A short proof of backward uniqueness for some geometric evolution equations

0.22 MB·English

by  Brett Kotschwar

A SHORT PROOF OF BACKWARD UNIQUENESS FOR SOME
GEOMETRIC EVOLUTION EQUATIONS.
5
1 BRETTKOTSCHWAR
0
2
Abstract. We give a simple,direct proof of the backward uniqueness of so-
n lutions toaclassof second-order geometricevolution equations includingthe
a Ricci and cross-curvature flows. The proof, based on a classical argument of
J
Agmon-Nirenberg,usesthelogarithmicconvexityofacertainenergyquantity

5 in the place of Carleman inequalities. We also demonstrate the applicability
ofthetechnique totheL2-curvatureflowandotherhigher-orderequations.

]
G
D
. 1. Introduction
h
at The invariance of a curvature flow ∂ g = E(g) on a smooth manifold M under
∂t
m the action of Diff(M) has the important consequence that the isometry group of a
[ solution remains unchanged within any class in which the uniqueness of solutions

forward and backward in time is assured. An operator E with such an invariance,
1
however, is never strongly elliptic, and so these same questions of uniqueness are
v
6 not merely consequences of standard parabolic theory.
4 Whereas the problem of forward uniqueness for curvature flows can often be
9 converted to one for a strictly parabolic system by the means of DeTurck’s trick
0 (see, e.g., [H], [CZ], [BH2]), the ill-posedness of terminal-value problems for par-
0
abolic equations prohibits the application of this same technique to the problem
.
1 of backward uniqueness. This is discussed in some detail in [Ko1] in the context
0
of the Ricci flow. In that paper we show that, nevertheless, the problem can be
5
recast as one for a prolonged system to which the usual method of weighted L2
1
: (Carleman-type) estimates can be extended.
v
The Carleman inequality approach to problems of backward uniqueness, which
i
X originates in the classical work of Lees and Protter [LP], [Pr], is extremely robust
r andisthebasisofavastliteratureofresultsontheuniquecontinuationofsolutions
a
to parabolic equations; for a picture of the current state-of-the-art, we refer the
reader to [E], [EF], [EKPV], [ESS], [EV], [F], [KT], [So], [T], and the references
therein. For the global problem of backward uniqueness, however, the simplest
approach is perhaps still the energy-quotient/logarithmic-convexity technique of
AgmonandNirenberg[AN1],[AN2]. This approachdates to afew yearsafter [LP]
and has since been clarified and extended by a number of authors (see, e.g., [BT],
[G], [Ku1], [Ku2], [O]). The analysis of the basic energy quotient on which it is
centered is related in turn to that of the parabolic frequency quantities of Poon
[Po] and others (e.g., [EKPV]).
Theprimarypurposeofthispaperistoshowthatthisenergyquotienttechnique
canalsobeappliedtobackwarduniquenessproblemsforgeometricflowsoncethey
Date:Aug2013.
Theauthor wassupportedinpartbyNSFgrantDMS-1160613.
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2 BRETTKOTSCHWAR
havebeenappropriatelyreformulated. InSection2, weusethis technique togivea
shorterandmoretransparentlyquantitativeproofofthe generalbackwardunique-
ness theorem for systems of mixed differential inequalities in [Ko1]. In Section 3,
we discuss the applicationof this resultto the problem of backwarduniqueness for
the Ricci and cross-curvature flows. In Section 4, we use this technique to prove
a backwarduniqueness theoremfor solutions to the fourth-orderL2-curvatureflow
and discuss extensions to a class of of higher-order equations.
We have already mentioned the connection of uniqueness results for geometric
flows to the preservation of the isometry group of a solution: for such flows, a
backwarduniqueness theorem implies that solutions do not acquire new isometries
duringtheirsmoothlifetimes. Thereareanumberofotherqualitativepropertiesof
geometric flows, however, which, when suitably formulated, amount to statements
of backward uniqueness for some system associated to the flow. Among them, for
example, are the impossibility of a classical solution to the Ricci flow becoming
spontaneously Einstein or self-similar at some point in its evolution [Ko1] or ac-
quiring a novel K¨ahler or local product structure [Ko2]. A backward uniqueness
problem(albeitasomewhatnonstandardone–see[ESS])isalsoattheheartofour
our recent work [KW] with Lu Wang on the asymptotic rigidity of shrinking Ricci
solitons. The connectionbetweenthisparticulartypeofproblemofelliptic unique-
continuation-at-infinity and backward uniqueness was first illustrated by Wang in
[W1] (cf. [W2]) in the context of the mean curvature flow.
