A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds

Greg Anderson School of Mathematics University of Minnesota gwanders@math.umn.edu

Bennett Chow^{∗} Department of Mathematics University of California at San Diego benchow@math.ucsd.edu

1 Introduction

An important component of Hamilton’s program for the Ricci flow on compact 3-manifolds is the classification of singularities which form under the flow for certain initial metrics. In particular, Type I singularities, where the evolving metrics have curvatures whose maximums are inversely proportional to the time to blow-up, are modelled on the 3-sphere and the cylinder S^{2} × ℝ and their quotients. On the other hand, Type II singularities (the complementary case) are much more difficult to understand. Despite this, it is known from the work of Hamilton that their singularity models are stationary solutions to the Ricci flow. This uses several techniques, including Harnack inequalities of Li-Yau-Hamilton type, the strong maximum principle for systems, dimension reduction, and the study of the geometry at infinity of noncompact stationary solutions (see§§14-26 of [ H2].) In terms of Hamilton’s program, at least two obstacles remain: obtaining an injectivity radius estimate for Type II solutions and ruling out the so-called cigar soliton (the unique complete stationary solution on a surface with positive curvature) as the dimension reduction of a Type II singularity model.^{1}^{2}

∗Research partially supported by NSF grant DMS-9971891.

1In fact, Hamilton and Yau have announced informally that these are the only two obstacles and that they both would follow from obtaining a suitable differential matrix Harnack inequality of Li-Yau-Hamilton type for arbitrary solutions of the Ricci flow on compact 3manifolds.

2Added in proof: Very recently, Perelman [P1] has given a proof of the Little Loop Conjecture in all dimensions without curvature restriction. See also [P2] for some further developments.

On the other hand, it is also conjectured by Hamilton that Type II singularities are not generic. If this conjecture can be proven with some definition of generic which implies that for any compact 3-manifold the Ricci flow with suitable surgeries (see [H5] for how to perform surgeries) does not form Type II singularities, then there would be no need for obtaining an injectivity radius estimate for Type II solutions or ruling out the so-called cigar soliton. Partly for these reasons we are motivated to study the linearized Ricci flow. Given an initial metric on a 3-manifold and the corresponding solution to the Ricci flow, one would like to understand the behavior of solutions with nearby initial metrics. The linearized Ricci flow system is the pair of equations (4)-(5) which we consider below. In this paper we obtain an apriori estimate for arbitrary solutions to the linearized Ricci flow on compact 3-manifolds which we hope may be useful in its study. The inspiration for this estimate comes from the works of Hamilton (§10 of [ H1] and §24 of [ H2]) and Gursky [ G].

Recall that if (M^{3}, g (t)) is a solution to the Ricci flow on a compact 3manifold with positive scalar curvature, then Hamilton obtained the following parabolic Bochner-type estimate

\frac{∂}{∂t} (\frac{|Rc|^{2}}{R^{2}} ) ≤ Δ (\frac{|Rc|^{2}}{R^{2}} ) + \frac{2}{R}∇R · ∇ (\frac{|Rc|^{2}}{R^{2}} ) .  (1)

See Lemmas 10.5 (with γ = 2) and 10.6 in [ H1] for the positive Ricci curvature case, and the equation for Y in the proof of Theorem 24.7 in [ H2] for the more general positive scalar curvature case. A sharpened form of this estimate is the main estimate in showing that the normalized Ricci flow evolves a closed 3-manifold with positive Ricci curvature into a spherical space form; see [ H1], Theorem 10.1, or [ H3], Theorem 5.3 for a simpler proof. A further extension of this estimate is used to show that for a Type I singularity of the Ricci flow on a closed 3-manifold not diffeomorphic to a spherical space form, there exists a sequence of dilations about points and times approaching the singularity time that limits to a quotient of the cylinder S^{2} × ℝ ; see Theorem 24.7, Corollary 24.8 and Theorem 26.5 in [ H2].

