Ricci Flow

Hamilton's flow that smooths the metric — the engine behind Perelman's proof

Ricci Flow

Ricci flow evolves a Riemannian metric by the equation ∂g/∂t = −2 Ric, deforming the geometry of a manifold proportional to its Ricci curvature. Introduced by Richard Hamilton in 1982, Ricci flow is one of the most powerful tools in modern differential geometry.

On surfaces (2D), Ricci flow simplifies to ∂g/∂t = −Rg where R is the scalar curvature. The flow smooths out curvature: bumpy regions with high curvature get flattened, while flat regions expand. Over time, Ricci flow drives a metric toward one of uniform curvature.

Interactive: Surface Ricci Flow

Watch Ricci flow smooth a 2D surface. The conformal factor u(x,y) determines the metric ds² = e²²ᵘ(dx² + dy²). High u (warm colors) represents hills of curvature; low u (cool colors) represents valleys. Under Ricci flow, the surface evolves toward uniform geometry.

Min R
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Avg R
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Max R
0.000
1.0x
step 0
Initial:
High u (hills)
Low u (valleys)
Grid lines show metric distortion

The Ricci Flow Equation

Hamilton's Ricci flow deforms the metric tensor g at a rate proportional to the Ricci curvature tensor:

∂gij/∂t = −2 Rij

Regions with positive Ricci curvature shrink (the metric contracts), while regions with negative curvature expand. This is directly analogous to the heat equation, but applied to geometry itself rather than a temperature distribution.

Ricci Flow on Surfaces

On a 2D surface with conformal metric ds² = e2u(dx² + dy²), Ricci flow reduces to the evolution of the conformal factor:

∂u/∂t = e−2u Δu

This is a nonlinear heat equation for the conformal factor — the Laplacian smooths u, but the exponential weight makes the smoothing geometry-dependent.

Normalized Ricci Flow

Standard Ricci flow can shrink surfaces to a point. The normalized version preserves total volume:

∂g/∂t = −2 Ric + (2r/n) g
  • Volume preserved: The average curvature term r = ∫R dV / ∫dV counteracts shrinking
  • Convergence: On surfaces, normalized Ricci flow always converges to constant curvature
  • Long-time behavior: The metric approaches a canonical geometry

Interactive: Curvature Evolution

Watch how the scalar curvature distribution evolves under normalized Ricci flow. Starting from a bumpy, irregular curvature profile, the flow drives R(x) toward a constant — the average value. The histogram shows curvature values concentrating into a spike.

t = 0.00
Profile:
Current R(x)
Initial R(x)
Average (target)

Ricci Flow as a Geometric Heat Equation

In harmonic coordinates, the Ricci flow equation becomes:

∂gij/∂t = Δgij + (lower order terms)

This reveals that Ricci flow is essentially a heat equation for the metric. Just as the heat equation smooths temperature distributions, Ricci flow smooths geometry. The "lower order terms" involve curvature and are what make Ricci flow geometrically interesting — and mathematically challenging.

Interactive: Ricci Flow & the Uniformization Theorem

The Uniformization Theorem states that every closed surface admits a metric of constant curvature. Ricci flow provides a constructive proof: it deforms any metric into one of three canonical geometries, determined entirely by topology (genus).

t = 0.00
Topology:
Target curvature:
Positive (R > 0)
Canonical geometry:
Round Sphere (S²)
Euler characteristic:
χ = 2

Ricci flow contracts to a round sphere with constant positive curvature.

Positive R
Negative R
Target constant curvature

The Gauss-Bonnet Constraint

Ricci flow respects a deep topological constraint. The Gauss-Bonnet theorem fixes the total curvature in terms of the Euler characteristic:

∫ R dA = 4π(1 − g)

where g is the genus (number of handles). This means the average curvature is fixed by topology. Ricci flow cannot change the total curvature — it can only redistribute it, evening out bumps and dips until the curvature is the same everywhere.

Key Takeaways

  • Ricci flow ∂g/∂t = −2 Ric evolves the Riemannian metric proportional to Ricci curvature, acting as a heat equation for geometry
  • Curvature smoothing: Ricci flow flattens bumps and fills valleys, driving curvature toward a constant value
  • Normalized Ricci flow preserves volume and always converges on surfaces to a metric of constant curvature
  • Uniformization: Ricci flow constructively proves that every surface admits one of three canonical geometries — spherical, flat, or hyperbolic
  • Gauss-Bonnet: ∫R dA = 4π(1−g) constrains total curvature by topology, so Ricci flow can only redistribute, not create or destroy curvature

Next: Ricci Flow with Surgery — how Perelman extended Hamilton's program to handle singularities and prove the Poincaré Conjecture.