Monotonicity of the Reduced Volume

12A.6 𝓛-Geodesics and the Variational Formulas proved Perelman's differential inequalities for the reduced distance at points where it is smooth: where the minimising $\mathcal L$-geodesic is unique and nondegenerate. This chapter removes that assumption and proves, with full care, the two consequences used in the rest of the proof of the Poincaré conjecture: the reduced volume is nonincreasing in $\tau$, with limit $1$ as $\tau \to 0$, and $\min\ell(\cdot, \tau) \leq \frac n2$ for every $\tau$.

The strategy is the one used for Bishop–Gromov (9B.2 Volume Comparison). There, one never differentiates the volume of a ball directly; one pulls the volume back to the tangent space by the exponential map, writes it as an integral of the Jacobian over the region inside the cut locus, and shows that the Jacobian, suitably normalised, is monotone along each geodesic and that the region shrinks. Perelman's proof of the monotonicity of $\tilde V$ does exactly this, with the $\mathcal L$-exponential map in place of the exponential map, the "injectivity domain" in place of the interior of the cut locus, and the quantity $\tau^{-n/2}e^{-\ell}J$ in place of $r^{1-n}J$. Nothing needs to be differentiated across the set where $\ell$ is not smooth, because that set is never seen: it is the image of the boundary of a region in the tangent space, and has measure zero. The bound $\min\ell \leq \frac n2$ needs a different technique, the maximum principle in the barrier sense, which is the space-time version of Calabi's trick for the Laplacian of the distance function.

By the end of this chapter you will be able to:

• define the $\mathcal L$-exponential map, the injectivity set in the tangent space and the injectivity domain, and prove that the injectivity sets shrink as $\tau$ increases;
• compute the Jacobian of $\mathcal L\exp$ on flat space and show that $\tau^{-n/2}e^{-\tilde\ell}J$ is invariant under parabolic rescaling;
• prove that $\tau^{-n/2}e^{-\tilde\ell}J \to 2^ne^{-|v|^2}$ as $\tau \to 0$, and hence $\tilde V(\tau) \to 1$;
• prove Perelman's Jacobian inequality (7.12) and the identity $\frac{d}{d\tau}\tilde\ell = -\frac12\tau^{-3/2}K$ along $\mathcal L$-geodesics, and conclude the monotonicity of $\tilde V$ and $\tilde V \leq 1$;
• explain the localised reduced volume of a subset, the version used for flows with surgery;
• define upper barriers and prove $\partial_\tau\ell + \Delta\ell \leq \frac{n/2 - \ell}{\tau}$ in the barrier sense, and deduce $\min\ell \leq \frac n2$;
• describe what is known about the regularity of $\ell$ on complete flows (local Lipschitz, smooth off a closed null set, the inequalities in the distributional sense).

7.1The $\mathcal L$-exponential map and its injectivity domain

Fix a base point $x = (p, T)$ in space-time and work in backward time $\tau = T - t$. For $v \in T_pM$, let $\gamma_v$ be the $\mathcal L$-geodesic with $\gamma_v(0) = p$ and $\lim_{\tau\to0}\sqrt\tau X(\tau) = v$ (12A.6 𝓛-Geodesics and the Variational Formulas), and define

These are smooth where defined, by smooth dependence of solutions of the (regular, in $s = \sqrt\tau$) geodesic ODE on initial data. The tildes mark functions on the tangent space, as opposed to $L(q, \tau)$ and $\ell(q, \tau)$, which are functions on space-time defined by minimisation.

The differential. The derivative of $\mathcal L\exp_\tau$ at $v$ in the direction $w$ is $Y_w(\tau)$, where $Y_w$ is the $\mathcal L$-Jacobi field along $\gamma_v$ with $Y_w(0) = 0$ and $\lim\sqrt\tau\nabla_XY_w = w$ (Morgan–Tian, Lemma 6.19). So $\mathcal L\exp_\tau$ is a local diffeomorphism at $v$ exactly when no nonzero $\mathcal L$-Jacobi field along $\gamma_v$ vanishes at $0$ and at $\tau$. Near $\tau = 0$ this always holds (their Lemma 6.23): in the variable $s$ the Jacobi equation is regular, and a Jacobi field with $Y(0) = 0$ and nonzero initial derivative is nonzero for small $s$.