2. A general backward uniqueness theorem for second-order systems
2.1. Definitions and conventions. In this section, M = Mn will denote a con-
nectedsmoothmanifoldequippedwithasmoothfamilyg(τ)ofRiemannianmetrics
definedforτ ∈[0,Ω];theLevi-Civitaconnectionandvolumedensityofg(τ)willbe
denotedby∇=∇(τ)anddµ=dµ ,respectively. LetX andY besmoothvector
g(τ)
bundles over M equipped with their own smooth families of bundle metrics. To
avoidhavingtointroduceseparatenotationforthesemetrics,wewillregardX and
Y as orthogonalsubbundles of the bundle W =X ⊕Y relative to the single family
ofmetricsγ =γ(τ)onW. Wewillwrite∂ g +b,∂ γ +β,andB +tr (b)=gijb .
τ τ g ij
We will also assume that X is equipped with a smooth family of connections
∇ˆ =∇ˆ(τ)compatiblewithγ(τ),anduse∇ˆ todenotealsothefamilyofconnections
induced by ∇ and ∇ˆ on tensor products of the bundles TM and T∗M with X.
With this convention, we introduce the elliptic operator (cid:3) + Λij∇ˆ ∇ˆ for some
i j
smooth, symmetric positive definite family of sections Λ = Λ(τ) of T2(M). Here
0
andelsewhereweusetheconventionthatrepeatedLatinindicesdenoteasumfrom
1 to n. Finally, we define the backwardand forward parabolic operators
L +∂ +(cid:3)+∇ Λij∇ , L +∂ −(cid:3)−∇ Λij∇
B τ i j F τ i j
acting on smooth families of sections of X.
2.2. Integral identities. The main ingredients of the proof of Theorem 3 below
aretheintegralidentitiesandinequalitiesinthenexttwolemmas. Exceptforsome
ultimatelyinsignificanterrortermsarisingfromthetime-dependencyofthemetrics
and connections involved, these identities are essentially specific instantiations of
those in Agmon-Nirenberg [AN1, AN2] (see also [Ku1], [Po]).
BACKWARD UNIQUENESS FOR SOME GEOMETRIC EVOLUTION EQUATIONS 3
For τ ∈ [0,Ω], we will let (·,·) denote the L2(dµ )-inner product induced by
g(τ)
γ(τ) on W, that is,
(U,V)+ hU,Vi dµ ,
γ(τ) g(τ)
ZM
andusek·ktodenotetheassociatednorm. Toreducetheclutterinourexpressions,
wewilluseanunadornednorm|·|todenoteallfiberwisenormsinducedbyg andγ
onthevariousbundleswewillencounterbelow;thecontextwilldeterminethenorm
unambiguously. For smooth compactly supported families of sections X = X(τ)
and Y =Y(τ) of X and Y, respectively, we define
E +E(X,Y)+kXk2+kYk2, F +F(X),+ |∇ˆX|2 dµ
Λ
ZM
where |∇ˆX|2 +Λijh∇ˆ X,∇ˆ Xi.
Λ i j
Lemma 1. Suppose X and Y are smooth families of sections of X and Y on
[0,Ω] such that, for all τ, suppX(·,τ) and suppY(·,τ) are contained in some fixed
compact subset of M. Then,
d
E(X,Y)=2F(X)+2(L X,X)+2(∂ Y,Y)+I (X,Y)
B τ 1
(1) dτ
=(L X,X)+(L X,X)+2(∂ Y,Y)+I (X,Y),
B F τ 1
1
(2) F(X)= ((L X,X)−(L X,X)),
F B
2
d 1
(3) F(X)= kL Xk2−kL Xk2 +I (X,∇ˆX),
F B 2
dτ 2
(cid:0) (cid:1)
for all τ ∈[0,Ω], where
B
I (X,Y)+ β(X ⊕Y,X⊕Y)+ (|X|2+|Y|2) dµ,
1
2
ZM(cid:18) (cid:19)
I (X,∇ˆX)+ ∂ Λijh∇ˆ X,∇ˆ Xi+2Λij ∂ ,∇ˆ X,∇ˆ X
2 τ i j τ i j
ZM(cid:18) Dh i E
B
+Λijβ(∇ˆ X,∇ˆ X)+ |∇ˆX|2 dµ.