On the other hand, also recall that Gursky (see [ G]) proved that if (M^{4}, g) is a closed, oriented, 4-manifold with positive scalar curvature such that

ΔR = −4 |W^{+}|^{2} − 2 |Rc − \frac{1}{4}Rg|^{2} + \frac{1}{6}R^{2}  (2)

and α is a self-dual harmonic 2-form, then

Δ (\frac{|α|^{2}}{R^{2}} ) + \frac{2}{R}∇R · ∇ (\frac{|α|^{2}}{R^{2}} ) ≥ 0.  (3)

There is a formal similarity between these two Bochner formulas. Note that Kähler-Einstein surfaces satisfy (2). On the other hand, the Ricci flow is the parabolic version of the equation for Einstein metrics (the fixed points of the volume normalized Ricci flow are the Einstein metrics). Furthermore, if (M^{2k}, g) is a Kähler manifold and α is a J-invariant 2-form, then h (X, Y) ≑ α (X, JY) is a J-invariant symmetric 2-tensor and (Δ_{d}α) (X, JY) = (Δ_{L}h) (X, Y) , where Δ_{L} is the Lichnerowicz laplacian (defined below). Thus if α is also harmonic, then Δ_{L}h = 0. The parabolic version of this last equation is the Lichnerowicz laplacian heat equation.

The above considerations partly motivate us to study the analogue of estimates (1) and (3) in the context of the Ricci flow

\frac{∂}{∂t}g_{ij} = −2R_{ij}  (4)

coupled to the Lichnerowicz laplacian heat equation

\frac{∂}{∂t}h_{ij} = (Δ_{L}h)_{ij} ≑ Δh_{ij} + 2R_{kijl}h_{kl} − R_{ik}h_{kj} − R_{jk}h_{ki}  (5)

for a symmetric 2-tensor h. This is the linearized Ricci flow system and arises from linearizing the Ricci flow using a version of DeTurck’s trick (see §2). A differential Harnack inequality of Li-Yau-Hamilton type, patterned after Hamilton’s trace inequality for the Ricci flow [H4] and Li-Yau’s seminal inequality for the heat equation [LY], for this coupled system was proved by Hamilton and one of the authors [ CH] and interpreted geometrically in terms of linearizing the Ricci flow by S.-C. Chu and one of the authors [ CC]. A complex version of this inequality was proven by Ni and Tam [NT] and applied to the study of the Kähler-Ricci flow. The linearized Ricci flow has been studied by Guenther, Isenberg and Knopf [GIK] at flat solutions from the point of view of maximal regularity theory. Additional work on differential Harnack inequalities of Li-Yau-Hamilton type which appear related to the linearized Ricci flow are in [C] and [CK].^{3}

Analogous to (1) and (3), we may consider the quantity \frac{|h|^{2}}{R^{2}} for solutions to the Ricci flow on 3-manifolds with positive scalar curvature. This is a pointwise measure of the size of h relative to the scalar curvature. More generally, since R_{min} (t) is a nondecreasing function for solutions to the Ricci flow, we may replace R by R + ρ, where ρ ∈ [0, ∞) is chosen so that R + ρ > 0 at t = 0.

Main Theorem. Let (M^{3}, g (t)) be a solution to the Ricci flow on a closed 3-manifold on a time interval [0, T) with T < ∞ and let ρ ∈ [0, ∞) be such that R_{min} (0) > −ρ. If the pair (g, h) is any solution to the linearized Ricci flow system (4)-(5), then there exists a constant C < ∞ such that

\frac{∂}{∂t} (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) ≤ Δ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) + \frac{2}{R + ρ}∇R · ∇ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} )  (6)

+ 4Cρ\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} .

3In the latter reference, a Li-Yau-Hamilton inequality for the Ricci flow is proved which generalizes Hamilton’s matrix inequality and has some formal similarities with linear inequalities.

Consequently, by direct application of the maximum principle, the norm of the solution to the linearized Ricci flow equation is comparable to the scalar curvature plus a constant:

|h| ≤ C (R + ρ)

on M × [0, T), where C depends only on g (0) , ρ and T. Furthermore, when

ρ = 0, C is independent of T.

Taking ρ = 0 and h_{ij} = R_{ij} , we obtain:

Corollary 1 (Hamilton, [ H1]) If (M^{3}, g (t)) , t ∈ [0, T), T < ∞ is a solution to the Ricci flow on a closed 3-manifold with positive scalar curvature, then there exists a constant C < ∞ such that

\frac{|Rc|}{R} ≤ C

on M × [0, T).

