ENTROPY AND REDUCED DISTANCE FOR RICCI EXPANDERS

Michael Feldman^{1}, Tom Ilmanen^{2}and Lei Ni^{3}

May 3, 2004

Abstract. Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂g_{ij}/∂t = −2R_{ij} and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W_{+} is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ_{+} and ν_{+}. The forward reduced volume θ_{+} is monotone in general and constant exactly on expanders.

A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/t^{n/2} (Hamilton) and λ̄ (Perelman), as well as our new quantities. In general, we show that if vol(g) grows like t^{n/2} (maximal volume growth) then W_{+}, θ_{+} and λ̄ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture.

Small, large and distant parts of a Ricci flow are known to be modelled by various kinds of Ricci solitons: steady, shrinking, and expanding. Perelman has discovered monotone quantities correponding to the first two of these. The functional F is monotone on the Ricci flow and constant precisely on steadies [P1, §1]. The shrinking, or localizing, entropy W is monotone in general and constant precisely on shrinking solitons [P1, §3]. Closely related is the notion of the (backward) reduced volume [P1, §7]. The latter two show that developing singularities and ancient histories are modelled on shrinkers, when an appropriate blowup or blowdown is taken [P1, S].

In this note, by tweaking some signs, we observe similar monotonicity formulae for the expanding case. In §1, analogous to [P1, §1], we define an expanding, or delocalizing entropy W_{+}, which is monotone in general and constant precisely on expanders. It follows from a

1Department of Mathematics, University of Wisconsin, Madison WI 53706. Research partially supported by NSF grant DMS-0200644.

2Departement Mathematik, ETH Zentrum, 8092 Zürich, Switzerland. Research partially supported by Schweizerische Nationalfonds grant 21-66743.01.

3Department of Mathematics, University of California, San Diego, La Jolla CA 92093. Research partially supported by NSF grant DMS-0328624 and an Alfred P. Sloan Fellowship.

differential inequality for the conjugate heat equation similar to [P1, §9]; see also [LY]. We also construct functionals μ_{+}(g, σ) and ν_{+}(g) from W_{+}.

In §2, analogous to [P1, §7], we present a forward reduced distance ℓ_{+} and its corresponding forward reduced volume, which is again monotone in general, and constant exactly on expanders.

Our formulae may be compared to those for vol(g)/t^{n/2} (Hamilton) and of λ̄(g) := vol(g)^{2/n}λ(g) (Perelman). All of these monotonicities have implications for the convergence of the rescaled Ricci flow as t → ∞. If g(t) is a compact Ricci flow that exists for all t > 0, and the volume vol(g) grows like t^{n/2} (maximal volume growth), we observe in Theorem 1.4 that W_{+} remains bounded. A similar result holds for Perelman’s scaled λ-functional, and for the forward reduced volume (Proposition 2.3). This suggests that (ignoring zero-volume collapsing parts) the long-term limit of g(t)/t is a negative Einstein manifold, possibly with singularities; see Conjecture 1.6.

Our results may be set against recently discovered entropy formulae and pointwise differential inequalities in other settings: for the standard heat equation on a manifold of nonnegative Ricci curvature [N1], and for the forward conjugate heat equation u_{t} = Δu + Ru on a 2-dimensional Ricci flow [N2] and (matrix Harnack!) on a Kähler-Ricci flow [N3].^{4}

It is also instructive to compare the Ricci flow to the mean curvature flow in Euclidean space. The monotonicity formulae for F, W, W_{+}, and for the corresponding reduced volumes, are analogous to the following three monotone energies on a mean curvature flow M_{t} in ℝ^{N}:

\frac{d}{dt} ∫_{M_{t}} \frac{e^{−|x−x_{0}|^{2}/4(t_{0}−t)}}{(−4πt)^{n/2}} ≤ 0, t < 0,

\frac{d}{dt} ∫_{M_{t}} e^{⟨x,x_{0}⟩−|x_{0}|^{2}t} ≤ 0,  t ∈ ℝ,  \frac{d}{dt} ∫_{M_{t}} \frac{e^{|x−x_{0}|^{2}/4(t−t_{0})}}{(4πt)^{n/2}} ≤ 0,  t > 0.

The first is Huisken’s well-known result [Hu] and is constant precisely on shrinkers. The second picks out steadies (translators) [I1], whereas the third is stationary on expanders [I4].

In the noncompact case, for both flows, the energy of a steady or expander is typically infinite, so in this case the monotonicity formulae will not be useful in proving convergence to an asymptotic soliton.^{5} This is unfortunate, because noncompact expanders are needed for modelling the relaxation out of a previously formed singularity of conic type; see for example [ACI, I3, FIK] and remarks after Proposition 2.3.