The injectivity set. Following Morgan and Tian (Definition 6.25), let $\tilde U(\tau) \subseteq T_pM$ be the set of $v$ such that $\mathcal L\exp_\tau$ is a local diffeomorphism at $v$ and, for all $v'$ near $v$, the geodesic $\gamma_{v'}|_{[0,\tau]}$ is the unique minimiser of $\mathcal L$ between its endpoints. Its image $U(\tau) = \mathcal L\exp_\tau(\tilde U(\tau)) \subseteq M$ is the injectivity domain at time $\tau$. On it:

• $\mathcal L\exp_\tau : \tilde U(\tau) \to U(\tau)$ is a diffeomorphism onto an open set, and $L(\cdot, \tau)$ and $\ell(\cdot, \tau)$ are smooth on $U(\tau)$ with $L\circ\mathcal L\exp_\tau = \tilde L$ (their Proposition 6.28);
• all the computations of 12A.6 𝓛-Geodesics and the Variational Formulas are valid, in particular (7.13)–(7.15).

The proposition is the analogue of the fact that the interior of the cut locus is star-shaped: a geodesic that is minimising and nondegenerate up to time $\tau'$ is so at every earlier time. For a Ricci flow on a closed manifold, or a complete one of bounded curvature, the complement of $U(\tau)$ in $M$ is closed and of measure zero (Morgan–Tian, Proposition 7.5), as the cut locus is in Riemannian geometry; minimising $\mathcal L$-geodesics exist from $x$ to every point (their Lemma 7.2); and $L$ and $\ell$ are locally Lipschitz on space-time (their Proposition 6.56). So the reduced volume can be computed by integrating over $U(\tau)$, and pulled back to $\tilde U(\tau)$ by the change of variables formula:

where $J(v, \tau)$ is the Jacobian determinant of $\mathcal L\exp_\tau$ at $v$, computed with the metric $g(T)$ on $T_pM$ and $g(\tau)$ on $M$.

7.2Flat space and rescaling

On static flat $\mathbb{R}^n$, $\mathcal L\exp_\tau(v) = p + 2\sqrt\tau v$, so $J = 2^n\tau^{n/2}$, and $\tilde\ell(v, \tau) = \frac{|2\sqrt\tau v|^2}{4\tau} = |v|^2$. Hence

independent of $\tau$ (Morgan–Tian, Claim 6.77). Since $\int_{\mathbb{R}^n}2^ne^{-|v|^2}dv = 2^n\pi^{n/2} = (4\pi)^{n/2}$, the normalised reduced volume of flat space is exactly $1$. This explains the normalisation: the guide's $\tilde V$ includes $(4\pi)^{-n/2}$ so that the model value is $1$; Perelman and Morgan–Tian omit it, and their model value is $(4\pi)^{n/2}$ (12A.1 How to Read Perelman).

Parabolic rescaling. Let $g_Q(t) = Qg(T + (t - T)/Q)$ be the flow rescaled by $Q$ about the time $T$, so $\tau_Q = Q\tau$. Then $\mathcal L$-lengths scale as $\mathcal L_Q = \sqrt Q\,\mathcal L$, the initial vectors as $v_Q = Q^{-1/2}v$ measured in $g_Q(T) = Qg(T)$ (so $|v_Q|_{g_Q} = |v|_g$), the reduced distance is invariant, and the Jacobian satisfies $J_Q = Q^{n/2}J$ (Morgan–Tian, Lemma 6.75). Consequently

the integrand of the reduced volume, and the reduced volume itself, are scale invariant. This is what makes $\tilde V$ useful for blow-up arguments: it is unchanged by the rescalings that produce blow-up limits, as $\mathcal W$ is (12A.3 The 𝓦-Entropy).

