No breathers theorem II

3.1  

In order to handle the shrinking case when $\lambda>0,$ we need to replace our functional $\mathcal {F}$ by its generalization, which contains explicit insertions of the scale parameter, to be denoted by $\tau.$ Thus consider the functional

$${\cal W}(g_{ij},f,\tau)=\int_M{[\tau(\vert\nabla f\vert^2+R)+f-n](4\pi\tau)^{-\frac{n}{2}}e^{-f}dV},$$ (3.1)

restricted to $f$ satisfying

$$\int_M{(4\pi\tau)^{-\frac{n}{2}}e^{-f}dV}=1,$$ (3.2)

$\tau>0.$ Clearly ${\cal W}$ is invariant under simultaneous scaling of $\tau$ and $g_{ij}.$ The evolution equations, generalizing (1.3) are

$$(g_{ij})_t=-2R_{ij} , f_t=-\triangle f+\vert\nabla f\vert^2-R+\frac{n}{2\tau} , \tau_t=-1$$ (3.3)

The evolution equation for $f$ can also be written as follows: $\Box^*u=0,$ where $u=(4\pi\tau)^{-\frac{n}{2}}e^{-f},$ and $\Box^*=-\partial/\partial t -\triangle +R$ is the conjugate heat operator. Now a routine computation gives

$$d{\cal W}/dt=\int_M{2\tau\vert R_{ij}+\nabla_i\nabla_j f-\frac{1}{2\tau}g_{ij}\vert^2(4\pi\tau)^{-\frac{n}{2}}e^{-f}dV} .$$ (3.4)

Therefore, if we let $\mu(g_{ij},\tau)=$inf$\ {\cal W}(g_{ij},f,\tau)$ over smooth $f$ satisfying (3.2), and $\nu(g_{ij})=$inf$\ \mu(g_{ij},\tau)$ over all positive $\tau,$ then $\nu(g_{ij}(t))$ is nondecreasing along the Ricci flow. It is not hard to show that in the definition of $\mu$ there always exists a smooth minimizer $f$ (on a closed $M$). It is also clear that $\lim_{\tau\to\infty}\mu(g_{ij},\tau)=+\infty$ whenever the first eigenvalue of $-4\triangle +R$ is positive. Thus, our statement that there is no shrinking breathers other than gradient solitons, is implied by the following

Claim For an arbitrary metric $g_{ij}$ on a closed manifold M, the function $\ \mu(g_{ij},\tau)$ is negative for small $\tau>0$ and tends to zero as $\tau$ tends to zero.

Proof of the Claim. (sketch) Assume that $\bar{\tau}>0$ is so small that Ricci flow starting from $g_{ij}$ exists on $[0,\bar{\tau}].$ Let $u=(4\pi\tau)^{-\frac{n}{2}}e^{-f}$ be the solution of the conjugate heat equation, starting from a $\delta$-function at $t=\bar{\tau}, \tau(t)=\bar{\tau}-t.$ Then ${\cal W}(g_{ij}(t),f(t),\tau(t))$ tends to zero as $t$ tends to $\bar{\tau},$ and therefore $\mu(g_{ij},\bar{\tau})\le{\cal W}(g_{ij}(0),f(0),\tau(0))<0$ by (3.4).

Now let $\tau\to 0$ and assume that $f^{\tau}$ are the minimizers, such that

$${\cal W}(\frac{1}{2}\tau^{-1}g_{ij},f^{\tau},\frac{1}{2}) ={\cal W}(g_{ij},f^{\tau},\tau)=\mu(g_{ij},\tau)\le c<0.$$

The metrics $\frac{1}{2}\tau^{-1}g_{ij}$ "converge" to the euclidean metric, and if we could extract a converging subsequence from $f^{\tau},$ we would get a function $f$ on $\mathbb{R}^n$, such that $\int_{\mathbb{R}^n}{(2\pi)^{-\frac{n}{2}}e^{-f}dx}=1$ and

$$\int_{\mathbb{R}^n}{[\frac{1}{2}\vert\nabla f\vert^2+f-n](2\pi)^{-\frac{n}{2}}e^{-f}dx}<0$$

The latter inequality contradicts the Gaussian logarithmic Sobolev inequality, due to L.Gross. (To pass to its standard form, take $f=\vert x\vert^2/2-2\log\phi$ and integrate by parts) This argument is not hard to make rigorous; the details are left to the reader.

3.2 Remark. Our monotonicity formula (3.4) can in fact be used to prove a version of the logarithmic Sobolev inequality (with description of the equality cases) on shrinking Ricci solitons. Indeed, assume that a metric $g_{ij}$ satisfies $R_{ij}-g_{ij}-\nabla_i b_j-\nabla_j b_i=0.$ Then under Ricci flow, $g_{ij}(t)$ is isometric to $(1-2t)g_{ij}(0),$ $\mu(g_{ij}(t),\frac{1}{2}-t)=\mu(g_{ij}(0),\frac{1}{2}),$ and therefore the monotonicity formula (3.4) implies that the minimizer $f$ for $\mu(g_{ij},\frac{1}{2})$ satisfies $R_{ij}+\nabla_i\nabla_j f-g_{ij}=0.$ Of course, this argument requires the existence of minimizer, and justification of the integration by parts; this is easy if $M$ is closed, but can also be done with more efforts on some complete $M$, for instance when $M$ is the Gaussian soliton.

3.3* The no breathers theorem in dimension three was proved by Ivey [I]; in fact, he also ruled out nontrivial Ricci solitons; his proof uses the almost nonnegative curvature estimate, mentioned in the introduction.

Logarithmic Sobolev inequalities is a vast area of research; see [G] for a survey and bibliography up to the year 1992; the influence of the curvature was discussed by Bakry-Emery [B-Em]. In the context of geometric evolution equations, the logarithmic Sobolev inequality occurs in Ecker [E 1].