In §2 we recall how the system (4)-(5) is obtained by linearizing the Ricci flow using a version of DeTurck’s trick with a time-dependent background metric. In §3 we give the proof of equation (6), from which the main theorem follows. This depends on the nonnegativity of a certain degree 4 polynomial in 6 variables (Lemma 5), which is proved in §4.

2 The linearized Ricci flow system

This section is mainly to motivate our study of the linearized Ricci flow system. The reader well familiar with DeTurck’s trick [ D] may skip this section.

As we stated in the introduction, a solution to the linearized Ricci flow system consists of a complete solution (M^{n}, g_{o} (t)) , t ∈ [0, T), to the Ricci flow

\frac{∂}{∂t}g_{ij} = −2R_{ij}  (7)

coupled with a solution h (t) , t ∈ [0, T), to the Lichnerowicz laplacian heat equation

\frac{∂}{∂t}h_{ij} = (Δ_{L}h)_{ij}  (8)

where

(Δ_{L}h)_{ij} ≑ Δh_{ij} + 2R_{kijl}h_{kl} − R_{ik}h_{kj} − R_{jk}h_{ki}.

The Lichnerowicz laplacian Δ_{L} is defined using the evolving metric g_{o} (t) . Our main interest is when the solution is compact. However, in view of compactness arguments in the category of pointed solutions, it may be of interest to study the linearized Ricci flow system for complete, noncompact solutions.

This system arises as follows. Given a solution (M^{n}, g_{o} (t)) , t ∈ [0, T_{o}), to the Ricci flow, consider the modified Ricci flow with time-dependent background metrics g_{o} (t) :

\frac{∂}{∂t}g_{ij} = −2R_{ij} + ∇_{i}W_{j} + ∇_{j}W_{i}  (9)

where the 1-forms W (t) are defined by

W (t)_{ℓ} ≑ g (t)_{ℓk} g (t)^{ij} (Γ [g (t)]_{ij}^{k} − Γ [g_{o} (t)]_{ij}^{k})  (10)

and the covariant derivatives are with respect to the metrics g (t) . This is De-Turck’s trick [ D] with a time-dependent background metric. Note that the solution g_{o} (t) to the Ricci flow is itself also a solution the modified Ricci flow with background metrics g_{o} (t) .

There exists a unique solution to the initial value problem for the modified Ricci flow for short time. The modified Ricci tensor, which we define to be the rhs of (9), depends only on g (t) and g_{o} (t) . We first compute the linearization of the modified Ricci flow about the solution g_{o} (t) . In this case we get the Lichnerowicz laplacian heat equation. Hence the modified Ricci flow is a parabolic equation, which in turn, implies uniqueness and short time existence. DeTurck gave this argument as a new proof of the short time existence and uniqueness of solutions to the Ricci flow originally proved in [ H1].

Let {g_{s,o}}_{s∈(−ε,ε)} be a smooth, one-parameter family of initial metrics with g_{0,o} = g_{o} (0) . Consider the one-parameter family {g_{s} (t) , t ∈ [0, T_{s})}_{s∈(−ε,ε)} of solutions to the modified Ricci flow:

\frac{∂}{∂t} (g_{s})_{ij} = −2 (Rc_{s})_{ij} + ∇_{i}W_{j} + ∇_{j}W_{i}  (11)
g_{s} (0) = g_{s,o},  (12)

where

W_{s} (t)_{ℓ} ≑ g_{s} (t)_{ℓk} g_{s} (t)^{ij} (Γ [g_{s} (t)]_{ij}^{k} − Γ [g_{o} (t)]_{ij}^{k}) ,  (13)

and the Ricci tensor and covariant derivative are with respect to the metrics g_{s} (t) . Recall that

g_{0} (t) ≡ g_{o} (t) and W_{0} (t) ≡ 0.

Define

v_{ij} (t) ≑ \frac{∂}{∂s}|_{s=0} g_{s} (t)_{ij} .  (14)

We shall call v the variation of the metric tensor.