In the compact case, a sharp disanalogy between the two flows arises. Every compact mean curvature flow disappears in finite time, so mean curvature flow has no compact steadies or expanders. On the other hand, in Ricci flow, negative curvature allows a compact solution to last forever, even grow. So compact steadies and expanders exist, but on sufferance: they are always Einstein [H3, §2; CK Prop. 5.20].

4It is conjugate to the backward heat equation u_{t} = −Δu.

5There are graphs over ℝ^{n} (due to Ecker) that evolve by mean curvature flow forever, but are asymptotic neither to expanders nor translators. There also appear to be noncompact curves in ℝ^{2} whose mean curvature flow exhibits nontrivial periodicity, i.e. breathers [BI, I4].

On the other hand, shrinkers typically have finite energy whether compact or noncompact. Furthermore, the soliton potential need not be constant. (So a Ricci shrinker g(t) need not be Einstein, and a mean curvature shrinker M_{t} need not be a minimal submanifold of a sphere.) There is accordingly a lively variational calculus for them (see [Hu, I2, I4, W, CHI] and others).

§1 Expander entropy formula.

Let (M, g(t)) be a solution to the Ricci flow

(1.1)  \frac{∂g_{ij}}{∂t} = −2R_{ij}.

We call g(t) a (gradient) expander if

g(t) = a(t)ϕ_{t}^{∗}(g_{0}),

where ϕ_{t} is a family of diffeomorphisms whose velocity is given by a gradient vector field, and a(t) is any increasing function. Then a(t) = t − T for some T and g solves the PDE

Rc + ∇^{2}F + \frac{g}{2(t − T)} = 0, t > T,

on each time slice, for some function F = F (·, t).

Now let M be closed, g a metric, u > 0 a function, and σ > 0. Dual to the shrinker entropy W [P1, Section 3], (which we call W_{−}) define the expander entropy by

\begin{aligned}W_{+}(g, u, σ) & := ∫_{M} [σ (\frac{|∇u|^{2}}{u} + Ru) + u log u] dv + \frac{n}{2} log(4πσ) + n, \\ & = ∫_{M} [σ(|∇f_{+}|^{2} + R) − f_{+} + n] u dv \end{aligned} (1.2)

restricted to u satisfying

∫_{M} u dv = 1.

Here f_{+} = f_{+}^{u,σ} is defined via

u = \frac{e^{−f_{+}}}{(4πσ)^{n/2}}

Now take u = u(x, t) to be a positive solution to the conjugate heat equation

(1.3)  \frac{∂u}{∂t} = −Δu + Ru,

satisfying the conserved condition ∫ u dv = 1. Note that f_{+} = f_{+}^{u,t−T} solves

\frac{∂f_{+}}{∂t} = −Δf_{+} + |∇f_{+}|^{2} − R − \frac{n}{2(t − T)}.

The following monotonicity formula for W_{+} was found by varying the signs in the shrinker entropy formula [P1, §3]. A more elegant derivation is given a few pages hence.

Theorem 1.1 (Expander entropy monotonicity). Under evolution equations (1.1), (1.3) we have

(1.4)  \frac{∂}{∂t}W_{+}(g(t), u(t), t − T) = ∫_{M} 2(t − T)u |R_{ij} + ∇_{i}∇_{j}f_{+} + \frac{g_{ij}}{2(t − T)}|^{2} . dv,

Note that the right hand side vanishes precisely on an expander with birth time T.

Formula (1.4) is a consequence of the following differential inequality, which is similar to the one in [P1, §9] (see also [LY]), but with sign changes.

Theorem 1.2. Let (M, g(t)), u, f_{+}, and T be as above. Define

v_{+} := [(t − T) (2Δf_{+} − |∇f_{+}|^{2} + R) − f_{+} + n] u, t > T.

Then

(1.5)  (\frac{∂}{∂t} + Δ − R) v_{+} = 2(t − T)u |R_{ij} + ∇_{i}∇_{j}f_{+} + \frac{g_{ij}}{2(t − T)}|^{2} ,  t > T.

