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Hamilton's Ricci Flow

Summary

Richard Hamilton introduced the evolution equation \(\partial_t g = -2\,\mathrm{Ric}(g)\) in 1982. It deforms a Riemannian metric so that curvature "averages out" like heat. With it Hamilton showed that every closed 3-manifold of positive Ricci curvature is a spherical space form – the blueprint for Perelman's proof.

1. The Idea

A metric carries curvature inhomogeneities – strong curvature here, weak there. In 1982 Hamilton asked: is there a natural evolution equation that redistributes a metric so that curvature becomes more uniform – analogous to the heat equation smoothing out temperatures? His proposal:

\[\boxed{\,\frac{\partial g(t)}{\partial t} = -2\,\mathrm{Ric}\bigl(g(t)\bigr)\,}\quad\text{(Hamilton 1982).}\]

Initial condition \(g(0) = g_0\). The equation is autonomous in \(g\) – the right-hand side depends on \(g\) only through its Ricci curvature (see Article 02).

2. Why "Minus Twice Ricci"?

Three observations explain the form.

(a) Differential-topological naturality. \(\mathrm{Ric}\) is the unique natural symmetric \((0,2)\)-tensor field of second order in \(g\) that depends pointwise only on derivatives of \(g\) up to order 2. The sign \(-2\) makes the equation parabolic in its linearisation – analogous to the heat equation \(\partial_t u = \Delta u\).

(b) Heat-equation heuristic. In harmonic coordinates (\(\Box x^k = 0\)) one has approximately

\[\frac{\partial g_{ij}}{\partial t} = \Delta_g g_{ij} + \text{lower-order terms}.\]

The metric diffuses like a scalar – with the effect that curvature spikes get smoothed.

(c) Variational principle. Hamilton motivated the equation through symmetry; later Perelman showed that the Ricci flow is the gradient flow of the \(\mathcal{F}\)-functional (see Article 05) – a retroactive variational principle.

3. Hamilton's Original Theorem (1982)

In the seminal Three-manifolds with positive Ricci curvature Hamilton proved:

Theorem (Hamilton 1982). Let \((M^3, g_0)\) be a closed 3-manifold with \(\mathrm{Ric}(g_0) > 0\). Then the Ricci flow \(g(t)\) exists for all \(t \in [0, T)\), and the normalised Ricci flow converges as \(t \to T\) to a metric of constant positive sectional curvature. In particular \(M\) is diffeomorphic to a spherical space form \(S^3/\Gamma\).

— Hamilton, J. Differential Geometry 17 (1982), 255–306.

The proof combines maximum principles for tensors, careful analysis of the curvature tensor under the flow, and the fact that in dimension 3 the Ricci tensor determines the full curvature tensor (Article 02, §6).

4. Short-Time Existence and Uniqueness

The equation is not strictly parabolic (it has a diffeomorphism gauge freedom), but DeTurck's trick converts it into a parabolic one by adding a Lie-derivative term that fixes a gauge. Hence:

Short-time existence. For every smooth initial metric \(g_0\) on a closed manifold there exists \(T > 0\) and a unique solution \(g(t)\) of the Ricci flow on \([0, T)\).

— Hamilton 1982 (existence), DeTurck 1983 (simplified proof).

5. Scaling Behaviour

The Ricci flow is not scale-invariant. Rescaling \(g \mapsto \lambda^2 g\) also rescales time: \(\mathrm{Ric}\) is scale-invariant, but \(\partial_t\) carries the factor \(\lambda^{-2}\). Consequences:

  • On an Einstein initial metric \(\mathrm{Ric}(g_0) = \lambda g_0\) the solution is \(g(t) = (1 - 2\lambda t)\, g_0\). For \(\lambda > 0\) the sphere collapses to a point in finite time; for \(\lambda < 0\) hyperbolic space expands without bound.
  • The normalised Ricci flow \(\partial_t g = -2\,\mathrm{Ric} + \tfrac{2}{n}\bar R\, g\) preserves volume; Einstein metrics become genuine fixed points.

6. Examples

Round sphere. On \((S^3, g_{\mathrm{round}})\), \(\mathrm{Ric} = 2\, g_{\mathrm{round}}\). Solution: \(g(t) = (1 - 4t)\, g_{\mathrm{round}}\), singular at \(T = 1/4\) (shrinking soliton).

Flat torus. On \(T^3\) with the flat metric, \(\mathrm{Ric} = 0\), so \(g(t) \equiv g_0\) – static.

Hyperbolic space form. \(\mathrm{Ric} = -2\, g\) gives \(g(t) = (1 + 4t)\, g\) – eternal expansion.

Cylinder \(S^2 \times \mathbb{R}\). The \(S^2\) factor shrinks, the \(\mathbb{R}\) factor stays still – the solution degenerates after finite time to a line. This "neck pinch" is the model case of a neckpinch singularity (see Article 04).

7. What the Flow Controls

Under the Ricci flow many curvature quantities satisfy their own evolution equations, inviting maximum principle arguments:

  • \(\partial_t R = \Delta R + 2\,\lvert\mathrm{Ric}\rvert^2\) – scalar curvature satisfies a heat equation with non-negative source term; in particular \(R \ge 0\) is preserved.
  • \(\partial_t \mathrm{Ric} = \Delta_L \mathrm{Ric}\) (Lichnerowicz Laplacian) – the Ricci tensor diffuses.
  • Diameter, volume and curvature bounds propagate through explicit comparison estimates.

This is the toolbox Hamilton used in 1982 and that Perelman expanded decisively in 2002–03.

8. What the Flow Cannot Do

Three problems remained open after Hamilton and shaped the research programme of the next twenty years:

  1. Singularity formation. The flow breaks down in finite time before a smooth limit metric is reached – e.g. the neckpinch.
  2. Collapse. Volume can vanish locally, invalidating comparison theorems.
  3. Topology change. To continue past a singularity one must cut the manifold ("surgery"), change topology and restart the flow.

The tools to address these – the \(\mathcal{F}\)- and \(\mathcal{W}\)-entropy, \(\kappa\)-non-collapse, reduced length and canonical neighbourhoods – follow in Articles 04–07.

Sources

  • Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306.
  • Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys Diff. Geom. 2 (1995), 7–136.
  • Bennett Chow & Dan Knopf, The Ricci Flow: An Introduction, AMS Math. Surveys 110 (2004).
  • John W. Morgan & Gang Tian, Ricci Flow and the Poincaré Conjecture, AMS (2007), §3.

Cross References