Table Of ContentAn End to End Gluing Construction for Metrics of Constant
Positive Scalar Curvature
2
Jesse Ratzkin
0
0
2 February 1, 2008
n
a
J
1 Introduction
7
1
The goalof this paper to to describe a generalprocess by which one can glue together metrics of
] constant positive scalar curvature on punctured spheres along their ends to obtain new metrics
G
of constant positive scalar curvature.
D
First let (M = Sn\{p ...p },g ) and (M = Sn\{q ...q },g ) be complete metrics
1 0 k1−1 1 2 0 k2−1 2
. ofscalarcurvaturen(n−1). ThesemetricsareasymptotictoDelaunaymetricsinsmall(standard
h
at spherical)punctured balls aboutpj and qj respectively. We will refer to these punctured balls as
the ends of M and M . The Delaunay metrics can be written as
m 1 2
[ 4
un−2(t+T)(dt2+dθ2)
ǫ
1
v where u is a periodic function which assumes its minimal value ǫ (called the necksize of the
ǫ
6 metric) at t = 0. These metrics are uniquely determined by their singular set on Sn, necksize
6
and the translation parameter T. Assume that we choose g and g such that the asymptotic
1 1 2
1 necksize of g1 at p0 is equal to the asymptotic necksize of g2 at q0; we will call this common
0 necksize ǫ. Then we can truncate M and M by removing small spherical balls around p and
1 2 0
2 q and patch these two metrics together at a neck (value of t where u achieves its minimum) to
0 ǫ
0 obtain a new metric g˜ on M =Sn\{p ...p ,q ...q }. This construction depends on two
/ 1 k1−1 1 k2−1
h parameters: R,whichwecanthinkofasthesizeoftheballsweexcisedinthetruncationprocess,
t andφ∈SO(n)whichspecifiesarotationintheSn−1 factorofthe secondsummand. Noticethat
a
m the parameterR is discrete. To indicate the dependence of g˜onthese parameterswe will denote
it as g˜ . Much of the analysis is independent of at least one of these parameters, and in this
: R,φ
v case we will suppress the appropriatesubscript. We will construct this metric, which we will call
i
X an approximate solution (because its scalar curvature is very close to n(n−1)), in section 4.
Themetricg˜ doesnothaveconstantscalarcurvature,butthedeviationψ =n(n−1)−S˜
r R,φ R,φ
a isgloballysmall. Moreprecisely,withoutanymodificationtog andg ,ψiscompactlysupported
1 2
and kψk = O(e−R). After modifying g and g by conformal transformations, we can
C0,α(M) 1 2
further arrangethat kψkC0,α(M) =O(e−γn+1(ǫ)R) where γn+1(ǫ) is a coefficient we will discuss in
section3. We wishto deformg˜ by a conformalfactorto obtainametric with scalarcurvature
R,φ
n(n−1). Recall how the scalar curvature transforms under a conformal change of metric: if
g′ =un−42g then
Sg′ =Sgu−n−42 − 4(n−1)u−nn−+22∆gu,
n−2
which we can rewrite as
n−2 n−2 n+2
∆gu− Sgu+ Sg′un−2 =0. (1)
4(n−1) 4(n−1)
1
If we normalize the scalar curvatures by setting Sg =n(n−1)−ψ and Sg′ =n(n−1) the above
equation becomes
n(n−2) n−2 n(n−2) n+2
∆gu− u+ (ψ/4)u+ un−2 =0.
4 n−1 4
(cid:18) (cid:19)
The linearized equation (linearized about u=1) is
n−2
L (u)=∆ u+nu+ (ψ/4)u=0. (2)
g g
n−1
(cid:18) (cid:19)
We will call L the Jacobi operator associated to g and solutions L u = 0 Jacobi fields of g.
g g
Notice that L = ∆ +n if ψ = 0, i.e. if we wish to deform a constant scalar curvature metric
g g
into another conformal constant scalar curvature metric. In the present case L is a small
g˜R,φ
perturbation of ∆ +n.
g˜R,φ
Our first step is to show that we can solve this linearized equation, with uniform (in R)
estimatesonthesizeofthesolutionoperator. Onecanalsothinkofthisstepasfindingapositive
lower bound on the spectrum of L as an operator between appropriate function spaces. We
g˜R,φ
will address precisely which function spaces are the proper ones for this problem in section 5.1.
Inordertoprovethiswewillneedtoassumethat(M ,g )arebothunmarkednondegenerate;i.e.
i i
there are no Jacobi fields which decay at a rate faster than e−tj on all ends. We will also need
to assume that one can adjust the necksize of the end corresponding to p in the moduli space
0
of constant scalar curvature metrics (see the statement of the theorem for a precise statement of
this condition). We need this last condition to exclude certain Jacobi fields which we could glue
together to yield an exponentially small eigenvalue. This linear analysis will occupy section 5.