Proof. We may assume T = 0. Set V := 2Δf_{+} − |∇f_{+}|^{2} + R and compute

(1.6)  (\frac{∂}{∂t} + Δ) V = 2|R_{ij} + ∇_{i}∇_{j}f_{+}|^{2} + 2⟨∇V, ∇f_{+}⟩.

and

(\frac{∂}{∂t} + Δ) (tV − f_{+} + n) = 2t|R_{ij} + ∇_{i}∇_{j}f_{+}|^{2} + 2⟨∇(tV − f_{+} + n), ∇f_{+}⟩ + 2Δf_{+} + 2R + \frac{n}{2t}

and do further calculation. □

Proof of Theorem 1.1. Note that ∫ v_{+} dv = W_{+}. To get the time derivative of W_{+}, integrate (1.5) over M. □

Remark. For the steady case, one can define e^{−f} = u, v_{0} := V u = (2Δf − |∇f|^{2} + R)u and verify

(\frac{∂}{∂t} + Δ − R) v_{0} = 2u |R_{ij} + ∇_{i}∇_{j}f|^{2} ,

which may be viewed as a “neutral” differential Harnack inequality. It may be integrated to get the monotonicity of F in [P1, §1]. See [CCLN]. □

We next consider the implications for long term behavior of the Ricci flow. Let M be compact and consider a Ricci flow g(t) that exists for all t > b. We construct a global solution of the conjugate heat equation. Let t^{i} → ∞, and let u^{i}, b < t ≤ t^{i} be the solution of (1.3) with final data

u^{i}(x, t^{i}) = \frac{1}{V (t^{i})}, x ∈ M.

Passing t^{i} → ∞ we obtain an immortal solution u > 0 of (1.3) with ∫_{M} u = 1 for all t > b. Now let α_{j} → ∞ and define the blowdown sequence

g_{j}(t) := α_{j}^{−1}g(α_{j}t), t > b/α_{j}.

The following theorem is classic. We give a new proof using W_{+}. The hypothesis will hold (subsequentially) if, for example, |Rm|, diameter, and injectivity radius of g(t)/t are controlled for all time [H2]. A similar theorem has been proven for the shrinking case [S].

Theorem 1.3 (Hamilton [H3]). Let M be compact. Suppose there are diffeomorphisms ϕ_{j} such that ϕ_{j}^{∗}(g_{j}(t)) → g_{∞}(t) smoothly, t > 0. Then g_{∞}(t) is an expanding soliton (hence negative Einstein).

Proof. Let u_{j}(t) := α_{j}^{−n/2}u(α_{j}t). Evidently we can extract a smoothly convergent subsequence ϕ_{j}^{∗}(u_{j}) → u_{∞} > 0 satisfying (1.3) for all time. The convergence is smooth, the manifold is compact, and W_{+} is monotone and scale invariant. So W_{+} is constant on the limit flow, which is therefore an expanding soliton. □

Remark. Note that an expanding breather can be expressed as a blowdown of itself, after extending it iteratively to a maximal interval of existence t > b. Applying the Theorem then yields an alternative proof that there are no nontrivial compact expanding breathers [P1, §2]. □

Theorem 1.3 may be proven using any monotone, scale invariant quantity that is constant only on expanders. Hamilton’s original method [H3] uses the quantity R_{min}(t) V (t)^{2/n}. Alternately (Hamilton) one can use the scaled volume

Ṽ (t) := \frac{V (t)}{t^{n/2}}.

Here V (t) is the volume of g(t) and Ṽ (t) is the volume of the rescaled metric g̃(t) := g(t)/t. Perelman’s proof [P1, §2] uses the scaled λ-functional

λ̄(g) := V (g)^{2/n}λ(g),

where λ(g) := inf{F(g, u)| ∫ u dv = 1}. One can also use the forward reduced volume (see §2). In each case, the right-hand side of the monotonicity formula vanishes identically only on expanders.

We now drop the hypothesis that the blowdowns converge smoothly to a compact limit. Can we still get convergence to expanders? The various monotonicity formulae will only be useful if the monotone quantity remains bounded for all time. Accordingly, we embark on a study of the long-term behavior of W_{+}, Ṽ, and λ̄.

We begin with the scaled volume. Recall from Hamilton [H3]

\frac{∂R}{∂t} = ΔR + 2|Rc|^{2},

from which follows via R^{2} ≤ n|Rc|^{2} and the maximum principle that

(1.7)  R + \frac{n}{2t} ≥ 0,  \frac{dṼ}{dt} = −\frac{1}{t^{n/2}} ∫ (R + \frac{n}{2t}) dv ≤ 0.

So Ṽ (t) is nonincreasing, and Ṽ_{∞} := lim_{t→∞} Ṽ (t) exists.^{6}

6The proof of Theorem 1.3 (Hamilton) then runs as follows: assume g_{j}(t) converges smoothly to a compact limit g_{∞}(t). Then Ṽ (g(t)) → Ṽ (g_{∞}(t)), which is therefore constant, so R_{∞} ≡ −n/2t, so Rc_{∞} is diagonal and equals −g_{∞}/2t. This shows that any such blowdown is negative Einstein. In particular, any compact expander is negative Einstein.

