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

Step 4 of 9: Perelman's W-entropy and monotonicity formula

In plain words

The F-functional from the previous step works well but changes in an uncontrolled way if the manifold is simply stretched or shrunk — it is not scale-invariant. Borrowing language from statistical mechanics, Perelman built a scale-invariant "entropy" W that still only ever increases.

Because W never decreases and is scale-invariant, taking its smallest possible value over all choices of the auxiliary function — called mu — gives a single number attached to the geometry at each time that also never decreases. This is the tool that will forbid the manifold from secretly thinning out into a lower-dimensional shape.

\mathcal{W}(g_{ij},f,\tau)=\int_M\Big[\tau\big(|\nabla f|^2+R\big)+f-n\Big](4\pi\tau)^{-n/2}e^{-f}\,dV,\quad \frac{d\mathcal{W}}{dt}=\int_M 2\tau\Big|R_{ij}+\nabla_i\nabla_j f-\frac{1}{2\tau}g_{ij}\Big|^2(4\pi\tau)^{-n/2}e^{-f}\,dV\ge 0

Detailed analysis

Coupling $\partial_t g_{ij}=-2R_{ij}$ with the backward conjugate heat equation $\partial_t f=-\Delta f+|\nabla f|^2-R+n/(2\tau),\ \tau_t=-1$ (keeping $\int_M(4\pi\tau)^{-n/2}e^{-f}dV=1$ ), Perelman defines the scale-invariant W-entropy (§3, arXiv:math/0211159) and proves $d\mathcal{W}/dt\ge 0$ in every dimension, with equality only for gradient shrinking Ricci solitons.

Taking $\mu(g,\tau) = \inf_f \mathcal{W}(g,f,\tau)$ over all valid $f$ produces a real number for each metric and each scale $\tau$ that is monotone nondecreasing along the flow (§4), independent of any particular choice of the auxiliary function.

This single monotone quantity is exactly the invariant used in the next step to rule out 'local collapsing' — regions where the manifold looks like it is degenerating to a lower dimension even though curvature stays bounded.

Terms in this step

Entropy functional ( $\mathcal{W}$ ): A scale-invariant quantity built from the metric, an auxiliary function $f$ , and a scale parameter $\tau$ , modeled on entropy in statistical mechanics, that is monotone nondecreasing along Ricci flow.
$\mu$ -invariant: The infimum of the $\mathcal{W}$ -entropy over all admissible auxiliary functions $f$ at a fixed scale $\tau$ ; a single real number attached to $(g,\tau)$ that inherits the monotonicity of $\mathcal{W}$ .

Knowledge used in this step

Riemannian geometry