Then in section 6.1 we will explicitly write down geometric deformations of the metric asso-
ciated to a parameter u ∈ Rk(2n+2), where k = k +k −2 is the number of ends of M. One
1 2
canthink ofthese deformationsasadjusting the necksizesandpositionofthe necksofthe metric
on the ends of M. Finally, in section 6.2 we will use the solution operator we found and these
geometric deformations to solve the nonlinear problem via the Contraction Mapping principle.
This will yield the existence part of the following theorem.
Theorem 1 Let (M = Sn\{p ...p },g ) and (M = Sn\{q ,...q },g ) be complete
1 0 k1−1 1 2 0 k2−1 2
metrics with scalar curvature n(n−1). Assume g and g are unmarked nondegenerate and that
1 2
the asymptotic necksizes associated to p and q are both ǫ. Assume also that there is a one-
0 0
parameter family of scalar curvaturen(n−1) metrics g on M , t∈(−δ,δ), where the asymptotic
t 1
necksize of g associated to p is ǫ+t. Then for η >0 there is an R such that for R≥R one
t 0 0 0
can deform the approximate solution (M ,g˜ ) first by a geometric parameter u with |u|<η and
R R
4
then by a conformal factor (1+v)n−2 (with v exponentially decaying) to obtain a metric with
scalar curvature n(n−1). Moreover, this metric is unmarked nondegenerate.
We first remark that the connect sum of two Delaunay metrics constructed in [MPU1] and
all the metrics constructed by Byde in [B] and by Mazzeo and Pacard in [MP] satisfy all the
hypotheses of this theorem. One particular application of this gluing construction is to take
(M ,g ) and (M ,g ) to be isometric and attach M to M along isometric ends. We will call
1 1 2 2 1 2
this construction doubling along an end.
One can think of this theorem as the scalar curvature analogueof a similar end to end gluing
construction for surfaces of constant mean curvature in Euclidean space (see [R] and [MPPR]).
In fact, most of the analysis is the same for the two constructions. This phenomenon has been
widely noted (compare, e.g., [MPU2] and [KMP]), but not completely explained.
This theoremis alsoverymuchinthe spiritofthe resultsofSchoenin[S], ofMazzeo,Pollack
andUhlenbeckin[MPU1]andofJoycein[J]. Inallcasesoneconstructsanapproximatesolution
to the gluing problem, solves the linearized equation with uniform estimates, and solves the
2
nonlinear problem using a fixed point theorem or an iteration technique. The constructions of
Mazzeo and Pacard in [MP] and of Byde in [B] are similar in spirit, but use a different method,
in that they solve boundary value problems on appropriate subdomains and then match Cauchy
data.
I would like to thank F. Pacard for suggesting this problem. I would also like to thank D.
Pollack, R. Mazzeo and F. Pacard for many useful suggestions as I was learning this subject.
2 Notation
In this section we establish some notation for the rest of the paper.
2.1 Notation for Delaunay Metrics
First we consider the Delaunay metrics. These can be written as
4
g =un−2(t+T)(dt2+dθ2)
ǫ,T ǫ
on R×Sn−1. In the case T =0 we will suppress it from the subscript. The function u satisfies
ǫ
the ordinary differential equation
u′′− (n−2)2u+ n(n−2)unn−+22 =0.
4 4
From this ODE one can show that u is a periodic function uniquely determined by its minimal
ǫ
value ǫ (once we normalize u so it achieves its minimum at t = 0). We will denote the period
ǫ
of u by T . As we will see in section 3, solutions of L (u) = 0 which lie outside a specific
ǫ ǫ gǫ
two-dimensionalspacesatisfy a boundwhich (upto the changeofvariablest7→−t)we canstate
as
≤ cet t<0
|u(t,θ)|
≥ ce−t t>0.
(cid:26)
For more discussion about the solutions to L u=0, see section 3.
gǫ
2.2 Notation for Everything Else in this Paper
Recallthatwearestartingwith(M =Sn\{p ,...,p },g )and(M =Sn\{q ,...,q },g )
1 0 k1−1 1 2 0 k2−1 2
two complete metrics with scalar curvature n(n−1). We will assume that the p are mutually
j
disjoint and that the qj are mutually disjoint (we will allow, however, pj = qj′ for some j and
j′). Let r be small enough so that the discs B (p ) in the usual round metric are pairwise
0 r0 j
disjoint, and also so that the discs B (q ) in the usual round metric are pairwise disjoint. Let
r0 j
Mc = Sn\(∪B (p )) and Mc = Sn\(∪B (q )). Next fix two cutoff functions χ and χ such
1 r0 j 2 r0 j 1 2
that
0 p∈Mc
χ (p)= 1
1 1 p∈B (p )\{p }
(cid:26) r0/2 j j
and
0 p∈Mc
χ (p)= 2
2 1 p∈B (q )\{q }.