7.3The Jacobian inequality

Fix $v \in \tilde U(\bar\tau)$ and the geodesic $\gamma = \gamma_v$. Choose a basis $\{w_\alpha\}$ of $T_pM$ whose images under $d\mathcal L\exp_{\bar\tau}$ form an orthonormal basis $\{Y_\alpha(\bar\tau)\}$ of $T_{\gamma(\bar\tau)}M$, and let $Y_\alpha(\tau)$ be the corresponding $\mathcal L$-Jacobi fields. Then $J(\tau)$ is a constant multiple of $\sqrt{\det\langle Y_\alpha(\tau), Y_\beta(\tau)\rangle}$, and at $\tau = \bar\tau$, where the Gram matrix is the identity,

By the identity for derivatives of inner products (12A.6 𝓛-Geodesics and the Variational Formulas), $\frac12\frac{d}{d\tau}|Y_\alpha|^2 = \langle\nabla_XY_\alpha, Y_\alpha\rangle + \operatorname{Ric}(Y_\alpha, Y_\alpha)$, and since $Y_\alpha$ is an $\mathcal L$-Jacobi field along a minimiser, $\langle\nabla_XY_\alpha, Y_\alpha\rangle = \frac{1}{2\sqrt{\bar\tau}}\operatorname{Hess}L(Y_\alpha, Y_\alpha)$ at $\bar\tau$. The Hessian comparison (7.9) then gives

which is Perelman's (7.11), with $\tilde Y_\alpha$ the adapted field with $\tilde Y_\alpha(\bar\tau) = Y_\alpha(\bar\tau)$. Summing over $\alpha$ with the trace identity of 12A.6 𝓛-Geodesics and the Variational Formulas:

Equality holds exactly when every adapted field is $\mathcal L$-Jacobi, which forces $2\operatorname{Ric} + \frac{1}{\sqrt{\bar\tau}}\operatorname{Hess}L = \frac1{\bar\tau}g$ at $\gamma(\bar\tau)$.

The reduced distance along a geodesic. Along $\gamma_v$, $\tilde\ell(v, \tau) = \frac{\tilde L(v, \tau)}{2\sqrt\tau}$ with $\frac{d}{d\tau}\tilde L = \sqrt\tau(R + |X|^2)$, so

using (7.4) in the form $R + |X|^2 = \frac{\tilde\ell}{\tau} - \tau^{-3/2}K$ (Morgan–Tian, Corollary 6.54; Exercise 7.8). Note the contrast with $\partial_\tau\ell$ computed on space-time in 12A.6 𝓛-Geodesics and the Variational Formulas: the derivative along a geodesic and the partial derivative at a fixed point of $M$ differ by $\langle\nabla\ell, X\rangle$.

The localised version. The proof is local in the following sense, which is how the reduced volume is used in flows with surgery. For an open set $A$ of space-time, define $\tilde V(A_\tau) = \int_{A_\tau}(4\pi\tau)^{-n/2}e^{-\ell}\,dV$ over its time-$\tau$ slice. If every minimising $\mathcal L$-geodesic from $x$ to a point of $A$ stays in $A$, then $\tilde V(A_\tau)$ is nonincreasing (Morgan–Tian, Theorem 6.80): the same pull-back argument, restricted to the preimage of $A$. In a flow with surgery, $A$ will be the set of points reachable by $\mathcal L$-geodesics that avoid the surgery regions, and the theorem lets the reduced volume be used without ever crossing a surgery (12B.9 Surgery, 12B.13 Ricci Flow with Surgery for All Time).