Next we motivate W_{+} as mentioned above. Analogously to [P1, §5] (see also [N2]), define

N (g, u) := ∫_{M} u log u dv, N_{+}(g, u, σ) := N (g, u) + \frac{n}{2} log(4πσ) + \frac{n}{2}.

The quantity N is the Nash entropy [Na]^{7}. Note that N_{+} is dual to the entropy used for shrinkers in [P1, §5]. Now let g(t) and u(t) evolve as above for all t > 0 and set σ(t) := t. Compute

\frac{dN}{dt} = F := ∫_{M} (\frac{|∇u|^{2}}{u} + Ru) dv, \frac{dN_{+}}{dt} = F_{+} := F + \frac{n}{2t},

and

\frac{∂}{∂t}(tN_{+}) = N_{+} + tF_{+}

= ∫ u [log u + \frac{n}{2} log(4πt) + \frac{n}{2} + \frac{n}{2} + t (\frac{|∇u|^{2}}{u} + Ru)] dv

= W_{+},

which motivates W_{+}.

Next we recall [P1, §2]

\frac{d}{dt}F ≥ \frac{2}{n}F^{2}.

By ODE comparison, if g(t) exists for all t > 0 we have

−\frac{n}{2t} ≤ F(g, u) ≤ 0, t > 0.

(Side remark: passing the starting time to −∞, we notice that a compact eternal solution has F ≡ 0, hence is a steady soliton.)

Next we study N, W_{+}, and λ̄ as t → ∞. By Jensen’s inequality, we have in general

∫ u log u dv ≥ log (\frac{1}{V} )

when ∫ u dv = 1, with equality precisely when u = 1/V. So we have the lower bound

N_{+}(g, u, t) ≥ − log(V (t)) + \frac{n}{2} log(4πt) + \frac{n}{2}.

To get a similar upper bound, we use monotonicity and the construction of u as asymptotically constant. From above we have

\frac{d}{dt}N_{+}(g, u, t) = F_{+}(g, u, t) = F(g, u) + \frac{n}{2t} ≥ 0,

7Our sign convention for N is the opposite of Nash’s.

so N_{+} is monotone nondecreasing. This also applies also to u^{i}. So fixing t, we have for all large t^{i},

N_{+}(g, u^{i}, t) ≤ N_{+}(g, u^{i}, t^{i}) = log (\frac{1}{V (t^{i})}) + \frac{n}{2} log(4πt^{i}) + \frac{n}{2}

Taking i → ∞, combining with the lower bound, and sending t → ∞, we obtain

lim_{t→∞} N_{+}(g, u, t) = − log(Ṽ_{∞}) + \frac{n}{2} (log 4π + 1) .

(This holds even if Ṽ_{∞} = 0.) If Ṽ_{∞} > 0, then since N_{+} is monotone, there exist t_{j} → ∞ such that

lim_{j→∞} t_{j}F_{+}(g, u, t_{j}) = 0.

Since W_{+} = tF_{+} + N_{+}, and since W_{+} is monotone, we obtain

lim_{t→∞} W_{+}(g, u, t) = lim_{t→∞} N_{+}(g, u, t).

(Again, it holds even if Ṽ_{∞} = 0.)

Next we turn to λ̄. It is straightforward to show that the minimizer in the definition of λ depends smoothly on g, so λ(g(t)) is smooth, and

\frac{dλ}{dt} ≥ \frac{2}{n}λ^{2}, −\frac{n}{2} ≤ tλ ≤ 0,

since g(t) exists for all t > 0. From t_{j}F_{+} → 0 and the lower bound for λ we get

lim_{t_{j}→∞} t_{j}λ(g) = lim_{t_{j}→∞} t_{j}F(g, u) = −\frac{n}{2}.

Using λ ≤ ∫ R dv/V, (1.7), and the fact that λ ≤ 0, we obtain Perelman’s monotonicity [P1, §2]

\frac{d}{dt}λ̄(g(t)) ≥ 0.

Since t^{−1}V (t)^{2/n} → Ṽ_{∞}^{2/n} we obtain

lim_{t→∞} λ̄(g) = −\frac{n}{2}Ṽ_{∞}^{n/2}.

Our results are summarized in the following Proposition, which by inspecting the above argument holds even if Ṽ_{∞} = 0.

