I am reading Ben Andrews book about Ricci flow and at the start of the chapter about Perelman's gradient flow formulation for Ricci flow he says Robert Bryant exposed that there are no functionals defined on the $L^2$-space of Riemannian metrics that promotes Ricci flow as a gradient flow.
Does anyone know the name of this paper? The book does not include it on their references.
If there were such a functional $\mathcal{F}$, observe that
Under Ricci flow the functional would have to decrease. That is, if $\partial_t g(t) = -2 Rc[g]$ then $\partial_t \mathcal{F}(g(t)) \leq 0$, with strict inequality of $-2 Rc[g] \neq 0$.
The functional would have to be invariant under diffeomorphisms, that is if $\Phi$ is a diffeomorphism of the manifold then $\mathcal{F}(\Phi^*g) = \mathcal{F}(g)$. This is because the Ricci flow is invariant under diffeomorphism.
The point is that Robert Bryant found a metric on $\mathbb R^n$ which moves under Ricci flow only by diffeomorphisms (which is called a steady soliton). That is, there is a metric $g_S$ and a family of diffeomorphisms $\Phi_t$ such that $g(t) = \Phi^*_t g_S$ is the solution of Ricci flow starting from $g_S$. By point 2 above, $\mathcal{F}(g(t)) = \mathcal{F}(g_S)$, which contradicts point 1.
The details of the constructing the soliton are written by Robert Bryant here. Many other gradient steady solitons have been discovered since then.
Edit: As pointed out by Rbega, this argument only shows that such a functional would have to be infinite on the Bryant soliton. We can say that mean curvature flow is the gradient flow for area. The gradient of area is defined even for surface with infinite area, by consider compact perturbations. Therefore, we need a stronger argument.
If there was a compact steady soliton, then the argument above would work (assuming that $\mathcal{F}$ is always finite on compact manifolds). However, there is no compact steady soliton (except for Ricci flat manifolds).
The stronger argument comes from the existence of shrinking solitons on compact manifolds. The first known example is the Koiso-Cao soliton. A shrinking soliton is a pair of a metric $g$ and a vector field $X$ which solves $$-2Rc = - g + \mathcal{L}_X g,$$ equivalently $$-2Rc_{ij} = - g _{ij}+ \nabla_i X_j + \nabla_j X_i.$$ Under Ricci flow, a steady soliton shrinks and also move by diffeomorphisms integrating $X$.
If $-2Rc$ where the gradient of $\mathcal{F}$ then, $$-Grad \mathcal{F} = - g + \mathcal{L}_X g$$ I claim this can't be true unless $\mathcal{L}_X g = 0$. Note that the two parts of this tensor field are orthogonal in $L^2$, because $$\int_{M} (g,\mathcal{L}_X g)_g dVol_g = \int_M tr(\mathcal{L}_X g) dVol_g = \int_M div_g(X) dVol_g = 0.$$ Therefore we have \begin{align} |\mathcal{L}_X g|^2_{L^2} &= (-Grad \mathcal{F}, \mathcal{L}_X g)_{L^2} \\ &=-\partial_{\epsilon} \mathcal{F}(g + \epsilon \mathcal{L}_X g) \\ &= -\partial_{\epsilon} \mathcal{F}(\Phi_{\epsilon}^* g) = 0 \end{align} The first equality is by the orthogonality of $-g$ and $\mathcal{L}_X g$, the second is the defining property of the gradient.
In this old blog post I showed that Perelman’s modified Ricci flow is a gradient flow for closed manifolds. I’m copying some of what I wrote there, edited a bit.
I want to interpret what Perelman might mean by Ricci flow being the gradient flow for the first eigenvalue of $-4\Delta +R$. First, what do we mean by gradient flow? Given a smooth manifold, the space of smooth Riemannian metrics is denoted $Riem(M)$, and is an open cone in the space of smooth symmetric bilinear forms $S^2M$ on $M$, so that for $g\in Riem(M)$, $T_g(Riem(M))\cong S^2M$. There is a natural inner product on $T_g(Riem(M))$, given by $(h,h')_g=\int_M h_{ik}h'_{lm}g^{il}g^{km} dv_g$ for $h,h'\in S^2M$ where we use the Einstein summation convention ($h_{ik}$ is the tensor written in local coordinates, and $g^{il}$ is the inverse tensor so that $g^{il}g_{lj}=\delta^i_j$). (Note: I think the convention is to usually assume that $g_{ij}=\delta_{ij}$ is diagonalized, in which case we can write $(h,h')_g=\int_M h_{ik}h'_{ik} dv_g$, summing over repeated indices). Thus, a flow $g_t=V(g)$, $V:Riem(M)\to S^2M$, is a gradient flow with respect to this $L^2$ inner product if there is a functional $\mathcal{F}:Riem(M)\to \mathbb{R}$ such that $\frac{\partial\mathcal{F}(g)}{\partial h}=(h,V(g))_g$, in which case $\frac{\partial\mathcal{F}(g_t)}{\partial t}=(V(g),V(g))_g\geq 0$, since $(,)_g$ is positive definite. This is of interest, since then the functional $\mathcal{F}$ is monotonic along the flow, and one might be able to use it to analyze the evolution of $g_t$ under the flow. It turns out Perelman has shown that flows related to the Ricci flow are gradient flows, but with respect to modified $L^2$ metrics on $Riem(M)$.