i j 2 Λ
(cid:19)
Proof. The identities in (1) follow from differentiating under the integral sign and
writing the term involving ∂ X in two ways. Writing
τ
2h∂ X,Xi dµ= 2 L X −(cid:3)X−∇ Λij∇ˆ X,X dµ
τ B i j
ZM ZM D E
and integrating by parts yields the first, while writing
2h∂ X,Xi dµ= hL X +L X,Xi dµ
τ B F
ZM ZM
yields the second.
For equation (2), we simply integrate by parts to obtain
1
F(X)=− (cid:3)X +∇ Λij∇ˆ X,X dµ= (hL X,Xi−hL X,Xi) dµ.
i j F B
2
ZMD E ZM
4 BRETTKOTSCHWAR
For equation (3), we differentiate under the integral sign to obtain
d
F(X)=2 Λij ∇ˆ ∂ X,∇ˆ X +Λij ∂ ,∇ˆ X,∇ˆ X dµ
i τ j τ i j
dτ
ZM (cid:16)D E Dh i E(cid:17)
B
+ ∂ Λijh∇ˆ X,∇ˆ Xi+Λijβ(∇ˆ X,∇ˆ X)+ |X|2 dµ.
τ i j i j
2
ZM(cid:18) (cid:19)
Rewriting the first term as
2 Λij ∇ˆ ∂ X,∇ˆ X dµ
i τ j
ZM D E
1
=−2 ∂ X,(cid:3)X+∇ Λij∇ˆ X = kL Xk2−kL Xk2
τ i j F B
2
then yields the desire(cid:16)d identity. (cid:17) (cid:0) (cid:1) (cid:3)
ThekeystepintheproofofTheorem3backwarduniquenesstheoremistobound
the frequency/energy-quotient quantity F/E, which controls the rate of vanishing
of logE. The following inequality (which follows [AN1], [AN2], [Po]) is the basis of
this bound.
Lemma 2. Suppose that E(τ) > 0 on (a,b) and that X(·,τ) and Y(·,τ) have
support in some fixed compact subset of M for each τ. Then N =F/E satisfies
kL Xk2+k∂ Yk2 I kL Xk2+k∂ Yk2 I
(4) − B τ + ≤N˙ ≤ F τ +
2E E2 2E E2
(cid:18) (cid:19)
where I +I (X,∇ˆX)E −I (X,Y)F.
2 1
Proof. Now, N˙ =(F˙E −E˙F)/E2 and, by (3), we have
1
F˙E = kL Xk2−kL Xk2 kXk2+ kL Xk2−kL Xk2 kYk2 +I E,
F B F B 2
2
(cid:26) (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
while, by (2) and (3), we have
1
E˙F = (L X,X)+(L X,X)+2(∂ Y,Y) · (L X,X)−(L X,X) +I F
F B τ F B 1
2
(cid:26) (cid:27) (cid:26) (cid:27)
1
= (L X,X)2−(L X,X)2 +(∂ Y,Y) (L X,X)−(L X,X) +I F.
F B τ F B 1
2
(cid:26) (cid:27) (cid:26) (cid:27)
Thus, taken together, we have
kL Xk2kXk2−(L X,X)2 (L X,X)2−kL Xk2kXk2
N˙ = F F + B B
2E2 2E2
kL Xk2−kL Xk2 (L X,X)−(L X,X)
(5) + F B kYk2+ B F (∂ Y,Y)
2E2 E2 τ
(cid:26) (cid:27) (cid:26) (cid:27)
I E −I F
2 1
+ .
