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Worked solution: Perelman's Ricci flow with surgery (2002–2003), condensed

Step 3 of 9: Ricci flow as a gradient-like flow: the F-functional
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In plain words

A ball rolling downhill only ever loses height — it can never climb back up to exactly where it started, unless it is already sitting at the bottom. In 2002 Perelman went looking for a hidden "landscape" that Ricci flow itself is always rolling down.

He showed that, once one corrects for how points are relabeled (a diffeomorphism), Ricci flow always moves in the direction that increases a functional called F. Because F can only go up, the flow can never repeat its own history exactly — the first clue that a singularity, not an endless cycle, is the flow's natural way of running out of room.

F(g,f)=∫M(R+∣∇f∣2)e−f dV,dFdt=2∫M∣Rij+∇i∇jf∣2e−f dV≥0\mathcal{F}(g,f) = \int_M (R+|\nabla f|^2)e^{-f}\,dV, \qquad \frac{d\mathcal{F}}{dt} = 2\int_M \big|R_{ij}+\nabla_i\nabla_j f\big|^2 e^{-f}\,dV \ge 0
Detailed analysis

Perelman's first breakthrough (arXiv:math/0211159, §1) pairs the Ricci flow ∂tgij=−2Rij\partial_t g_{ij} = -2R_{ij} with an auxiliary function ff evolving by a backward heat-type equation, and studies the functional F(g,f)=∫M(R+∣∇f∣2)e−f dV\mathcal{F}(g,f) = \int_M (R+|\nabla f|^2)e^{-f}\,dV.

After absorbing a diffeomorphism into the flow of ff (§1.2-1.3), the coupled system becomes exactly the L2L^2-gradient flow of F\mathcal{F}, and a direct computation gives dFdt=2∫M∣Rij+∇i∇jf∣2e−f dV≥0\frac{d\mathcal{F}}{dt} = 2\int_M |R_{ij}+\nabla_i\nabla_j f|^2 e^{-f}\,dV \ge 0, with equality exactly at steady or soliton solutions.

A functional that strictly increases along the flow (except at solitons) cannot return to its starting value, so Ricci flow admits no nontrivial periodic solutions ("breathers"). This gradient-flow structure is the germ that Perelman refines, in the next step, into a scale-invariant entropy strong enough to control singularities directly.

Terms in this step
Gradient flow
A flow that moves in the direction that most increases (or decreases) some functional, so the value of that functional changes monotonically along the flow.
Breather / gradient Ricci soliton
A "breather" is a nontrivial solution to Ricci flow that repeats its geometry, up to scaling and diffeomorphism, periodically in time; a gradient Ricci soliton is the borderline case where the metric evolves purely by scaling and diffeomorphism, with no genuinely new geometry appearing.
Knowledge used in this step
Riemannian geometry