Let V (t) = g_{0} (t)^{ij} v (t)_{ij} . A standard computation yields (see for example [ H1])

Lemma 2 The variation of the Ricci tensor is

\frac{∂}{∂s}|_{s=0} (−2Rc [g_{s} (t)]_{ij})
= Δv_{ij} + 2R_{kijl}v_{kl} − R_{ik}v_{kj} − R_{jk}v_{ki} + ∇_{i}∇_{j}V − ∇_{i}∇^{k}v_{kj} − ∇_{j}∇^{k}v_{ki}
= (Δ_{L}v)_{ij} + ∇_{i}∇_{j}V − ∇_{i}∇^{k}v_{kj} − ∇_{j}∇^{k}v_{ki}  (15)

Recall the algebraic Einstein operator

G (v)_{ij} ≑ v_{ij} − \frac{1}{2}V g_{ij}

(which takes the Ricci tensor to the Einstein tensor: G (Rc)_{ij} = R_{ij} − \frac{1}{2}Rg_{ij}) and the divergence

δ : C^{∞} (S^{2}T^{∗}M) → C^{∞} (T^{∗}M)

where

δ (T)_{i} ≑ −g^{jk}∇_{k}T_{ij}.

The L^{2} adjoint of δ

δ^{∗} : C^{∞} (T^{∗}M) → C^{∞} (S^{2}T^{∗}M)

is the same as the Lie derivative operator acting on the metric:

δ^{∗} (v)_{ij} = \frac{1}{2} (∇_{i}v_{j} + ∇_{j}v_{i}) = \frac{1}{2} (L_{v^{∗}}g)_{ij}

where (v^{∗})^{i} = g^{ij}v_{j}. The last three terms on the rhs of equation (15) may be rewritten as

∇_{i}∇_{j}V − ∇_{i}∇^{k}v_{kj} − ∇_{j}∇^{k}v_{ki} = 2 [δ^{∗} (δ [G (v)])]_{ij} .

Hence we have the following well-known identity:

Lemma 3 The variation of the Ricci tensor has the form

\frac{∂}{∂s}|_{s=0} (−2Rc [g_{s} (t)]_{ij}) = (Δ_{L}v)_{ij} + 2 [δ^{∗} (δ [G (v)])]_{ij} .

We may rewrite the 1-form W_{s} (t) as

W_{s} (t)_{ℓ} = −\frac{1}{2}g_{s} (t)_{ℓk} g_{0} (t)^{kp} g_{s} (t)^{ij} (∇_{i} [g_{0}]_{jp} + ∇_{j}[g_{0}]_{ip} − ∇_{p} [g_{0}]_{ij})

= W_{s} (t)_{ℓ} = \frac{1}{2}g_{s} (t)_{ℓk} g_{0} (t)^{kp} (δ [G (g_{0})])_{p} ,

where the covariant derivatives are with respect to the metrics g_{s} (t) . Define

(g_{0})^{−1} : C^{∞} (T^{∗}M) → C^{∞} (T^{∗}M)

by

(g_{0})^{−1} T ≑ g_{s} (t)_{ℓk} g_{0} (t)^{kp} T_{p}.

Then

W_{s} (t)_{ℓ} = \frac{1}{2} [(g_{0})^{−1} (δ [G (g_{0})])]_{ℓ} .

Hence the last two terms of the modified Ricci tensor can be expressed as

∇_{i}W_{j} + ∇_{j}W_{i} = 2 (δ^{∗}W)_{ij} = (δ^{∗} [(g_{0})^{−1} (δ [G (g_{0})])])_{ij} ,

so that

−2R_{ij} + ∇_{i}W_{j} + ∇_{j}W_{i} = −2R_{ij} + (δ^{∗} [(g_{0})^{−1} (δ [G (g_{0})])])_{ij} .

Thus the only change to DeTurck’s modification of the Ricci flow that we have made is that we allow the background metric g_{0} to depend on time. In particular, we take g_{0} to be the solution of the Ricci flow that we are linearizing about.

The motivation for studying the linearized Ricci flow system is the following.

Proposition 4 The variation v (t) of the metric tensor g (t) corresponding to a one-parameter family {g_{s} (t)}_{s∈(−ε,ε)} of solutions to the modified Ricci flow (11)-(12) is a solution to the Lichnerowicz laplacian heat equation:

\frac{∂}{∂t}v = Δ_{L}v.