E2
Now,
1 1
(6) kL Xk2kYk2−(L X,X)(∂ Y,Y)≥− k∂ Yk2kXk2,
F F τ τ
2 2
and
1 1
(7) (L X,X)2+(L X,X)(∂ Y,Y)≥− k∂ Yk2kYk2,
B B τ τ
2 2
BACKWARD UNIQUENESS FOR SOME GEOMETRIC EVOLUTION EQUATIONS 5
thus, since the first term in (5) is nonnegative, we have
(kL Xk2+k∂ Yk2)(kXk2+kYk2) I
N˙ ≥− B τ + ,
2E2 E2
which is the left-hand inequality in (4).
The right-hand inequality in (4) can be obtained in the same way, using the
non-positivity of the second term in (5) and substituting the inequalities
1 1
− kL Xk2kYk2+(L X,X)(∂ Y,Y)≤ k∂ Yk2kXk2,
B B τ τ
2 2
and
1 1
− (L X,X)2−(L X,X)(∂ Y,Y)≤ k∂ Yk2kYk2,
F F τ τ
2 2
in place of (6) and (7), respectively. (cid:3)
2.3. A general backward uniqueness theorem. The following theorem is a
slight generalization of one proven in [Ko1] using Carleman-type estimates. We
now give a simpler proof. The notation ∂ ,∇ˆ denotes the family of sections
τ
∂ ∇ˆ −∇ˆ∂ ∈C∞(T∗M ⊗End(X)). h i
τ τ
Theorem 3 (cf. [Ko1], Theorem 3.1). Assume that (M,g(τ)) is complete for all
τ ∈ [0,Ω], and let r (x) = d (x,x ) for some fixed x ∈ M. Suppose that
0 g(0) 0 0
Λ(τ) ∈ C∞(T2(M)) satisfies the lower bound λgij ≤ Λij for some λ > 0 and,
0
together with g and ∇ˆ, the uniform bounds
(8) sup |b|+|∇b|+|Λ|+|∇Λ|+|∂ Λ|+ ∂ ,∇ˆ +|Rm| ≤L ,
τ τ 0
M×[0,Ω]
n (cid:12)h i(cid:12) o
for some λ, L0 > 0. Then, if the smooth families(cid:12)(cid:12)of sectio(cid:12)(cid:12)ns X(τ) ∈ C∞(X) and
Y(τ)∈C∞(Y) vanish identically for τ =0, satisfy the growth condition
(9) sup e−L1(r0(x)+1){|X|+|∇X|+|Y|}<∞,
M×[0,Ω]
and the system of inequalities
(10) |∂τX +(cid:3)X|≤C0(|X|+|∇X|+|Y|), |∂τY|≤C0(|X|+|∇X|+|Y|).
on M ×[0,Ω], they must vanish identically.
We first give the proof in the technically simpler case that M is compact.
Proof. (Compact Case.) The quantities E = E(X,Y) and F = F(X) are differen-
tiable on [0,Ω] and E(0) = 0. Consequently, the set A = {τ ∈ (0,Ω]|E(τ) > 0}
is open; if A = ∅, there is nothing to prove. Otherwise, writing α = infA, there
is ω > α such that E(τ) > 0 on (α,ω]. Note that, since E(0) = 0, we must have
E(α)=0. On (α,ω], we define N +F/E as in the preceding section.
By Lemma 2, the derivative of N satisfies
kL Xk2+k∂ Yk2 I
(11) N˙ ≥− B τ + ,
2E E2
(cid:18) (cid:19)
where I = I E −I F and I , I are the error terms appearing in Lemma 1. Since
2 1 1 2
M is compact, and k∇ˆXk2 ≤ λ−1F, there is a constant C = C(λ,L ) such that
0
6 BRETTKOTSCHWAR
I ≥ −C(E +F) and I ≤ CE on [0,Ω]. Thus, using (10), for a further enlarged
2 1
C >0 depending on C , we have
0
N˙ ≥−C(N +1).
It follows that N(τ)≤eCω(N(ω)+1)+N for all τ ∈(α,ω].
0
Onthe other hand, using (10) withthe firstidentity in(1) andthe compactness
of M, we have
E˙ ≤CE +2F +kL Xk2+k∂ Yk2 ≤C(E +F)
B τ
[˙
on (α,ω] for some C = C(λ,C ,L ). It follows that logE ≤ C(N +1), whence
0 0 0
E(ω) ≤ E(τ)eC(N0+1)(ω−τ) for any α < τ ≤ ω. Sending τ ց α, we conclude that
E(α)>0, a contradiction. Therefore A=∅ and E(τ)≡0 on [0,Ω]. (cid:3)
2.4. Noncompact case. The crux of the proof in this case is again Lemma 2; we
will just need to cut-off the sections X and Y in space and condition them with a
suitably rapidly decaying function to ensure the finiteness of the various integral
quantities and the validity of the manipulations we need to perform.
