Let \((M, g)\) be an n-dimensional compact Riemannian manifold. In [3], Perelman introduced the functional
$$ \mathcal{F}(g,f)= \int _{M}\bigl(|\nabla f|^{2}+R\bigr)e^{-f} \,dv $$
(1.1)
and proved that the \(\mathcal{F}\)-functional is nondecreasing under the Ricci flow coupled to a backward heat-type equation
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{\partial}{\partial t}g_{ij}=-2R_{ij},\\ f_{t}=-\Delta f+|\nabla f|^{2}-R, \end{array}\displaystyle \right . $$
(1.2)
where R is the scalar curvature depending on the metric g. More precisely, they proved that under the system (1.2),
$$ \frac{d}{d t}\mathcal{F}=2 \int _{M}|R_{ij}+f_{ij}|^{2}e^{-f} \,dv\geq0. $$
(1.3)
If we define
$$ \lambda(g)=\inf_{f}\mathcal{F}(g,f), $$
(1.4)
where the infimum is taken over all smooth functions f which satisfy
$$ \int _{M}e^{-f}\,dv=1, $$
(1.5)
then the nondecreasing of the \(\mathcal{F}\)-functional implies the nondecreasing of \(\lambda(g)\). In particular, \(\lambda(g)\) defined in (1.4) is the lowest eigenvalue of the operator
In [4], Cao considered the eigenvalues of the operator \(-\Delta+\frac{R}{2}\) on manifolds with nonnegative curvature operator and showed that the eigenvalues are nondecreasing along the Ricci flow. Using the same technique, Li [2] also obtained the same monotonicity of the first eigenvalue of the operator \(-\Delta+\frac{R}{2}\) by removing the assumption on a nonnegative curvature operator.
Later, Cao [1] proved the first eigenvalues of the operator \(-\Delta+bR\) with the constant \(b\geq1/4\) are nondecreasing along the Ricci flow. That is, they assume \(u=u(x, t)\) is the corresponding positive eigenfunction of \(\lambda(t)\):
$$ (-\Delta+bR)u=\lambda^{b} u $$
(1.7)
with \(\int_{M}u^{2}\,dv=1\), then
$$ \frac{d}{d t}\lambda^{b}=\frac{1}{2} \int _{M} |R_{ij}+f_{ij}|^{2}e^{-f} \,dv+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \geq0 $$
(1.8)
by letting \(f=-2\log u\). Multiplying both sides of (1.7) with u and integrating on M, we see that the first eigenvalue given in (1.7) satisfies
$$ \lambda(t)=\inf\tilde{\mathcal{F}}^{b}(g,u), $$
(1.9)
where
$$ \tilde{\mathcal{F}}^{b}(g,u)= \int _{M}\bigl(|\nabla u|^{2}+bRu^{2} \bigr)\,dv. $$
(1.10)
In particular,
$$ \tilde{\mathcal{F}}^{b}(g,u)=\frac{1}{4} \mathcal{F}^{4b}(g,f), $$
(1.11)
where
$$\mathcal{F}^{c}(g,f)= \int _{M}\bigl(|\nabla f|^{2}+cR \bigr)e^{-f}\,dv $$
if we let \(f=-2\log u\). It is easy to see from (1.11) that the nondecreasing of the \(\tilde{\mathcal{F}}^{b}\)-functional is equivalent to the nondecreasing of \(\lambda(t)\).
In this paper, we consider the monotonicity along the Ricci flow of lowest constant \(\lambda_{a}^{b}(g)\) such that to the following nonlinear equation there exist positive solutions:
$$ -\Delta u+au\log u+bRu=\lambda_{a}^{b} u $$
(1.12)
with
$$ \int _{M}u^{2}\,dv=1, $$
(1.13)
where a is a real constant. In particular, (1.7) can be seen a special case of (1.12) when \(a=0\). For the lowest constant \(\lambda_{a}^{b}(g)\) such that to the nonlinear equation (1.12) there exist positive solutions, we prove the following.
Theorem 1.1
Let
\(g(t)\), \(t\in[0,T)\)
be a solution to the Ricci flow
$$ \frac{\partial}{\partial t}g_{ij}=-2R_{ij} $$
(1.14)
on a compact Riemannian manifold
M. Then for
\(b\geq\frac{1}{4}\), the lowest constant
\(\lambda_{a}^{b}(g)\)
such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies
$$\begin{aligned} \frac{d}{d t} \biggl(\lambda_{a}^{b}(t)+ \frac{na^{2}}{8}t \biggr)=&\frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv+ \biggl(2b- \frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \\ \geq&0, \end{aligned}$$
(1.15)
where
\(f=-2\log u\).
For the normalized Ricci flow, we can obtain the following.
Theorem 1.2
Let
\(g(t)\), \(t\in[0,T)\)
be a solution to the normalized Ricci flow
$$ \frac{\partial}{\partial t}g_{ij}=-2 \biggl(R_{ij}- \frac{r}{n}g_{ij} \biggr) $$
(1.16)
on a compact Riemannian manifold
M, where
\(r=(\int_{M}R \,dv)/(\int_{M} \,dv)\)
is the average scalar curvature. Then the lowest constant
\(\lambda_{a}^{b}(g)\)
such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies
$$\begin{aligned} \frac{d}{d t} \biggl(\lambda_{a}^{b}+ \frac{na^{2}}{8}t \biggr)+\frac{2r}{n}\lambda^{b} ={}& \frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv \\ &{}+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv, \end{aligned}$$
(1.17)
where
\(f=-2\log u\)
and
\(\lambda^{b}\)
is the lowest eigenvalue of (1.7).