That is, the pair (g, v) is a solution to the linearized Ricci flow system (7)-(8).

Proof. We compute

\frac{∂}{∂t}v_{ij} (t) = \frac{∂}{∂t} \frac{∂}{∂s}|_{s=0} g_{s} (t)_{ij} = \frac{∂}{∂s}|_{s=0} \frac{∂}{∂t}g_{s} (t)_{ij}

= \frac{∂}{∂s}|_{s=0} (−2R_{ij} + ∇_{i}W_{j} + ∇_{j}W_{i})

= \frac{∂}{∂s}|_{s=0} (−2Rc [g_{s} (t)]_{ij}) + ∇_{i} ( \frac{∂}{∂s}|_{s=0} W_{s} (t)_{j}) + ∇_{j} ( \frac{∂}{∂s}|_{s=0} W_{s} (t)_{i})

= (Δ_{L}v)_{ij} + ∇_{i}∇_{j}V − ∇_{i}∇^{k}v_{kj} − ∇_{j}∇^{k}v_{ki}

+ ∇_{i} (\frac{1}{2}g_{0} (t)^{kℓ} (∇_{k}v_{ℓj} + ∇_{ℓ}v_{kj} − ∇_{j}v_{kℓ}))

+ ∇_{j} (\frac{1}{2}g_{0} (t)^{kℓ} (∇_{k}v_{ℓi} + ∇_{ℓ}v_{kj} − ∇_{i}v_{kℓ}))

= (Δ_{L}v)_{ij} ,

where we used W_{0} (t) ≡ 0 to obtain the fourth equality, and the fact that ∇_{i}∇_{j}V = ∇_{j}∇_{i}V for the last equality.

3 Proof of the pinching estimate

In this and the following section, we give the derivation of equation (6), which implies the main theorem. Using the Ricci flow equation, (5), and the standard equation \frac{∂}{∂t}R = ΔR + 2 |Rc|^{2} , we compute

\frac{∂}{∂t} (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) = −\frac{2}{(R + ρ)^{2}} \frac{∂}{∂t}g_{ij} · h_{ij}^{2} + \frac{2}{(R + ρ)^{2}} (\frac{∂}{∂t}h) · h − 2\begin{matrix}|h| & 2 \\ (R + ρ) & 3 \end{matrix} \frac{∂}{∂t}R

= \frac{4Rc · h^{2}}{(R + ρ)^{2}} + \frac{2}{(R + ρ)^{2}} (Δh_{ij} + 2R_{kijl}h_{kl} − R_{ik}h_{kj} − R_{jk}h_{ki}) h_{ij}

− 2\begin{matrix}|h| & 2 \\ (R + ρ) & 3 \end{matrix} (ΔR + 2 |Rc|^{2})

= Δ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) − 2\begin{matrix}|∇h| & 2 \\ (R + ρ) & 2 \end{matrix} − 6\begin{matrix}|h| & 2 \\ (R + ρ) & 4 \end{matrix} |∇R|^{2}

+ \frac{8}{(R + ρ)^{3}} h∇_{i}R · ∇_{i}h + \frac{4}{(R + ρ)^{2}} R_{ijkl}h_{il}h_{jk} − \frac{4}{(R + ρ)^{3}} |h|^{2} |Rc|^{2} .

Since

\frac{1}{R + ρ}∇R · ∇ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) = −2\begin{matrix}|h| & 2 \\ (R + ρ) & 4 \end{matrix} |∇R|^{2} + \frac{2}{(R + ρ)^{3}} h∇_{i}R · ∇_{i}h,

we may rewrite the above evolution equation as

\frac{∂}{∂t} (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) = Δ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) + \frac{2}{R + ρ}∇R · ∇ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} )  (16)

− 2\begin{matrix}|(R + ρ) ∇_{i}h_{jk} − ∇_{i}Rh_{jk}| & 2 \\ (R + ρ) & 4 \end{matrix} + \frac{4}{(R + ρ)^{2}} R_{ijkl}h_{il}h_{jk}

− \frac{4}{(R + ρ)^{3}} |h|^{2} |Rc|^{2} .