First,weobservethat,undertheassumptionsofTheorem3,Rc(g(0))isbounded
below and |b(x,τ)| ≤ L on M ×[0,Ω]. Therefore, the metrics g(τ) are uniformly
0
boundedand,bytheBishop-Gromovtheorem,thereisaconstantV =V (n,L ,Ω)
0 0 0
such that
(12) vol (B (x ,r))≤V eV0r
g(τ) g0 0 0
for any x ∈M, r >0, and τ ∈[0,Ω].
0
Next,werecordaslightextensiontoalemmafrom[ChRF](cf. [Ta])concerning
the existence of smooth distance-like functions with controlled derivatives up to a
certain prescribed order.
Lemma 4. Let m ≥0 and (M,g(τ)) be a smooth family of complete Riemannian
manifolds with ∂ g =b. Assume that there is a constant L such that
τ
m−1 m−2
sup |∇(k)b|≤L, sup |∇(k)Rm|≤L.
M×[0,Ω] k=0 M×[0,Ω] k=0
X X
Then, there exists a smooth function ρ : M → (0,∞) and positive constants C =
1
C (n,L,Ω), C =C (n,m,L,Ω) such that
1 2 2
m
(13) C−1(1+r (x))≤ρ(x)≤C (1+r (x)), sup |∇(k)ρ|≤C .
1 0 1 0 2
M×[0,Ω]k=0
X
Here, as before, r (x)=d (x,x ) for some fixedpoint x of M, and |·|=|·| ,
0 g(0) 0 0 g(τ)
∇=∇ , and Rm=Rm(g(τ)).
g(τ)
Proof. In Section 26.4 of [ChRF] it is shown that, under the assumptions on the
curvature tensor alone, there is a smooth function ρ satisfying the first condition
in (13) and the second condition at τ =0. The uniform bound on |b| then implies
the uniform equivalence of the metrics g(τ) and the bounds on |∇ρ| for all τ.
g(τ)
We can then control |∇(k)ρ| for 1 < k ≤ m inductively with bounds on ∇(l)ρ
g(τ)
and ∇(l)b for l ≤ k − 1, since the latter quantities control the evolution of the
connection. (cid:3)
BACKWARD UNIQUENESS FOR SOME GEOMETRIC EVOLUTION EQUATIONS 7
2.4.1. ProofofTheorem3inthenoncompactcase. Wewilluseacut-offargumentto
boundalocalizedversionofN =F/E withsomeadditionalerror. Ourassumptions
donot,apriori,precludethepossibilitythatX(τ )andY(τ )mayvanishidentically
0 0
onsomeopensetatsomeτ ,sowemustchoosethesizeofthesupportofourcut-off
0
function with a little care.
Proof. Define B + B (x ,R) and let ρ be the function guaranteed by Lemma
R g(0) 0
4 with m = 2. Using ρ we can construct a cut-off function φ ∈ C∞(M,[0,1])
R c
satisfying
φ ≡1 on B , suppφ ⊂B , sup {|∇φ |+|∇∇φ |}≤C .
R R R 2R R R 3
M×[0,Ω]
As we have observedin (12), there exists a constant V such that vol (B )≤
0 g(τ) R
V0eV0R for all τ. Define B =max{L1,V0}, and put
X˜ +e−3BρX, Y˜ +e−3BρY, X˜ +φ X˜, Y˜ +φ Y˜.
R R R R
Then, X˜ (·,0)=0 and Y˜ (·,0)=0 for any R>0,
R R
sup sup e2Bρ |X˜ |+|∇ˆX˜ |+|Y˜ | <C ,
R R R 4
R>0M×[0,Ω]
n o
and
(14) ∂τX˜ +(cid:3)X˜R + ∂τY˜R ≤C(|X˜R|+|∇ˆX˜R|+|Y˜R|)+Ce−2BRχsupp∇φR
(cid:12) (cid:12) (cid:12) (cid:12)
for som(cid:12)e C which de(cid:12)pen(cid:12)ds on(cid:12)n, B, C0, C3 and C4, but is independent of R.