In particular, when \(n=2\), we have \(R_{ij}=\frac{R}{2}g_{ij}\) and the normalized Ricci flow (1.16) becomes \(\frac{\partial}{\partial t}g_{ij}=-(R-r)g_{ij}\). Hence, \(\frac{d}{d t}r=0\), which implies that r is a constant (or see p.455 in [5] for an alternative proof). Then from the estimate (1.17), we obtain the following.
Theorem 1.3
Let
\(g(t)\), \(t\in[0,T)\)
be a solution to the normalized Ricci flow (1.16) on a compact surface
\(M^{2}\). Then for
\(b\geq\frac{1}{4}\), the lowest constant
\(\lambda_{a}^{b}(g)\)
such that to the nonlinear equation (1.12) with (1.13) there exist positive solutions satisfies
$$\begin{aligned} \frac{d}{d t} \biggl(\lambda_{a}^{b}+ \frac{a^{2}}{4}t+r \int _{0}^{t}\lambda^{b}(s)\,ds \biggr) ={}&\frac{1}{2} \int _{M} \biggl|R_{ij}+f_{ij}+ \frac{a}{2}g_{ij}\biggr|^{2}e^{-f}\,dv \\ &{}+ \biggl(2b-\frac{1}{2} \biggr) \int _{M} |R_{ij}|^{2}e^{-f}\,dv \\ \geq{}&0, \end{aligned}$$
(1.18)
where
\(f=-2\log u\)
and
\(\lambda^{b}\)
is the lowest eigenvalue of (1.7).
Remark 1.1
In particular, when \(a=0\), our estimate (1.15) reduces to Theorem 1.5 of Cao in [1] and the estimate (1.18) reduces to the Corollary 2.4 of Cao in [1], respectively.
On the other hand, under the transformation \(f=-2\log u=-\log v\) with \(u^{2}=v\), equation (1.2) becomes
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{\partial}{\partial t}g_{ij}=-2R_{ij},\\ v_{t}=-\Delta v+Rv. \end{array}\displaystyle \right . $$
(1.19)
In particular, the second equation in (1.19) is exactly the conjugate heat equation introduced by Perelman. In [6], Cao and Zhang obtained differential Harnack inequalities for positive solutions of the nonlinear parabolic equation of the type \(v_{t}=\Delta v-v\log v+Rv\). Extending the second equation in (1.19) to the following nonlinear version:
$$ v_{t}=-\Delta v+a v\log v+Sv, $$
(1.20)
Guo and Ishida [7, 8] studied Harnack inequalities for positive solutions of equation (1.20) on a compact Riemannian manifold with a family of \(g(t)\) evolving by a geometric flow \(\frac{\partial}{\partial t}g_{ij}=-2S_{ij}\), where \(S_{ij}\) is a family of smooth symmetric two-tensor and \(S=g^{ij}S_{ij}\). Clearly, there is a one-to-one relation for the following two equations:
$$ \frac{\partial}{\partial t}v=-\Delta v+a v\log v+Rv\quad\Longleftrightarrow \quad\frac{\partial}{\partial t}f=-\Delta f+|\nabla f|^{2}+af-R $$
(1.21)
under \(f=-\log v\). Therefore, a natural problem is to consider the monotonicity of
$$ \overline{\mathcal {F}}^{c}_{d}(g,f)= \int _{M}\bigl[|\nabla f|^{2}+cR+d(f+1) \bigr]e^{-f}\,dv $$
(1.22)
under the Ricci flow coupled to a nonlinear backward heat-type equation
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{d}{d t}g_{ij}=-2R_{ij},\\ f_{t}=-\Delta f+|\nabla f|^{2}+af-R, \end{array}\displaystyle \right . $$
(1.23)
where c, d are two real constants.
For the functional \(\overline{\mathcal{F}}^{c}_{d}(g,f)\), we derive the following monotonicity formula.
Theorem 1.4
Let
\(g(t)\), \(t\in[0,T)\)
be a solution to the Ricci flow (1.14) on a compact Riemannian manifold
M. Then all functionals
\(\overline{\mathcal{F}}^{c}_{d}(g,f)\)
defined by (1.22) under the system (1.23) satisfy
$$\begin{aligned} \frac{d}{d t}\overline{\mathcal{F}}_{\frac{nak}{8}}^{k}(g,f) ={}&2 \int _{M}\biggl|R_{ij}+f_{ij}- \frac{a}{4}fg_{ij}\biggr|^{2}e^{-f}\,dv +2(k-1) \int _{M}\biggl|R_{ij}-\frac{a}{4}fg_{ij}\biggr|^{2}e^{-f} \,dv \\ &{}+\frac{na}{8}k\mathcal{F}^{1}(g,f)+a\mathcal{F}^{0}(g,f). \end{aligned}$$
(1.24)
In particular, if
\(R(t)\geq0\)
for all
t
and
\(a\geq0\), \(k\geq1\), then
\(\frac{d}{d t}\overline{\mathcal{F}}_{\frac{nak}{8}}^{k}(g,f)\geq0\).
Remark 1.2
Choosing \(a=0\) in (1.24), we obtain Theorem 4.2 of Li in [2].