When n = 3, we have the identity

R_{ijkl} = R_{il}g_{jk} + R_{jk}g_{il} − R_{ik}g_{jl} − R_{jl}g_{ik} − \frac{1}{2}R (g_{il} g_{jk −}g_{ik} g_{jl}) .

Hence

\frac{∂}{∂t} (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) = Δ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) + \frac{2}{R + ρ}∇R · ∇ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} )

− \frac{2}{(R + ρ)^{4}} |(R + ρ) ∇_{i}h_{jk} − ∇_{i}Rh_{jk}|^{2} + 4P

where

P = \frac{1}{(R + ρ)^{2}} [R_{il}g_{jk} + R_{jk}g_{il} − R_{ik}g_{jl} − R_{jl}g_{ik} − \frac{1}{2}R (g_{il} g_{jk −}g_{ik} g_{jl})] h_{il}h_{jk}

− \frac{1}{(R + ρ)^{3}} |h|^{2} |Rc|^{2}

= \frac{1}{(R + ρ)^{3}} [2 (R + ρ) Rc · hH − 2 (R + ρ) Rc · h^{2} + \frac{R}{2} (R + ρ) (|h|^{2} − H^{2}) − |h|^{2} |Rc|^{2}]

= \frac{1}{(R + ρ)^{3}} [2RRc · hH − 2RRc · h^{2} + \frac{1}{2}R^{2} (|h|^{2} − H^{2}) − |h|^{2} |Rc|^{2}]

+ \frac{ρ}{(R + ρ)^{3}} [2Rc · hH − 2Rc · h^{2} + \frac{1}{2}R (|h|^{2} − H^{2})] .

The main theorem is now a consequence of the following inequality, which we shall prove in the next section. Lemma 5 We have for any metric g and symmetric 2-tensor h, the inequality

|h|^{2} |Rc|^{2} − 2RHRc · h + 2RRc · h^{2} + \frac{1}{2}R^{2} (H^{2} − |h|^{2}) ≥ 0.

This is because, then

\frac{∂}{∂t} (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) ≤ Δ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) + \frac{2}{R + ρ}∇R · ∇ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} )

+ 4\frac{ρ}{(R + ρ)^{3}} [2Rc · hH − 2Rc · h^{2} + \frac{1}{2}R (|h|^{2} − H^{2})] .

On the other hand, we have the estimate |Rc| ≤ C (R + ρ) (see Theorem 24.4 of [ H2]), which implies

\frac{ρ}{(R + ρ)^{3}} [2Rc · hH − 2Rc · h^{2} + \frac{1}{2}R (|h|^{2} − H^{2})] ≤ Cρ\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} .

Hence

\frac{∂}{∂t} (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) ≤ Δ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) + \frac{2}{R + ρ}∇R · ∇ (\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} ) + 4Cρ\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} .

If t ∈ [0, T), then applying the maximum principle implies

\begin{matrix}|h| & 2 \\ (R + ρ) & 2 \end{matrix} (t) ≤ C_{0} exp (4CρT) ,

where C_{0} = max_{t=0} |h|^{2} / (R + ρ)^{2} . q.e.d.

4 Nonnegativity of a degree 4 homogeneous polynomial in 6 variables

Proof of Lemma 5. Since h is symmetric, we may assume h is diagonal. Let h_{1}, h_{2}, h_{3} denote the eigenvalues of h and let r_{1} = R_{11}, r_{2} = R_{22}, r_{3} = R_{33} denote the diagonal entries of R_{ij}. Then

−R^{3}P = |h|^{2} |Rc|^{2} − 2RHRc · h + 2RRc · h^{2} + \frac{1}{2}R^{2} (H^{2} − |h|^{2})
≥ Q ≑ (h_{1}^{2} + h_{2}^{2} + h_{3}^{2}) (r_{1}^{2} + r_{2}^{2} + r_{3}^{2})
− 2 (r_{1} + r_{2} + r_{3}) (h_{1} + h_{2} + h_{3}) (r_{1}h_{1} + r_{2}h_{2} + r_{3}h_{3})

+ 2 (r_{1} + r_{2} + r_{3}) (r_{1}h_{1}^{2} + r_{2}h_{2}^{2} + r_{3}h_{3}^{2}) + (r_{1} + r_{2} + r_{3})^{2} (h_{1}h_{2} + h_{1}h_{3} + h_{2}h_{3}) ,

where we used the inequality (throwing away the off-diagonal entries of R_{ij})

|Rc|^{2} ≥ r_{1}^{2} + r_{2}^{2} + r_{3}^{2}.

