Reduced Length and Reduced Volume

The L-Length Functional

The second major ingredient in Perelman's work is a space-time distance function adapted to the Ricci flow. Fix a basepoint $(p, t_0)$ and consider curves $\gamma : [0, \bar{\tau}] \to M$parametrised by backward time $\tau = t_0 - t$. The L-length is:

$$\mathcal{L}(\gamma) = \int_0^{\bar{\tau}} \sqrt{\tau}\,\bigl(R(\gamma(\tau), t_0 - \tau) + |\dot{\gamma}(\tau)|^2_{g(t_0 - \tau)}\bigr)\,d\tau$$

The $\sqrt{\tau}$ weight is chosen so that $\mathcal{L}$-geodesics in flat space reduce to straight lines at constant speed, and the reduced distance (defined below) equals the squared Euclidean distance divided by $4\tau$. The inclusion of scalar curvature $R$ couples the length to the evolving geometry.

Reduced Distance

The reduced distance (or $\ell$-distance) from the basepoint $(p, t_0)$ to a point $(x, t_0 - \tau)$ is defined by minimising the L-length over all paths:

$$\ell(x, \tau) = \frac{1}{2\sqrt{\tau}}\,\inf_\gamma \mathcal{L}(\gamma)$$

where the infimum is taken over all curves from $p$ at $\tau = 0$to $x$ at backward time $\tau$. The normalisation by $1/(2\sqrt{\tau})$ ensures that $\ell$ is dimensionless and equals $|x - p|^2/(4\tau)$ in flat space.

Reduced Volume

The reduced volume integrates $e^{-\ell}$ against a Gaussian-type measure:

$$\widetilde{V}(\tau) = \int_M (4\pi\tau)^{-n/2}\,e^{-\ell(x,\tau)}\,d\mu_{g(t_0 - \tau)}(x)$$

Theorem (Perelman, 2002)

The reduced volume $\widetilde{V}(\tau)$ is non-increasing in $\tau$. Moreover, $\widetilde{V}(\tau) \leq 1$ for all $\tau > 0$, with equality if and only if $(M, g(t))$ is flat Euclidean space.

The monotonicity of $\widetilde{V}$ is proved by computing the first variation using the differential inequalities for $\ell$ and applying integration by parts with the L-exponential map. This result provides an alternative route to non-collapsing and is essential for the analysis of ancient solutions.

Key Differential Inequalities for &ell;

The reduced distance satisfies two fundamental differential inequalities that encode all the monotonicity information. At points where $\ell$ is smooth:

$$\frac{\partial \ell}{\partial \tau} \;\leq\; \frac{n}{2\tau} - \frac{\ell}{\tau} + R$$

$$\Delta \ell \;\leq\; \frac{n}{2\tau} - R + \frac{\ell}{2\tau}$$

In the barrier sense (since $\ell$ may not be smooth everywhere due to the cut locus of the L-exponential map), these hold as distributional inequalities. Combining them gives:

$$\frac{\partial \ell}{\partial \tau} - \Delta \ell + |\nabla \ell|^2 + R - \frac{n}{2\tau} \;\leq\; 0$$

This combined inequality is the engine behind the monotonicity of $\widetilde{V}$: when integrated against $(4\pi\tau)^{-n/2} e^{-\ell}$, it yields $d\widetilde{V}/d\tau \leq 0$.

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