7.4The barrier sense and $\min\ell \leq \frac n2$

Perelman's argument for $\min\ell \leq \frac n2$ is one line: from (7.15), $\bar L = 2\sqrt\tau L$ satisfies $\partial_\tau\bar L + \Delta\bar L \leq 2n$, so at a minimum point, where $\Delta\bar L \geq 0$, $\partial_\tau\bar L \leq 2n$, and $\min\bar L - 2n\tau$ is nonincreasing; since $\bar L \to 0$ at $p$ as $\tau \to 0$, $\min\bar L \leq 2n\tau$, which is $\min\ell \leq \frac n2$. The difficulty is that the minimum point may be where $\ell$ is not smooth. The remedy is the device used by Calabi (1958) for the Laplacian of the distance function.

Upper barriers. A smooth function $\varphi$ defined near a point $y$ is an upper barrier for a continuous function $h$ at $y$ if $\varphi(y) = h(y)$ and $\varphi \geq h$ nearby. A differential inequality such as $\partial_\tau h + \Delta h \leq F$ holds in the barrier sense at $y$ if for every $\varepsilon > 0$ there is an upper barrier $\varphi$ at $y$ with $\partial_\tau\varphi + \Delta\varphi \leq F + \varepsilon$ there. The point is that at a minimum of $h$ in space, an upper barrier is also minimised there in space (it lies above $h$ and touches it), so $\Delta\varphi \geq 0$, and the inequality gives information about $\partial_\tau\varphi$, hence about the one-sided time derivative of $h$.

There is also a lower bound, much easier. In backward time, $R_\tau = -\Delta R - 2|\operatorname{Ric}|^2 \leq -\Delta R - \frac2nR^2$, and if the flow exists on $[0, \tau_0]$ the maximum principle (11A.7 Maximum Principles under Ricci Flow) gives $R(\cdot, \tau) \geq -\frac{n}{2(\tau_0 - \tau)}$, the forward statement $R \geq -\frac{n}{2t}$ for a flow that started at time $0$. Then $L \geq \int_0^\tau\sqrt\sigma R \geq -\frac{n}{2(\tau_0 - \tau)}\cdot\frac23\tau^{3/2}$ is bounded below for $\tau$ away from $\tau_0$, and so is $\ell$.

The other inequalities. Morgan and Tian also prove (their Theorem 7.13, adapting an argument of Ye) that (7.13) and (7.14) hold in the sense of distributions on all of $M$, and that if equality holds in either, then $\ell$ is smooth and the flow is a gradient shrinking soliton. This gives a second proof of the monotonicity of $\tilde V$, by integrating (7.13) against the volume form, which is the one Perelman mentions in parentheses after (7.12).

7.5History

Perelman stated the monotonicity of the reduced volume and $\min\ell \leq \frac n2$ in §7.1 of his first preprint (2002), with the remark that inequalities at non-smooth points hold "in the barrier sense, or in the sense of distributions". Rugang Ye's notes (2004) gave a careful treatment of the regularity of $\ell$ and the distributional inequalities, which Morgan and Tian follow in their chapter 7. Calabi's barrier argument dates from 1958, and Crandall and Lions's viscosity solutions from 1983. The analogy with Bishop–Gromov, whose classical proof integrates the Jacobian of the exponential map over the interior of the cut locus, goes back to Bishop (1963) and Gromov (1980).

7.6Exercises

Solution

(a) The map is linear with matrix $2\sqrt\tau I$, determinant $2^n\tau^{n/2}$. The $\mathcal L$-length of $\gamma_v|_{[0,\tau]}$ is $\int_0^\tau\sqrt\sigma|v\sigma^{-1/2}|^2d\sigma = 2\sqrt\tau|v|^2$, so $\tilde\ell = |v|^2$. (b) The integrand is $(4\pi\tau)^{-n/2}e^{-|v|^2}2^n\tau^{n/2} = \pi^{-n/2}e^{-|v|^2}$, whose integral is $1$. (c) $dq = 2^n\tau^{n/2}dv$ and $|v|^2 = \frac{|q - p|^2}{4\tau}$.