We expand and simplify this as

Q = r_{1}^{2}h_{1}^{2} + r_{2}^{2}h_{2}^{2} + r_{3}^{2}h_{3}^{2} + r_{1}^{2}h_{2}^{2} + r_{1}^{2}h_{3}^{2} + r_{2}^{2}h_{1}^{2} + r_{2}^{2}h_{3}^{2} + r_{3}^{2}h_{1}^{2} + r_{3}^{2}h_{2}^{2}
+ r_{3}^{2}h_{1}h_{2} + r_{1}^{2}h_{2}h_{3} + r_{2}^{2}h_{1}h_{3}
− r_{1}^{2}h_{1}h_{2} − r_{1}^{2}h_{1}h_{3} − r_{2}^{2}h_{1}h_{2} − r_{2}^{2}h_{2}h_{3} − r_{3}^{2}h_{1}h_{3} − r_{3}^{2}h_{2}h_{3}
− 2r_{1}r_{2}h_{1}h_{2} − 2r_{1}r_{3}h_{1}h_{3} − 2r_{2}r_{3}h_{2}h_{3}.

Writing Q as a bilinear form in h = (h_{1}, h_{2}, h_{3}) with coefficients in r = (r_{1}, r_{2}, r_{3}) , we have

Q = (r_{1}^{2} + r_{2}^{2} + r_{3}^{2}) h_{1}^{2} + (r_{1}^{2} + r_{2}^{2} + r_{3}^{2}) h_{2}^{2} + (r_{1}^{2} + r_{2}^{2} + r_{3}^{2}) h_{3}^{2}
+ (−r_{1}^{2} − r_{2}^{2} + r_{3}^{2} − 2r_{1}r_{2}) h_{1}h_{2} + (r_{1}^{2} − r_{2}^{2} − r_{3}^{2} − 2r_{2}r_{3}) h_{2}h_{3}
+ (−r_{1}^{2} + r_{2}^{2} − r_{3}^{2} − 2r_{1}r_{3}) h_{1}h_{3}.

The lemma follows from the claim that the polynomial Q takes nonnegative values for all real values of r_{1}, r_{2}, r_{3}, h_{1}, h_{2}, h_{3}. To prove the claim, it is enough to show that the Hessian matrix

\begin{aligned}H & := & [\frac{∂^{2}Q}{∂h_{i}∂h_{j}} ] \\ & = & [ \begin{matrix}2r_{1}^{2} + 2r_{2}^{2} + 2r_{3}^{2} & r_{3}^{2} − r_{1}^{2} − r_{2}^{2} − 2 r_{1}r_{2} & r_{2}^{2} − r_{1}^{2} − r_{3}^{2} − 2 r_{1}r_{3} \\ r_{3}^{2} − r_{1}^{2} − r_{2}^{2} − 2 r_{1}r_{2} & 2r_{1}^{2} + 2r_{2}^{2} + 2r_{3}^{2} & r_{1}^{2} − r_{2}^{2} − r_{3}^{2} − 2 r_{2}r_{3} \\ r_{2}^{2} − r_{1}^{2} − r_{3}^{2} − 2 r_{1}r_{3} & r_{1}^{2} − r_{2}^{2} − r_{3}^{2} − 2 r_{2}r_{3} & 2r_{1}^{2} + 2r_{2}^{2} + 2r_{3}^{2} \end{matrix} ] \end{aligned}

is positive semidefinite for all real values of r_{1}, r_{2}, r_{3}. In turn, it is enough to show that the determinants

Δ_{1} := H_{11}, Δ_{2} := | \begin{matrix}H_{11} & H_{12} \\ H_{21} & H_{22} \end{matrix} | , Δ_{3} := | \begin{matrix}H_{11} & H_{12} & H_{13} \\ H_{21} & H_{22} & H_{23} \\ H_{31} & H_{32} & H_{33} \end{matrix} |

take nonnegative values for all real values of r_{1}, r_{2}, r_{3}. The nonnegativity of