The (cid:12)energy quant(cid:12)ities(cid:12) E +(cid:12)E(X˜,Y˜), F + F(X˜), E + E(X˜ ,Y˜ ), and F +
R R R R
F(X˜ ) are then well-defined and differentiable on [0,Ω]. As before, the set A =
R
{τ ∈ (0,Ω]|E(τ) > 0} is open, and, if it is nonempty, we can find 0 ≤ α < ω ≤ Ω
such that E(τ)>0 on (α,ω] and E(α)=0.
Now fix a ∈ (α,ω). We will continue to use C to denote a sequence of large
constants that may depend on the parameters in the assumptions of Theorem 3
butareindependent ofa andR. Note thatthere is R ≥1suchthatE (τ)>0 for
a R
all τ ∈ [a,ω] provided R ≥ R . Using (14) and that vol (supp∇φ ) ≤ Ce2BR,
a g(τ) R
we have
kL X˜ k2+k∂ Y˜ k2 ≤C(E +F )+Ce−2BR.
B R τ R R R
Using this inequality and controlling I via (8) as in the compact case, Lemma 2
implies that N +F /E satisfies
R R R
N˙ ≥−C(N +1)+Ce−2BRE−1.
R R R
and hence that
d
eCτ(N (τ)+1) ≥−CeCτ−2BRE−1(τ),
dτ R R
on [a,ω] for all R ≥ R(cid:0). Since N (ω) →(cid:1) N(ω) as R → ∞, we have N (ω) ≤ N
a R R 0
for some N and this implies the estimate
0
ω
N (τ)+1≤eC(ω−τ)(N (ω)+1)+Ce−2BR eC(s−τ)E−1(s)ds
R R R
Zτ
ω
≤C(N +1)+Ce−2BR E−1(s)ds,
0 R
Zτ
8 BRETTKOTSCHWAR
for τ and R in the same range. We must control the new error term. Define
Q (τ)+ ωE−1(s)ds, and observe that Q (τ)≤Q (a)<∞ and
R τ R R R
ω
R lim Q (τ)= E−1(s)ds<∞
R
R→∞ Zτ
on [a,ω] for all R ≥ R . Also, define P + N +1+e−2BRQ (a) and note for
a R,a 0 R
later that limR→∞PR,a =N0+1, a finite nonzero limit independent of a.
If we apply the first identity in (1) to E˙ and use (14) again with our above
R
inequality for N˙ , we then have
R
d
e−CPR,aτE (τ) ≤Ce−CPR,aτ−2BR
R
dτ
(cid:26) (cid:27)
on [a,ω] and hence
e−2BR
e−CPR,a(ω−a)E (ω)≤E (a)+ 1−e−CPR,a(ω−a) .
R R
P
R,a
(cid:16) (cid:17)
for all R≥R . Sending R→∞, we obtain
a
e−C(N0+1)(ω−a)E(ω)≤E(a)
which implies E(ω)=0 upon sending aցα. Therefore we must have A=∅. (cid:3)
3. Applications to geometric flows: the prolongation procedure
As we have already mentioned, the lack of short-time existence for backward
parabolic problems means that the problem of backward uniqueness for geometric
flows such as the Ricci flow cannot be simply reduced to one for an associated
strictlyparabolicsystembyastandardimplementationoftheDeTurcktrick. Inour
prior work on the Ricci flow [Ko1], [Ko2], and building on an idea of Alexakis [Al]
(see also [WY]), we demonstrated that, appropriately reformulated, the problem
cannevertheless be attackedby essentially the same methods for strictly parabolic
equations.
Thephilosophybehindthisreformulationisthatthecurvaturetensors(andtheir
covariantderivatives)associatedtothesolutionsofgeometricflowsoftenthemselves
satisfy strictly parabolic equations. Since the elliptic operators in these curvature
equations typically depend in a nonlinear fashion on the solution metrics, the dif-
ference of the curvature tensors associated to two different solutions will not, in
general, satisfy a strictly parabolic equation itself. Nevertheless, it is possible in
many cases to add logically redundant lower-order differences of the metric and
higher-orderdifferences of the covariantderivatives of the curvature tensors to ob-
tain a prolongedsystem which satisfies a closedandeffectively parabolic system of
inequalities. We illustrate this procedure now on three examples: on the Ricci and
cross-curvatureflows in this section, and on the L2-curvature flow in the next.