Proposition 1.4. Let M be compact, g(t) a Ricci flow for all t ≥ 0, and u(t) the solution of the conjugate heat equation constructed above. Then

lim_{t→∞} W_{+}(g(t), u(t), t) = − log Ṽ_{∞} + \frac{n}{2}(1 + log 4π),

and

lim_{t→∞} λ̄(g(t)) = −\frac{n}{2}Ṽ_{∞}^{2/n}.

Remark. The upper bound on W_{+} can be proven without using N_{+}. If Ṽ_{∞} > 0 then from the monotonicity of t^{−n/2}V (t) we get the estimate

0 ≤ ∫_{0}^{∞} t^{−n/2} ∫_{M} (R + \frac{n}{2t}) dv dt < ∞.

So there exist t^{i} → ∞ so that

(t^{i})^{−n/2} ∫ t^{i}R dv → −\frac{n}{2}Ṽ_{∞}.

Employ this particular sequence t^{i} in the construction of u (it should not affect the value of u, but we have not proven this) yields for each fixed t and large t^{i},

W_{+}(g(t), u^{i}(t), t) ≤ W_{+}(g(t^{i}), u^{i}(t^{i}), t^{i})

= \frac{1}{V (t^{i})} ∫ t^{i}R dv − log V (t^{i}) + \frac{n}{2} log(4πt^{i}) + n

→ − log Ṽ_{∞} + \frac{n}{2}(1 + log(4π)), t^{i} → ∞.

The result follows by passing u^{i} → u. □

As a corollary to Proposition 1.4, we obtain the following estimate.

Proposition 1.5. Let M be compact and let g(t), u(t) solve (1.1), (1.3) for all t > 0. Assume Ṽ_{∞} > 0. Then the rescaled solution g̃(t̃) := g(t)/t, ũ(t̃) := t^{n/2}u, f̃_{+}(t̃) := f_{+}(t), t̃= log t satisfies

(1.8)  ∫_{0}^{∞} ∫_{M} ũ |R̃c + ˜∇^{2}f̃_{+} + \frac{g̃}{2}|^{2} dṽ dt̃< ∞.

This estimate may provide enough control to eventually deduce convergence of a blowdown sequence in a suitable weak sense. We formulate the following natural conjecture as sharply as possible.

Conjecure 1.6 (Long-term behavior). Let g(t) be a Ricci flow that exists for all t > 0 on a compact manifold M. Then there is an open set U ⊂ M, times t_{j} → ∞, and a metric g̃_{∞} such that

g(t_{j})/t_{j} → g̃_{∞} in U,

where g̃_{∞} is smooth and negative Einstein, the metric completion of g̃_{∞} has a singular set of dimension at most n − 4, and

vol(g̃_{∞}) = Ṽ_{∞}.

The metric completion of g̃_{∞} need not be compact or connected, but has finite volume, so it should be negative Einstein and not merely an expander. The set U need not be dense in M. If Ṽ_{∞} = 0, then take U to be empty and the statement holds vacuously; more generally, the assertion is that the scaled volume of the collapsing parts tends to zero. The rescaling is designed not to see these parts: they recede to infinity or hide beyond the singular set.

The conjecture is inspired by the convergence picture in the 3-dimensional case [H3, P2]; a good exercise would be to employ the W_{+} monotonicity as an alternative way of recognizing the hyperbolic pieces in that case. The codimension of the singular set is motivated by the fact that the right-hand side ∫ u|Rm + ⋯ |^{2} is unitless for n = 4, and is supported by familiar convergence results for Einstein 4-manifolds ([A, CC, CT] and many others), as well as analogous results for mean curvature flow in the critical case [I2] and other flows.

Formulation of μ_{+} and ν_{+} functionals. Define, analogously to [P1, §3],

μ_{+}(g, σ) := inf_{u} W_{+}(g, u, σ), ν_{+}(g) := sup_{σ>0} μ_{+}(g, σ),

where u varies over functions satisfying ∫_{M} u dv = 1. We investigate the existence, smoothness, and monotonicity of these quantities.

Theorem 1.7. Let M be a closed manifold.

(a) The inf in the definition of μ_{+} is attained by a unique u. μ_{+}(g(t), t−T) is monotone nondecreasing under the Ricci flow, and is constant only on an expander with starting time t = T.

(b) If λ(g) < 0, then the sup in the definition of ν_{+} is attained by a unique σ. ν_{+}(g(t)) is monotone nondecreasing under the Ricci flow, and is constant only on an expander.

