Gradient Flow Structure and the Morse–Bott Picture

The space of Riemannian metrics $\mathrm{Met}(M)$ on a closed manifold $M$is an infinite-dimensional open cone in the space of symmetric 2-tensors$\Gamma(S^2 T^*M)$. To define a gradient, we equip it with the Perelman-weighted $L^2$ inner product. For tangent vectors$h, k \in T_g\mathrm{Met}(M)$:

This is a DeWitt-type metric weighted by the Perelman measure $e^{-f}d\mu$. The weighting ensures that the gradient of $\mathcal{F}$ produces the modified Ricci flow. Without the weight, one would obtain only the unmodified flow$-2R_{ij}$, which does not preserve the volume constraint.

One can also write this in index-free notation. Denoting the pointwise inner product of symmetric 2-tensors by $\langle h, k\rangle_g = g^{ia}g^{jb}h_{ij}k_{ab}$:

where the trace is over both pairs of indices. This metric is positive definite since$g$ is positive definite and $e^{-f} > 0$.

For a one-parameter family $g(t)$ with $\dot{g}_{ij} = h_{ij}$ and$f(t)$ adjusted to preserve $\int e^{-f}d\mu$, the first variation of$\mathcal{F}$ is (derived in the previous lecture):

The gradient $\mathrm{grad}_g\mathcal{F}$ is defined by the relation$d\mathcal{F}(h) = \langle\mathrm{grad}_g\mathcal{F},\, h\rangle_{g,f}$ for all variations $h$. Comparing the two expressions, we read off:

The factor of $-2$ arises from matching the convention $\partial_t g = -2\,\mathrm{Ric}$. Modulo diffeomorphisms (on the quotient $\mathrm{Met}(M)/\mathrm{Diff}(M)$), the $\nabla_i\nabla_j f$ term is pure gauge (a Lie derivative $\mathcal{L}_{\nabla f}g$) and the gradient reduces to $-2R_{ij}$.

Hamilton's Ricci flow, augmented by the backward heat equation for $f$, is the ascending gradient flow of $\mathcal{F}$:

The backward heat equation for the dilaton is the compatibility condition:

Together, these ensure that $\frac{d}{dt}\int_M e^{-f}d\mu = 0$ (the weighted volume is preserved). The flow is ascending because $\mathcal{F}$ increases; this is the opposite of the usual Morse theory convention but matches the physics of entropy increase.

The DeTurck trick: the diffeomorphism $\phi_t$ generated by $V^i = g^{ij}\partial_j f$transforms the modified flow into pure Ricci flow. Explicitly, if $\tilde{g}(t) = \phi_t^* g(t)$:

The monotonicity formula follows immediately from the gradient flow structure. Along the coupled system:

Computing the squared norm explicitly:

Equality holds if and only if $R_{ij} + \nabla_i\nabla_j f = 0$ everywhere, which is the steady gradient Ricci soliton equation. The monotonicity is thus a tautological consequence of the gradient flow structure, not a deep analytic estimate.

This mirrors the fact that for any gradient flow $\dot{x} = \nabla F(x)$ on a Riemannian manifold, one has $\dot{F} = |\nabla F|^2 \geq 0$ — the function increases along its own gradient flow. The non-trivial content of Perelman's work is establishing that the Ricci flow is such a gradient flow.

The critical points of $\mathcal{F}$ on $\mathrm{Met}(M)\times C^\infty(M)$satisfy $\mathrm{grad}\,\mathcal{F} = 0$, i.e.:

These are precisely the steady gradient Ricci solitons. The critical set is not isolated: it forms an orbit under $\mathrm{Diff}(M)$ (since if $(g,f)$ is critical, so is $(\phi^*g, f\circ\phi)$ for any diffeomorphism $\phi$).

Examples include Ricci-flat metrics ($f = \text{const}$), the Hamilton cigar soliton on $\mathbb{R}^2$, and the Bryant soliton on $\mathbb{R}^3$. In three dimensions, any compact steady soliton must be Ricci-flat (hence flat), so the interesting examples are non-compact.

The Hessian of $\mathcal{F}$ at a critical point $(g_0, f_0)$ is the linearization of the gradient. For a variation $h_{ij}$:

where $\mathrm{Rm} * h * h$ denotes the curvature operator applied to $h$(involving $R_{ikjl}h^{kl}$ contractions). The Morse–Bott condition requires:

where $\mathcal{C}$ is the critical manifold (the $\mathrm{Diff}(M)$-orbit of $(g_0, f_0)$). When this holds, the critical manifold is clean in the sense of Bott, and the stable/unstable manifold decomposition is well-defined.

The stable manifold of a critical point $g_0$ is the set of initial metrics that flow to $g_0$ under the Ricci flow:

By the Lunardi–Simonett infinite-dimensional stable manifold theorem, $W^s(g_0)$is a smooth submanifold of $\mathrm{Met}(M)$. Its codimension equals the Morse index of $g_0$ (the number of unstable directions of the Hessian).

The tangent space to the stable manifold at $g_0$ is the negative eigenspace of$\mathrm{Hess}\,\mathcal{F}$ (since the flow ascends, "stable" means converging to the critical point as $t \to +\infty$, which requires being in a direction where$\mathcal{F}$ is already near its critical value).

Perelman's $\kappa$-non-collapsing theorem states: along the Ricci flow, if the curvature satisfies $|\mathrm{Rm}| \leq r^{-2}$ on a geodesic ball$B(x, r)$, then the volume ratio is bounded below:

for some $\kappa > 0$ depending only on initial data. In the Morse–Bott picture, this is a uniform lower bound on the size of the stable manifold. The proof uses the$\mathcal{W}$-entropy (the scale-invariant version of $\mathcal{F}$):

Its monotonicity $d\mathcal{W}/dt \geq 0$ provides a uniform lower bound on the reduced volume $\mu(g,\tau) = \inf_f \mathcal{W}$, which translates to volume non-collapsing via the log-Sobolev inequality: