Ricci Flow: The Hamilton Equation

The Fundamental PDE

Hamilton's Ricci flow is a geometric evolution equation that deforms a Riemannian metric on a smooth manifold $M$ in the direction of its Ricci curvature. Given an initial metric $g_0$, the flow is defined by the PDE:

$$\frac{\partial}{\partial t}\, g_{ij} = -2\,R_{ij}, \qquad g(0) = g_0$$

Here $g_{ij}(t)$ is a one-parameter family of Riemannian metrics and $R_{ij}$ denotes the Ricci curvature tensor of $g(t)$. The factor of $-2$ is conventional and ensures that the flow is a weakly parabolic system in the direction of decreasing curvature.

Intuitively, regions of positive Ricci curvature shrink, while regions of negative Ricci curvature expand. In dimension $n = 2$, we have $R_{ij} = \frac{1}{2}R\,g_{ij}$, so the Ricci flow reduces to a scalar equation and the metric evolves conformally. In dimension $n = 3$, the Ricci tensor determines the full Riemann tensor, which is why Hamilton originally introduced the flow in this setting.

Short-Time Existence: The DeTurck Trick

The Ricci flow equation is only weakly parabolic because the Ricci tensor is invariant under the diffeomorphism group. DeTurck (1983) resolved this degeneracy by introducing a modified flow that breaks the diffeomorphism invariance, yielding a strictly parabolic system.

Fix a background metric $\tilde{g}$ and define the DeTurck vector field:

$$W^k = g^{pq}\bigl(\Gamma^k_{pq} - \tilde{\Gamma}^k_{pq}\bigr)$$

The DeTurck–Ricci flow is:

$$\frac{\partial}{\partial t}\, g_{ij} = -2\,R_{ij} + \nabla_i W_j + \nabla_j W_i$$

This modified equation is strictly parabolic, so standard PDE theory (via the Nash–Moser inverse function theorem or direct energy estimates) gives short-time existence and uniqueness. A solution of the DeTurck flow pulls back via a time-dependent family of diffeomorphisms to give a solution of the original Ricci flow.

Normalised vs. Unnormalised Flow

The unnormalised Ricci flow does not preserve the volume of the manifold. Hamilton introduced the normalised Ricci flow, which adjusts the metric by a scalar factor to keep the total volume constant:

$$\frac{\partial}{\partial t}\, g_{ij} = -2\,R_{ij} + \frac{2}{n}\,r(t)\,g_{ij}$$

where $r(t)$ is the average scalar curvature:

$$r(t) = \frac{\displaystyle\int_M R\,d\mu}{\displaystyle\int_M d\mu}$$

The normalised and unnormalised flows differ only by rescaling in space and reparametrisation in time. They carry the same geometric information.

Convention in Perelman's Work

Throughout his three preprints, Perelman works exclusively with the unnormalised Ricci flow $\partial_t g = -2\,\mathrm{Ric}$. This simplifies many computations, particularly for the entropy functionals and reduced geometry, at the cost of allowing the volume and diameter to change with time. All estimates are then transferred to the normalised setting as needed.

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