Proof. (a) Write

W_{+} = ∫ [σ(4|∇w|^{2} + Rw^{2}) + w^{2} log w^{2}] dv + C,

where w^{2} = u. This functional is lower semicontinous and coercive on W^{1,2}, so it possesses a minimizer w subject to ∫ w^{2} = 1. Without loss of generality, w is nonnegative. By the strong maximum principle (taking into account the sign of the nonlinearity), w > 0 and therefore w is smooth. The corresponding u = w^{2} minimizes the strictly convex functional

W_{+} = ∫ [σ(4|∇u^{1/2}|^{2} + Ru) + u log u] dv + C

on the cone of smooth functions with u > 0, ∫ u = 1, and therefore u is unique. It is straightforward to verify that u varies smoothly when (g, σ) varies smoothly. Write u_{g,σ} for this minimizer.

Now let g(t) be a Ricci flow, fix T ∈ ℝ, and let ũ(t) := u_{g(t),t−T} be the minimizer for each t. Fix t_{0} > T and let u(t) t ≤ t_{0} be the solution of the conjugate heat equation (1.3) with final value u(t_{0}). Since the first variation of W_{+} at t = t_{0} with respect to u vanishes, we can compute (smoothly) at t = t_{0}

\frac{d}{dt}μ_{+}(g(t), t − T) = \frac{d}{dt}W_{+}(g(t), ũ(t), t − T) = \frac{d}{dt}W_{+}(g(t), u(t), t − T),

so

\frac{d}{dt}μ_{+}(g(t), t − T) = ∫ u |Rc + ∇^{2}f_{+} + \frac{g}{2(T − t)}|^{2} dv,

where u realizes the minimum at t = t_{0}. Since t_{0} was arbitrary, this formula actually holds for each t, where u realizes the minimum at time t. If μ_{+}(g(t), t − T) is constant on any interval, the right hand vanishes, so g(t) is an expander with starting time T.

(b) By the above, μ_{+}(g, σ) is continuous in σ, σ > 0. We show that it goes to −∞ at the endpoints. Recall [P1, §1]

F(g, u) := ∫ \frac{|∇u|^{2}}{u} + Ru dv, λ(g) := inf_{u} F(g, w),

where u ranges over functions with ∫ u = 1. Fix u and estimate

μ_{+}(g, σ) ≤ W_{+}(g, u, σ) = σF(g, u) + ∫ u log u dv + \frac{n}{2} log(4πσ) + n.

Thus μ_{+}(g, σ) → −∞ as σ → 0. If λ(g) < 0, we select u so that F(g, u) = λ(u) and find that μ_{+}(g, σ) → −∞ as σ → ∞. In addition it is strictly concave in σ. Therefore the supremum in ν_{+} is realized by some unique σ = σ_{g}. Evidently σ_{g} varies smoothly with g.

Now let g(t) be a Ricci flow and fix t_{0} > 0. Since the first variation of μ_{+}(g(t_{0}), σ) is zero at σ = σ_{g(t_{0})}, we may compute (smoothly) at t = t_{0}

\frac{d}{dt}ν_{+}(g(t)) = \frac{d}{dt}μ_{+}(g(t), σ_{g(t)}) = \frac{d}{dt}μ_{+}(g(t), σ + t − t_{0}).

Thus

\frac{d}{dt}ν_{+}(g(t)) = ∫ u |Rc + ∇^{2}f_{+} + \frac{g}{2σ} |^{2} dv,

where (u, σ) realizes the minimax at t = t_{0}. Since t_{0} was arbitrary, this formula actually holds for each t, where (u, σ) realizes the minimax at time t. If ν_{+}(g(t)) is ever constant, the right hand vanishes, so g(t) is an expander. □

§2 Forward reduced distance and reduced volume.

In this section we derive the dual version of Perelman’s monotonicity on the reduced volume in [P1, §7].

Let g(t) solve the Ricci flow on M × [0, T]. Fix x_{0} and let γ be a path (x(η), η) joining (x_{0}, 0) to (y, t). Analogous to [P1], we define

(2.1)  L_{+}(γ) = ∫_{0}^{t} \sqrt{η} (R + |γ′(η)|^{2}) dη.

Let X ≡ γ′(t) and let Y be a variational vector field along γ. One calculates the first variation of L_{+} to be

(2.2)  δL_{+} = 2\sqrt{t}⟨X, Y ⟩|_{0}^{t} + ∫_{0}^{t} \sqrt{η}⟨Y, ∇R − 2∇_{X}X + 4Rc(X, ·) − \frac{1}{η} X⟩ dη.

From this one can write the L_{+}-geodesic equation:

(2.3)  ∇_{X}X − \frac{1}{2}R + \frac{1}{2t}X − 2Rc(X, ·) = 0.