3.1. The Ricci flow. Assume that g and g˜ are complete solutions of uniformly
bounded curvature to the Ricci flow
∂
(15) g =−2Rc(g)
∂t
on M ×[0,Ω] which agree at t = 0. For our first example, we will review the
construction in Section 2 of [Ko1]. It will be convenient to define τ = Ω−t and
BACKWARD UNIQUENESS FOR SOME GEOMETRIC EVOLUTION EQUATIONS 9
consider g = g(τ) and g˜ = g˜(τ) as solutions to the backward Ricci flow with
g(0)=g˜(0).
ThedifferenceX(k) =∇(k)Rm−∇(k)Rmofthecurvaturetensorswillsatisfyan
equation of the form
(∂ +∆)X(k) =(∆−∆e)∇g(k)Rm+Pk(Rm)−P˜k(Rm)
τ 2 2
wherePk(U)(respectively,P˜k(U))representsalinearcombinationoftermsformed
l l e e g g
by contraction by g (respectively, g˜) from tensor products of the form ∇(i1)U ⊗
∇(i2)U⊗···⊗∇(il)U (respectively, ∇(i1)U⊗∇(i2)U⊗···⊗∇(il)U)wherei1+i2+
···+i =k.
l
The key observation is that, for aell k, theefirst term on tehe right-hand side can
be expressed as a polynomial of the form
(∆−∆)∇(k)Rm=Y(0)∗ T +Y(1)∗ T +Y(2)∗ T
g 0 g 0 g 3
whereY(0) =g−g˜,Y(1) =∇−∇,Y(2) =∇Y(2) andtheT are(bounded)catch-all
i
e g
factors involving Y(1) and ∇(l)Rm for l = 0,1,2,...,k+2. The quantities Y(i),
in turn, for i = 0,1,2, can be ceontrolled in an ordinary differential sense by X(0),
X(1) and ∇X(1). Thus we eaddgX(1), Y(0), Y(1), and Y(2) to our original system
consisting of X(0) to obtain a closed system of inequalities.
This is made precise in [Ko1], where it is shown that, if two solutions g and g˜
agree at τ = 0 then, for each δ > 0, the family of sections X = (X(0),X(1)) and
Y =(Y(0),Y(1),Y(2))ofX +T1(M)⊕T1(M)andY +T0(M)⊕T1(M)⊕T1(M),
3 4 2 2 3
together satisfy the system
|∂ X +∆X|≤C (|X|+|Y|), |∂ Y|≤C (|X|+|∇X|+|Y|)
τ δ τ δ
onany[0,Ω−δ]. Herethedependence oftheconstantsC onδ is anartifactofthe
δ
use ofShi’s estimates [Shi]to bound the coefficients (andcanbe eliminatedif M is
compact). These coefficients depend on the derivatives of the curvature tensors of
thesolutionsofuptothird-order,andundertheassumptionofboundedcurvature,
Shi’s estimates ensure that these derivatives are bounded uniformly for t ∈ (δ,Ω],
that is, for τ ∈ [0,Ω−δ). To prove that g ≡ g˜, we apply Theorem 3 to the above
system on M ×[0,Ω−δ] for any fixed δ, and then send δ ց 0. We obtain the
following result.
Theorem 5 (Theorem 1.1, [Ko1]). If g and g˜ are complete solutions to (15) with
uniformly bounded curvature, then g(Ω)=g˜(Ω) implies that g ≡g˜ on [0,Ω].
In[Ko3],weprovethatacompletesolutiontotheRicciflowofboundedcurvature
is in fact analytic in time for t>0 (see also [Sha] for a related result).