Δ_{1} = 2r_{1}^{2} + 2r_{2}^{2} + 2r_{3}^{2}

is clear. The nonnegativity of

Δ_{2} = (2r_{1} + 2r_{2} + 2r_{3})^{2} − (r_{1}^{2} − r_{2}^{2} − 2r_{2}r_{3} − r_{3}^{2})^{2}

follows from the inequality

2r_{1}^{2} + 2r_{2}^{2} + 2r_{3}^{2} ≥ r_{1}^{2} + r_{2}^{2} + 2|r_{1}r_{2}| + r_{3}^{2} ≥ |r_{1}^{2} − r_{2}^{2} − 2r_{2}r_{3} − r_{3}^{2}|.

One has

3Δ_{3} = 2 X^{6} − 6 X^{4}Y − 24 X^{2}Y^{2} + 24 Y^{3} + 16 X^{3}Z,

where

X := r_{1} + r_{2} + r_{3}, Y := r_{1}^{2} + r_{2}^{2} + r_{3}^{2}, Z := r_{1}^{3} + r_{2}^{3} + r_{3}^{3}.

The reader will have no difficulty verifying this identity with the help of a computer algebra package. Now on any circle in (r_{1}, r_{2}, r_{3})-space defined by holding X and Y constant, the extremal values of Z occur at points with two coordinates equal, as one verifies by a straightforward Lagrange multiplier argument. Since for fixed X and Y, Δ_{3} depends linearly on Z, it follows that on any circle defined by holding X and Y constant, the extreme values of Δ_{3} are taken at points with two coordinates equal. After making the evident reductions, the inequality

Δ_{3}(x, x, 1) = 4(8x^{2} + 1)(x − 1)^{2} ≥ 0

suffices to prove the nonnegativity of Δ_{3}, and in turn the claim. With hindsight, the reader can see that it was overkill to actually write out the expression of Δ_{3} in terms of X, Y and Z; all that was used in the proof of nonnegativity of Δ_{3} was the fact that for fixed X and Y, Δ_{3} depends linearly on Z. q.e.d.

Acknowledgment. We would like to thank Jiaping Wang for bringing to our attention Matt Gursky’s work which partly inspired the result in this paper.

References

[C] B. Chow, Interpolating between Li–Yau’s and Hamilton’s Harnack inequalities on a surface, J. Partial Diff. Equations (China) 11 (1998) 137–140.

[ CC] B. Chow and S.-C. Chu, A geometric approach to the linear trace Harnack inequality for the Ricci flow, Math. Res. Lett. 3 (1996) 549-568.

[ CH] Chow, B. and R. S. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math. 129 (1997) 213–238.

[CK] B. Chow and D. Knopf, New Li-Yau-Hamilton inequalities for the Ricci flow via the space-time approach, J. Differ. Geom. 60 (2002) 1–51.

[ D] D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom., 18 (1983) 157–162.; ibid. (improved version), in Selected papers on Ricci flow, edited by H.-D. Cao, etal., International Press, to appear.

[GIK] C. Guenther, J. Isenberg, and D. Knopf, Stability of the Ricci flow at Ricci-flat metrics, Comm. Anal. Geom. 10 (2002) 741–777.

[ G] M. Gursky, The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics, Ann. of Math. 148 (1998) 315-337.

[ H1] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255-306.

[ H2] R. S. Hamilton, The formation of singularities in the Ricci flow, in Surveys in Differential Geometry 2 (1995) International Press, 7–136.

[ H3] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153-179.

[H4] R. S. Hamilton, The Harnack estimate for the Ricci flow, J. Diff. Geom. 37 (1993) 225-243.

[H5] R. S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997) 1–92.

[LY] P. Li and S. T. Yau, On the parabolic kernel of the Shrödinger operator, Acta Math. 156 (1986) 153–201.

[NT] L. Ni and L.-F. Tam, Kähler-Ricci flow and the Poincaré-Lelong equation, Comm. Anal. Geom., to appear.

[P1] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159.

[P2] G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109.