1)$, $(3.2)$ is obtained.

Taking covariant derivatives and using the commutation formula for the covariant derivatives, we have

$\begin{equation} \nabla_iR_{jk}-\nabla_jR_{ik}-\alpha(\nabla_j\phi\nabla_i\nabla_k\phi-\nabla_i\phi\nabla_j\nabla_k\phi)+R_{ijkl}\nabla_lf=0.\notag \end{equation}$

Taking the trace on $j$ and $k$, we have

$ \begin{equation} \nabla_iR-\nabla_jR_{ij}-\alpha\nabla_j\phi\nabla_i\nabla_j\phi+\alpha\nabla_i\phi\tau_g\phi-R_{il}\nabla_lf=0.\notag \end{equation}$

Using the fact that

$\begin{eqnarray*} &&\nabla_iR-2\nabla_jR_{ij}=0, \\ &&\nabla_i\nabla_j\phi\nabla_j\phi=\frac{1}{2}\nabla_i(\nabla_j\phi\nabla_j\phi)=\frac{1}{2}\nabla_i|\nabla\phi|^2\end{eqnarray*}$

and

$\alpha\nabla_i\phi\tau_g\phi-R_{il}\nabla_lf=\nabla_i\nabla_jf\nabla_jf-\frac{1}{2}g_{ij}\nabla_jf=\frac{1}{2}\nabla_i(|\nabla f|^2-f), $

we obtain $\frac{1}{2}\nabla(R-\alpha|\nabla \phi|^2+|\nabla f|^2-f)=0, $ hence $R-\alpha|\nabla\phi|^2+|\nabla f|^2-f={\rm Const.}.$ Normalizing $f$ by by adding a constant, $(3.3)$ follows.

Lemma 3.2  Let $(M^n, g)$ be the complete noncompact shrinking Ricci-harmonic soliton structure (3.1) and $\textbf{V}(r):=\displaystyle\int_{\{f<r\}}dV$ and $\textbf{V}_R(r):=\displaystyle\int_{\{f< r\}}R-\alpha|\nabla\phi|^2dV, $ we have

$\begin{equation} n\textbf{V}(r)-2r\textbf{V}'(r)=2\textbf{V}_R(r)-2\textbf{V}'_R(r). \end{equation}$

(3.4)

Proof  Integrating by parts and using eq. (3.2),

$\begin{eqnarray} \frac{n}{2}\textbf{V}(r)-\textbf{V}_R(r)&=&\int_{\{f<r\}}\triangle fdV=\int_{\{f=r\}}\nabla f\cdot\frac{\nabla f}{|\nabla f|}dA=\int_{\{f=r\}}|\nabla f|dA, \end{eqnarray}$

(3.5)

which implies

$\begin{equation} \frac{n}{2}\textbf{V}(r)\geq\textbf{V}_R(r). \end{equation}$

(3.6)

By co-Area formula (see [11]), we have

$\begin{equation} \textbf{V}'(r)=\int_{\{f=r\}}\frac{1}{|\nabla f|}dA \end{equation}$

(3.7)

and

$\begin{equation} \textbf{V}_R'(r)=\int_{\{f=r\}}\frac{R-\alpha|\nabla f|^2}{|\nabla f|}dA. \end{equation}$

(3.8)

Using $(3.3)$ and combining $(3.5)$, $(3.7)$ and $(3.8)$, we obtain

$\begin{equation} \frac{n}{2}\textbf{V}(r)-\textbf{V}_R(r)=r\textbf{V}'(r)-\textbf{V}_R'(r).\notag \end{equation}$

Proof of Theorem 1.2  Calculating directly,

$\begin{equation} \frac{d}{dr}(\log(r^{-\frac{n-2\delta}{2}}\textbf{V}(r)))=\frac{r\textbf{V}'(r)-\frac{n-2\delta}{2}\textbf{V}(r)}{r\textbf{V}(r)}\leq\frac{\textbf{V}'_R(r)}{r\textbf{V}(r)}, \end{equation}$

(3.9)

where the last inequality comes from $(3.4)$ and $\textbf{V}_R(r)\geq\delta \textbf{V}(r)$.

Fixed $r_0>0$, for any $r_1>r_0$, integrating $(3.9)$ by parts on $[r_0, r_1]$ yields

$\begin{eqnarray} &&\log\frac{r_1^{-\frac{n-2\delta}{2}}\textbf{V}(r_1)}{r_0^{-\frac{n-2\delta}{2}}\textbf{V}(r_0)}\leq\int_{r_0}^{r_1}\frac{1}{r\textbf{V}(r)}d\textbf{V}_R(r)\notag\\ &\leq&\frac{\textbf{V}_R(r)}{r\textbf{V}(r)}|_{r_0}^{r_1}+\int_{r_0}^{r_1}\frac{\textbf{V}_R(r)}{r^2\textbf{V}(r)}dr+\int_{r_0}^{r_1}\frac{\textbf{V}_R(r)\textbf{V}'(r)}{r\textbf{V}^2(r)}dr\notag\\ &\leq&\frac{n}{2r_0}+\frac{n}{2}(\frac{\log(\textbf{V}(r_1))}{r_1}-\frac{\log(\textbf{V}(r_0))}{r_0})+\frac{n}{2}\int_{r_0}^{r_1}\frac{\log(\textbf{V}(r))}{r^2}dr. \end{eqnarray}$

(3.10)

When $r_0$ is large enough, by the proof of Theorem 1.1 in Yang and Shen [11], we have

$\begin{equation} \textbf{V}(r)\leq C_1r^n \end{equation}$

(3.11)

for some positive constant $C_1$ and for $r\geq r_0$. Plugging inequality (3.11) into (3.10), we have

$\begin{equation} \log\frac{r_1^{-\frac{n-2\delta}{2}}\textbf{V}(r_1)}{r_0^{-\frac{n-2\delta}{2}}\textbf{V}(r_0)}<+\infty \end{equation}$

(3.12)

for any $r_1>r_0$. Moreover, there is a positive constant $C_2$ such that

$\begin{equation} \textbf{V}(r_1)\leq C_2r_1^{\frac{n-2\delta}{2}}. \end{equation}$

(3.13)

For $f(x)\leq\frac{1}{4}(d(x_0, x)+2\sqrt{f(x_0)})^2$ (see Proposition 4.1 in [12]), $B_{x_0}(r)\subset\{f\leq\frac{1}{4}(r+C_2)^2\}$. We have

$\begin{equation} Vol(B_{x_0}(r))\leq\textbf{V}(\frac{1}{4}(d(x_0, x)+2\sqrt{f(x_0)})^2)\leq C(r+1)^{n-2\delta} \end{equation}$

(3.14)

for some positive constant C depends only on $g$ and $x_0$.