This differs from [P1] only by the sign in front of Rc(X, ·). Let L_{+}(y, t) denote the length of a shortest L_{+}-geodesic joining (x_{0}, 0) to (y, t). From (2.2) we know that

(2.4)  ∇L_{+} = 2\sqrt{t}X(t).

Following closely the computation of [P1, §7] we obtain

|∇L_{+}|^{2} = −4tR + 4t(R + |X|^{2}) (2.5)

\frac{∂L_{+}}{∂t} = 2\sqrt{t}R − \sqrt{t}(R + |X|^{2}). (2.6)

A calculation using the geodesic equation shows that

\frac{d}{dt} (R + |X|^{2}) = H(X) − \frac{1}{t} (R + |X|^{2}) ,

where H(X) := ∂R/∂t + 2⟨∇R, X⟩ + 2Rc(X, X) + R/t is the exactly twice the traced Li-Yau-Hamilton differential Harnack expression in [H1]. (Notice that the H in [P1, §7] also equals the LYH expression, but evaluated at a negative time t = −τ.) This gives

(2.7)  t^{3/2} (R + |X|^{2}) = K + \frac{1}{2}L_{+}

where

K := ∫_{0}^{t} η^{3/2}H(X) dη.

We then have

|∇L_{+}|^{2} = −4tR + \frac{2}{\sqrt{t}}L_{+} + \frac{4}{\sqrt{t}}K, (2.8)

\frac{∂L_{+}}{∂t} = 2\sqrt{t}R − \frac{1}{t}K − \frac{1}{2t}L_{+}. (2.9)

These differ from [P1] only by the sign in front of K (but recall that the interpretation of H also differs).

12 FELDMAN, ILMANEN, AND NI

We can similarly do the second variation computation. For the a variation vector Y satisfying Y (0) = 0, we get

\begin{aligned}δ_{Y}^{2} L_{+} & = 2\sqrt{t}⟨∇_{Y} Y, X⟩ + ∫_{0}^{t} \sqrt{η} (Y · Y · R − 2⟨∇_{Y} Y, ∇_{X}X⟩ \\ & +4Y (Rc(Y, X)) − 2X(Rc(Y, Y)) − 2R(X, Y, X, Y) \\ & +2|∇_{X}Y |^{2} − \frac{1}{η} ⟨∇_{Y} Y, X⟩) dη. \end{aligned} (2.10)

Using the L_{+}-geodesic equation we get

\begin{aligned}δ_{Y}^{2} L_{+} & = 2\sqrt{t}⟨∇_{Y} Y, X⟩ + ∫_{0}^{t} \sqrt{η} (∇^{2}R(Y, Y) − 2R(X, Y, X, Y) \\ & +2|∇_{X}Y |^{2}4∇_{Y} Rc(Y, X) − 2∇_{X}Rc(Y, Y)) dη. \end{aligned} (2.11)

We choose Y such that

(2.12)  ∇_{X}Y = Rc(Y, ·) + \frac{1}{2t}Y,

which in particular implies that |Y (η)|^{2} = η/t. Then we obtain

\begin{aligned}∇^{2}L_{+}(Y, Y) & ≤ \frac{1}{\sqrt{t}} + ∫_{0}^{t} \sqrt{η} (∇^{2}R(Y, Y) − 2R(X, Y, X, Y) + 2|Rc(·, Y)|^{2} + \frac{2}{η} Rc(Y, Y) \\ & (4∇_{Y} Rc(Y, X) − 4∇_{X}Rc(Y, Y)) + 2∇_{X}Rc(Y, Y)) dη. \end{aligned} (2.13)

Now we use the fact that

\frac{d}{dt} (Rc(Y, Y)) = \frac{∂Rc}{∂t}(Y, Y) + ∇_{X}Rc(Y, Y) + 2|Rc(Y, Y)|^{2} + \frac{1}{t}Rc(Y, Y)

which implies

\begin{aligned}2 ∫_{0}^{t} \sqrt{η} & ∇_{X}Rc(Y, Y) dη = 2\sqrt{t}Rc(Y, Y) \\ & − ∫_{0}^{t} \sqrt{η} (3\frac{Rc(Y, Y)}{η} − 2\frac{∂Rc}{∂t}(Y, Y) − 4|Rc(Y, ·)|^{2}) dη. \end{aligned} (2.14)

Plugging (2.14) into (2.13) we have that

(2.15)  ∇^{2}L_{+}(Y, Y) ≤ \frac{|Y |^{2}}{\sqrt{t}} + 2\sqrt{t}Rc(Y, Y) − ∫_{0}^{t} \sqrt{η}H(X, Y) dη.