Solution

$\frac{d}{d\tau}\tilde\ell = \frac{\sqrt\tau(R + |X|^2)}{2\sqrt\tau} - \frac{\tilde L}{4\tau^{3/2}} = \frac12(R + |X|^2) - \frac{\tilde\ell}{2\tau}$. By (7.4), $\tau^{3/2}(R + |X|^2) = \frac12\tilde L - K = \sqrt\tau\tilde\ell - K$, so $R + |X|^2 = \frac{\tilde\ell}{\tau} - \tau^{-3/2}K$, and $\frac{d}{d\tau}\tilde\ell = -\frac12\tau^{-3/2}K$. On flat space $K = 0$ and $\tilde\ell = |v|^2$ is constant.

Solution

(a) The $R$ and $K$ terms cancel: $\partial_\tau\ell + \Delta\ell \leq -\frac\ell\tau + \frac{n}{2\tau}$. (b) Adding gives $\partial_\tau\ell + \Delta\ell + \frac{n}{2\tau} + \frac{\ell - n}{\tau} = -\delta$, so $\partial_\tau\ell + \Delta\ell = \frac{n/2 - \ell}{\tau} - \delta \leq \frac{n/2 - \ell}{\tau}$. (Adding the inequalities (7.13) and (7.14) themselves would not work, since they point in opposite directions; the defect $\delta$ is what links them.) (c) $\partial_\tau\bar L + \Delta\bar L = 4\ell + 4\tau(\partial_\tau\ell + \Delta\ell) \leq 4\ell + 4(\frac n2 - \ell) = 2n$, and conversely.

Solution

Suppose $m(\tau_1) > c$ for some $\tau_1$. Since $m \to 0 < c$ as $\tau \to 0$, by continuity there is $\tau_0 < \tau_1$ with $m(\tau_0) = c$ and $m > c$ on $(\tau_0, \tau_1]$. On that interval $D^+m < 0$, and a continuous function with negative upper right Dini derivative everywhere on an interval is nonincreasing there (a standard lemma of real analysis); so $m(\tau_1) \leq m(\tau_0) = c$, a contradiction.

Solution

The proof pulls $A_{\tau'}$ back to a set $\tilde A \subseteq \tilde U(\tau')$ and uses $\int_{\tilde A}f(v, \tau') \leq \int_{\tilde A}f(v, \tau) = \tilde V(\mathcal L\exp_\tau(\tilde A))$, then needs $\mathcal L\exp_\tau(\tilde A) \subseteq A_\tau$ to conclude $\leq \tilde V(A_\tau)$. If some geodesic $\gamma_v$, $v \in \tilde A$, is outside $A$ at time $\tau$, its contribution at time $\tau$ is not counted in $\tilde V(A_\tau)$, and the last inclusion fails. In a flow with surgery, geodesics that pass through a surgery region are exactly such paths, which is why Morgan and Tian must show that the relevant minimisers avoid surgery caps (12B.13 Ricci Flow with Surgery for All Time).

Solution

With $\tau_Q = Q\tau$: $R_Q = Q^{-1}R$, $|\dot\gamma_Q|^2_{g_Q} = Q\cdot Q^{-2}|\dot\gamma|^2_g = Q^{-1}|\dot\gamma|^2$ (the metric gains $Q$, the velocity loses $Q$), $\sqrt{\tau_Q} = \sqrt Q\sqrt\tau$ and $d\tau_Q = Q\,d\tau$. So $\mathcal L_Q = \int\sqrt Q\sqrt\tau\,Q^{-1}(R + |\dot\gamma|^2)Q\,d\tau = \sqrt Q\,\mathcal L$. Then $\ell_Q = \frac{\sqrt QL}{2\sqrt{Q\tau}} = \ell$. And $(4\pi\tau_Q)^{-n/2}dV_{g_Q} = Q^{-n/2}(4\pi\tau)^{-n/2}Q^{n/2}dV_g$, so $\tilde V_Q(Q\tau) = \tilde V(\tau)$.