Perelman's F-Functional as an Action

Let $(M^n, g)$ be a closed Riemannian manifold and $f: M \to \mathbb{R}$ a smooth function. Perelman introduced the $\mathcal{F}$-functional as an action principle governing the Ricci flow. Define the action:

Here $R$ is the scalar curvature of $g$, $|\nabla f|^2 = g^{ij}\partial_i f\,\partial_j f$, and $d\mu = \sqrt{\det g}\,d^nx$ is the Riemannian volume form. The measure$e^{-f}d\mu$ is called the weighted volume element; it plays the role of the path-integral measure in the gravitational analogy.

The functional $\mathcal{F}$ combines the Einstein–Hilbert action$\int R\,d\mu$ with a dilaton kinetic term $\int|\nabla f|^2 e^{-f}d\mu$, mirroring the low-energy string effective action in the string frame. An integration by parts shows the equivalence to a more familiar form:

On a closed manifold $M$, the second integral vanishes by the divergence theorem, leaving $\mathcal{F} = \int(R + 2\Delta f - |\nabla f|^2)e^{-f}d\mu$.

The substitution $u = e^{-f/2}$ (so $f = -2\ln u$) linearizes the functional. Computing the kinetic piece step by step: since $\nabla f = -2\,u^{-1}\nabla u$ and $e^{-f} = u^2$, the product simplifies as:

The curvature term transforms as $R\,e^{-f} = R\,u^2$. Combining:

Integrating the gradient term by parts on the closed manifold $M$ (no boundary terms):

Therefore the action can be written as a quadratic form:

This expresses $\mathcal{F}$ as a Rayleigh quotient for the Yamabe (conformal Laplacian) operator, establishing $\mathcal{F}$ as a spectral quantity whose minimization is an eigenvalue problem.

The conformal Laplacian (Yamabe operator) in dimension $n$ is:

In $n = 3$ spatial dimensions this reduces to $\mathcal{H}_{\rm Yamabe} = -4\Delta + R$. Since $\mathcal{H}_{\rm Yamabe}$ is elliptic and self-adjoint on $L^2(M, d\mu)$, it has a discrete spectrum $\lambda_1 \leq \lambda_2 \leq \cdots$ with orthonormal eigenfunctions$\{u_k\}_{k=1}^\infty$.

Perelman's $\lambda$-invariant is the infimum of $\mathcal{F}$ over all normalized $u$:

The eigenvalue equation for the ground state is:

Positivity of $u_0$ follows from the Perron–Frobenius theorem for elliptic operators. The sign of $\lambda(g)$ determines the Yamabe class:$\lambda(g) > 0$ means positive scalar curvature is achievable conformally,$\lambda(g) = 0$ means scalar-flat, and $\lambda(g) < 0$ means the conformal class admits only negative scalar curvature.

For the round $S^3$ of radius $r$, one can compute $R = 6/r^2$and $\lambda_1 = 6/r^2$ (the constant function is the ground state). More generally, the Rayleigh quotient gives:

We vary $\mathcal{F}$ with respect to $f$ at fixed $g$, subject to the constraint $\int_M e^{-f}d\mu = \text{const}$. Under $f \to f + \delta f$:

For the scalar curvature term: $\delta_f(R\,e^{-f}) = -R\,e^{-f}\,\delta f$. Collecting and integrating the $2\nabla^i f\,\nabla_i\delta f$ term by parts:

Summing all contributions:

Setting $\delta_f\mathcal{F} = 0$ with the constraint enforced by a Lagrange multiplier $\lambda$:

Under $g_{ij} \to g_{ij} + \delta g_{ij}$ we use the Palatini identity$\delta R = -R_{ij}\,\delta g^{ij} + \nabla_i\nabla_j\,\delta g^{ij} - g_{ij}\,\Delta\,\delta g^{ij}$, together with $\delta\sqrt{g} = \frac{1}{2}\sqrt{g}\,g_{ij}\,\delta g^{ij}$ and$\delta_g(|\nabla f|^2) = -\nabla_i f\,\nabla_j f\,\delta g^{ij}$. After integration by parts against the weighted measure $e^{-f}d\mu$:

The key cancellations use $\nabla_i(e^{-f}) = -(\nabla_i f)\,e^{-f}$ to convert total divergences with the weighted measure into $\nabla_i\nabla_j f$ terms. The stationarity condition $\delta_g\mathcal{F} = 0$ yields:

This is the tensorial Euler–Lagrange equation. Notice the structural similarity to the vacuum Einstein equation $R_{ij} = 0$: the Hessian of $f$ plays the role of a matter source.

Combining both Euler–Lagrange equations gives the steady gradient Ricci soliton. From $R_{ij} + \nabla_i\nabla_j f = 0$, take the trace with $g^{ij}$:

Substitute $\Delta f = -R$ into $2\Delta f - |\nabla f|^2 + R = \lambda$:

The on-shell value is $\mathcal{F}_{\rm on-shell} = \lambda\cdot\int e^{-f}d\mu$. The full on-shell system:

This is the steady gradient Ricci soliton equation. A shrinking soliton instead satisfies$R_{ij} + \nabla_i\nabla_j f = \frac{1}{2\tau}g_{ij}$, corresponding to the$\mathcal{W}$-entropy. An expanding soliton has $R_{ij} + \nabla_i\nabla_j f = -\frac{1}{2\tau}g_{ij}$.

The contracted second Bianchi identity $\nabla^j(R_{ij} - \frac{1}{2}Rg_{ij}) = 0$ applied to the soliton equation gives an integrability condition:

which is automatically satisfied on-shell and constrains the topology of solitons.

To obtain the Ricci flow dynamically, introduce a time-extended action integrating$\mathcal{F}$ over $[0,T]$:

Varying $I_P$ with respect to the metric path $g(\tau)$ yields the modified Ricci flow:

Under the DeTurck diffeomorphism $\phi_\tau$ generated by $\nabla^i f$, this is gauge-equivalent to Hamilton's Ricci flow $\partial_\tau g_{ij} = -2R_{ij}$. The variation with respect to $f(\tau)$ yields the backward heat equation:

In terms of $u = e^{-f/2}$, this becomes the conjugate heat equation:

The coupled system preserves $\int e^{-f}d\mu = \text{const}$ and makes $\mathcal{F}$monotonically non-decreasing. The monotonicity formula (proved in the next lecture) gives:

Equality holds precisely at steady gradient Ricci solitons. This monotonicity is the analogue of the second law of thermodynamics for the geometric flow, and will be reinterpreted as a gradient flow identity in the next section.