We'll use $R$ to denote the scalar curvature (suppressing the Riemannian structure $(M,g)$). The operator $-4\Delta+R$ is a Schrodinger operator. It turns out to have a unique normalized positive eigenvector, that is there is a unique function $\Phi:M\to \mathbb{R}$ such that $\Phi(x)>0$ for all $x\in M$, $-4\Delta \Phi+R\Phi=\lambda \Phi$, and $\|\Phi\|_2^2=\int_M \Phi^2 dv_g=1$. In fact, $\Phi$ has minimal eigenvalue for $-4\Delta+R$. For example, if $R$ is constant (as for a Yamabe metric), then $\Phi$ is harmonic, so it must be the constant positive function such that $\|\Phi\|_2=1$. Kleiner and Lott claim that the eigenvalue (and presumably the eigenfunction) depend smoothly on $g$ (see after Definition 6.4). We will denote the eigenvalue $\lambda$ and the eigenfunction $\Phi=e^{-f/2}$, following Perelman. Now, define an $L^2$ metric on $T_g(Riem(M))$ by $[h,h']_g=\frac12\int_M h_{ik}h'_{lm}g^{il}g^{km} e^{-f} dv_g$. I claim that the flow $g_t=-2(Ric+\nabla^2f)$, or written locally as $(g_{ij})_t=-2(R_{ij}+\nabla_i\nabla_j f)$, is the gradient flow for the functional $\mathcal{F}(g)=\int_M (R+|\nabla f|^2)e^{-f} dv_g=\int_M \Phi(-4\Delta \Phi+R\Phi) dv_g=\lambda$ with respect to the metric $[,]$.
To see this, note that the eigenvalue equation $-4\Delta\Phi+R\Phi=\lambda\Phi$ is equivalent to the equation $2\Delta f-|\nabla f|^2+R=\lambda$ (this is claimed in 2.4, and follows easily from the computations in section 4 of Kleiner-Lott). The Rayleigh Ritz quotient method for obtaining the lowest eigenvalue of an operator $L=-4\Delta+R$ is given by $$\lambda(g)=\inf_{\Phi} \frac{(\Phi,L\Phi)_2}{\|\Phi\|_2^2},$$ or normalizing $\lambda=\inf \{ (\Phi,L\Phi)_2 | \|\Phi\|_2=1 \}$ (where $(,)_2$ is the inner product on $L^2(M)$), in which case the eigenvalue equation can be seen as a generalization of Lagrange multipliers). On p. 5, we see the formula for the variation of $\mathcal{F}(g)$ with $\delta g_{ij}=v_{ij}$ and $\delta f=h$ as $$\delta \mathcal{F}(v_{ij},h)=\int_M e^{-f}[-v_{ij}(R_{ij}+\nabla_i \nabla_j f)+(v/2-h)(2\Delta f-|\nabla f|^2+R)]dv_g,$$ so that if we let $f$ satisfy $\|\Phi\|_2^2=\int_M e^{-f}dv_{g_t}=1$ and $2\Delta f-|\nabla f|^2+R=\lambda(g_t)$, then $\delta \|\Phi\|_2^2 = \int_M (v/2-h) dv_g=0$ (from Proposition 4.3 of Kleiner-Lott), and we have $\delta \mathcal{F}= [v_{ij},-2 (R_{ij}+\nabla_i\nabla_j f)]_{g} +\lambda \int_M (v/2-h)e^{-f} dv_g = [v_{ij},-2 (R_{ij}+\nabla_i\nabla_j f)]_{g} $. Now, the flow $g_t=-2(Ric+\nabla^2 f)$ is equivalent to Ricci flow $g_t=-2Ric$ up to a diffeomorphism, since the term $\nabla^2 f$ only modifies the metric by the Lie derivative of the vector field $\nabla f$.