Here

\begin{aligned}H(X, Y) := & − ∇^{2}R(Y, Y) + 2R(X, Y, X, Y) + 2|Rc(·, Y)|^{2} \\ & + \frac{1}{t}Rc(Y, Y) + 2\frac{∂Rc}{∂t}(Y, Y) \\ & − 4∇_{Y} Rc(Y, X) + 4∇_{X}Rc(Y, Y). \end{aligned} (2.16)

This is exactly twice Hamilton’s matrix LYH expression. We get the following.

EXPANDER ENTROPY 13

Corollary 2.1. If (M, g(t)) has nonnegative curvature operator, then

∇^{2}L_{+}(Y, Y) ≤ \frac{|Y |^{2}}{\sqrt{t}} + 2\sqrt{t}Rc(Y, Y)

In the general curvature case, tracing (2.15) in Y yields

(2.17)  ΔL_{+} ≤ \frac{n}{\sqrt{t}} + 2\sqrt{t}R − \frac{1}{t}K.

Now defining

ℓ_{+}(y, t) := \frac{1}{2\sqrt{t}}L_{+}(y, t), L̄_{+}(y, t) := 4tℓ_{+}(y, t),

we get

|∇ℓ_{+}|^{2} = −R + \frac{ℓ_{+}}{t} + \frac{K}{t^{3/2}} , (2.8′)

\frac{∂ℓ_{+}}{∂t} = R − \frac{K}{2t^{3/2}} − \frac{ℓ_{+}}{t} , (2.9′)

Δℓ_{+} ≤ R + \frac{n}{2t} − \frac{K}{2t^{3/2}} . (2.17′)

These then imply

\frac{∂ℓ_{+}}{∂t} + Δℓ_{+} + |∇ℓ_{+}|^{2} − R − \frac{n}{2t} ≤ 0, (2.18)

(\frac{∂}{∂t} − Δ) (L̄_{+} + 2nt) ≥ 0, (2.19)

and negativity of a quantity resembling −v_{+}:

(2.20)  t(2Δℓ_{+} + |∇ℓ_{+}|^{2} − R) − ℓ_{+} − n ≤ 0.

Note that ℓ_{+} resembles −f_{+}, in contrast to the shrinker case, where ℓ is like f [P1, §9, CHI]. We have the following theorem from (2.18), (2.19). Theorem 2.2. Let g(t) solve the Ricci flow on M × [0, T]. Then

û(x, t) := \frac{e^{ℓ_{+}(x,t)}}{(4πt)^{n/2}}

is a super-solution to the conjugate heat equation, that is, ∂û/∂t + Δû − Rû ≤ 0. In particular, when M is compact, the forward reduced volume

(2.21)  θ_{+}(t) = θ_{+}^{(x_{0},0)}(t) := ∫_{M} û dv

is monotone non-increasing along the flow. Furthermore, L̄_{+} + 2nt is a super-solution to the standard heat equation.

Remarkably, the entropies W and W_{+} both increase in forward time, whereas the reduced volumes increase as t approaches the reference time t = 0.

The forward reduced volume is a constant precisely when g(t) is a compact expanding soliton and (x_{0}, 0) is the “vertex” of the soliton. In this case we get

log θ_{+}(t) = −ν_{+}(g),

a constant independent of time. Details will be given in future work; see [CHI] for the shrinker case. The following proposition says that θ_{+} is bounded away from zero when Ṽ_{∞} > 0, in the same way as the other monotone quantities.

Proposition 2.3. The forward reduced volume is bounded below by

log θ(t) ≥ log Ṽ (t) − \frac{n}{2}(1 + log 4π).

Proof. Since R ≥ −n/2t, we get for any curve γ from (x_{0}, 0) to (p, t),

L_{+}(γ) ≥ ∫_{0}^{t} \sqrt{η} (−\frac{n}{2η} ) dη = −n\sqrt{t},

so ℓ_{+}(p, t) ≥ −n/2 and

θ_{+}(t) ≥ \frac{V (t)e^{−n/2}}{(4πt)^{n/2}} = \frac{Ṽ (t)}{(4πe)^{n/2}} . □

This also shows that θ_{+}(t) → ∞ as t → 0 unless V (t) ∼ t^{n/2} for small t (maximal volume decay). In particular, when the initial metric is smooth except for a conic singularity x_{0}, as may arise by evolution from a smooth metric defined for negative times [FIK], then θ_{+}(t) will not be useful in proving that a blowup about the vertex (x_{0}, 0) for small t > 0 yields an expander.

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