No breathers theorem I

2.1  

A metric $g_{ij}(t)$ evolving by the Ricci flow is called a breather, if for some $t_1<t_2$ and $\alpha>0$ the metrics $\alpha g_{ij}(t_1)$ and $g_{ij}(t_2)$ differ only by a diffeomorphism; the cases $\alpha=1 , \alpha<1 , \alpha>1$ correspond to steady, shrinking and expanding breathers, respectively. Trivial breathers, for which the metrics $g_{ij}(t_1)$ and $g_{ij}(t_2)$ differ only by diffeomorphism and scaling for each pair of $t_1$ and $t_2$, are called Ricci solitons. (Thus, if one considers Ricci flow as a dynamical system on the space of riemannian metrics modulo diffeomorphism and scaling, then breathers and solitons correspond to periodic orbits and fixed points respectively). At each time the Ricci soliton metric satisfies an equation of the form $R_{ij}+cg_{ij}+\nabla_i b_j +\nabla_j b_i=0,$ where $c$ is a number and $b_i$ is a one-form; in particular, when $b_i=\frac{1}{2}\nabla_i a$ for some function $a$ on $M,$ we get a gradient Ricci soliton. An important example of a gradient shrinking soliton is the Gaussian soliton, for which the metric $g_{ij}$ is just the euclidean metric on $\mathbb{R}^n$, $c=1$ and $a=-\vert x\vert^2/2.$

In this and the next section we use the gradient interpretation of the Ricci flow to rule out nontrivial breathers (on closed $M$). The argument in the steady case is pretty straightforward; the expanding case is a little bit more subtle, because our functional $\mathcal {F}$ is not scale invariant. The more difficult shrinking case is discussed in section 3.

2.2  

Define $\lambda(g_{ij})=$inf$\ \mathcal {F}(g_{ij},f) ,$ where infimum is taken over all smooth $f,$ satisfying $\int_M{e^{-f}dV}=1 .$ Clearly, $\lambda(g_{ij})$ is just the lowest eigenvalue of the operator $-4\triangle+R.$ Then formula (1.4) implies that $\lambda(g_{ij}(t))$ is nondecreasing in $t,$ and moreover, if $\lambda(t_1)=\lambda(t_2),$ then for $t\in [t_1,t_2]$ we have $R_{ij}+\nabla_i\nabla_j f=0$ for $f$ which minimizes $\mathcal {F}.$ Thus a steady breather is necessarily a steady soliton.

2.3  

To deal with the expanding case consider a scale invariant version $\bar{\lambda}(g_{ij})=\lambda(g_{ij})V^{2/n}(g_{ij}).$ The nontrivial expanding breathers will be ruled out once we prove the following

Claim $\bar{\lambda}$ is nondecreasing along the Ricci flow whenever it is nonpositive; moreover, the monotonicity is strict unless we are on a gradient soliton.

(Indeed, on an expanding breather we would necessarily have $dV/dt>0$ for some $t {\in } [t_1,t_2].$ On the other hand, for every $t$, $-\frac{d}{dt}$log$V=\frac{1}{V}\int{RdV}\ge\lambda(t),$ so $\bar{\lambda}$ can not be nonnegative everywhere on $[t_1,t_2],$ and the claim applies.)

Proof of the claim.

$${\small\begin{array}{cc}d\bar{\lambda}(t)/dt\ge2V^{2/n}\int{\vert... ...(R+\triangle f)^2e^{-f}dV}-(\int{(R+\triangle f)e^{-f}dV})^2)]\ge0,\end{array}}$$

where $f$ is the minimizer for $\mathcal {F}.$

2.4  

The arguments above also show that there are no nontrivial (that is with non-constant Ricci curvature) steady or expanding Ricci solitons (on closed $M$). Indeed, the equality case in the chain of inequalities above requires that $R+\triangle f$ be constant on $M$; on the other hand, the Euler-Lagrange equation for the minimizer $f$ is $2\triangle f-\vert\nabla f\vert^2+R=const.$ Thus, $\triangle f-\vert\nabla f\vert^2=const=0$, because $\int{(\triangle f-\vert\nabla f\vert^2)e^{-f}dV}=0.$ Therefore, $f$ is constant by the maximum principle.

2.5* 1  

A similar, but simpler proof of the results in this section, follows immediately from [H 6,$\S 2$], where Hamilton checks that the minimum of $RV^{\frac{2}{n}}$ is nondecreasing whenever it is nonpositive, and monotonicity is strict unless the metric has constant Ricci curvature.