3.2. Cross-curvature flow. On a closed Riemannian three-manifold (M3,g) of
sectionalcurvatureofnegative(respectivelypositive)sign,theEinsteintensorE =
ij
R −(R/2)g is negative (respectively positive) definite. Write Eij = gikgjlE .
ij ij kl
If V = V is the two-tensor satisfying V Ekj = δj and P = det(Ei), the cross-
ij ik i j
curvature tensor of g is defined to be the tensor
Ξ +PV =−(1/2)EpqR .
ij ij pijq
For such manifolds, Hamilton and Chow [CH] define the cross-curvature flow as
the evolution equation
(16) ∂ g =−2σΞ(g),
t
10 BRETTKOTSCHWAR
for a family of metrics g(t) where σ = ±1 depending as the sectional curvature of
g(0) is positive or negative. Short-time existence for the flow is proven in [Bu].
As an equation for the metric, (16) is fully nonlinear, however, the parabolic
portion of the prolonged system we derive will be effectively quasilinear. For our
purposes,thekeyisthatthecurvaturequantitiesassociatedtothesolutionssatisfy
strictlyparabolicequationswith respectto the family ofstronglyelliptic operators
(cid:3) + −σEij∇ ∇ . (Since ∇ Eij = 0, these operators are even self-adjoint.) In
i j i
particular (see Lemma 17 of [Ko4]), the tensor V satisfies the equation
∂ V =(cid:3)V +(EalEkb−2EabEkl)∇ V ∇ V −Pgkl(V V +V V ).
t ij ij k ai l bj ik jl ij kl
with analogous equations for its covariantderivatives ∇(k)V.
Neglecting the constant σ, the difference of the operators (cid:3) and (cid:3) satisfy a
formula of the form
((cid:3)−(cid:3))W =Eab∇ (∇ −∇ )W +Eab(∇ −∇ )∇ W +(Eab−E˜aeb)∇ ∇ W
a b b a a b a b
=Y(1)∗T(cid:16) +Y(2)∗T +(cid:17)X(0)∗T
e 1 e 2 3 e e e e
where X(0) + V −V˜, Y(1) + ∇−∇, Y(2) + ∇Y(1) and the T denote bounded
i
tensors depending on g, g˜ and W. (For the third term on the right, we have used
the identity Eab−E˜ab =−EakE˜blVe.)
kl
Defining X(1) + ∇V −∇V˜ and Y(0) + g −g˜, and carrying out computations
either containedin orsimilar to those inour paper[Ko4], one canverifythat there
is a constant C such that Xe=(X(0),X(1)) and Y =(Y(0),Y(1),Y(2)) satisfy
|∂ X +(cid:3)X|≤C(|X|+|∇X|+|Y|), |∂ Y|≤C(|X|+|∇X|+|Y|)
τ τ
on M ×[0,Ω]. The following theorem is thus a consequence of Theorem 3 above,
taking Λij =−σEij.
Theorem 6. If g, g˜ are solutions to (16) on M3 ×[0,Ω] with either positive or
negative sectional curvature and g(Ω)=g˜(Ω), then g(t)=g˜(t) for all t∈[0,Ω].
4. The L2-curvature flow and higher-order equations
The L2-curvatureflow,introducedbyJ.Streets in[St1],is thenegativegradient
flow of the functional g 7→ |Rm(g)|2dµ . The metric evolves according to the
M g
fourth-order equation
R
1 1
(17) ∂ g =2 ∆R − ∇ ∇ R +2RpqR −RpR +RpqrR − |Rm|2g ,
t ij ij 2 i j ipqj i pj i jpqr 4 ij
(cid:18) (cid:19)
and the curvature tensor and its covariant derivatives satisfy a strictly parabolic
equation of the form
(18) ∂ ∇(k)Rm=−∆(2)∇(k)Rm+Pk+2(Rm)+Pk(Rm).
t 2 3
See Proposition 4.3 in [St1]. For our purposes, the key aspect of this equation is
thatits principalpart,the tensorbi-Laplacian,isasymmetricoperator. Moreover,
thefollowingresultofStreets[St2]guaranteestheinstantaneousboundednessofall
derivatives of the curvature under the assumption of a uniform curvature bound.
Theorem ([St2], Theorem 1.3). If (Mn,g(t)) is a complete solution to (17) with
|Rm|≤K on M ×[0,Ω], then, for any m≥1,
(19) sup tm4+2|∇(m)Rm|g(t) ≤C
M×[0,Ω]