(cid:26) r0/2 j j
InsideB (p )letr (p)be the distanceinthe sphericalmetric top andlett =−log(r /r ).
r0 j j j j j 0
Similarly,inB (q )letρ (p)bethedistanceisthesphericalmetrictoq andletτ =−log(ρ /r ).
r0 j j j j j 0
Then with respect to these coordinates the asymptotics theorem (see [CGS] or [KMPS]) states
that we can write the metric g in B (p ) as
1 r0 j
g1 =(u1,j +uǫj(·+Tj))n−42(tj,θj)(dt2j +dθj2)
3
where
ku k =O(e−tˆ)
1,j C2,α((tˆj−1,tˆj+1)×Sn−1)
for tˆ ≥1. Similarly, with respect to the coordinates (τ ,θ ), one can write
j j j
g2 =(u2,j +uǫ′j(·+Tj′))n−42(τj,θj)(dτj2+dθj2)
where
ku2,jkC2,α((τˆj−1,τˆj+1)×Sn−1) =O(e−τˆj)
forτˆ ≥1. Wewillseelaterthatwecanimprovetheseestimatesonu andu usingconformal
j 1,0 2,0
transformationsofSn. We willassume thatǫ =ǫ′ =ǫ is fixedthroughoutthe restofthe paper.
0 0
3 Delaunay Metrics
In this section we will discuss some of the important features of the Sn−1 invariant, complete,
scalar curvature n(n−1) metrics on R×Sn−1, which are known as Delaunay metrics. Most
importantly, we discuss the spectral behavior of the operator L = ∆ + n where g is a
gǫ gǫ ǫ
Delaunay metric.
Recall that we can write the Delaunay metrics as
4
g =un−2(t)(dt2+dθ2)
ǫ ǫ
where u solves the ordinary differential equation
ǫ
u′′− (n−2)2u+ n(n−2)unn−+22 =0.
4 4
We remark that solutions to this ODE exist for all time because the equation has a conserved
energy
(n−2)2 (n−2)2 2n
H =(u′)2− u2+ un−2,
ǫ 4 ǫ 4 ǫ
which would become unbounded if u were to become unbounded. These metrics are uniquely
ǫ
determined by their singular set on Sn, the minimum value ǫ of the conformal factor and a
translation parameter T. In fact, varying either parameter yields a one-parameter family of
Delaunay metrics. Taking the derivative of this one parameter family, we obtain two linearly
independent solutions to the Jacobi equation:
(∆ +n)v0,± =0.
gǫ ǫ
More precisely, we can write
d d
v0,+ = u (·+T) v0,− = u .
ǫ dT ǫ ǫ dη ǫ+η
(cid:12)T=0 (cid:12)η=0
(cid:12) (cid:12)
(cid:12) (cid:12)
From this construction we see that(cid:12)v0,± are independent of θ, a(cid:12)nd thus they satisfy the ordinary
ǫ
differential equation
u′ 4
(v0,±)′′+2 ǫ(v0,±)′+nun−2v0,± =0.
ǫ u ǫ ǫ ǫ
ǫ
We will normalize v0,± by choosing the initial conditions
ǫ
v0,+(0)=1 v0,−(0)=0 (v0,+)′(0)=0 (v0,−)′(0)=1.
ǫ ǫ ǫ ǫ
4
Indeed, if we try to separate variables for a general solution v of
∆ v+nv =0,
gǫ
we find that
v(t,θ)= v (t)η (θ)
j j
where ηj is the jth eigenfunction of ∆Sn−1 witXh eigenvalue λj (counted with multiplicity) and vj
satisfies the ordinary differential equation
u′ 4
v′′+2 ǫv′ +(n−λ )un−2v =0. (3)
j u j j ǫ j
ǫ
The functions v0,± form a basis for the solution space to this ODE when j = 0. We will again
ǫ
choose a normalized pair of solutions to this ODE, vj,+ and vj,−, normalized so that
ǫ ǫ
vj,+(0)=1 vj,−(0)=0 (vj,+)′(0)=0 (vj,−)′(0)=1.
ǫ ǫ ǫ ǫ
In fact, we can also find the a basis for the solution space when j =1,...,n, again by taking
explicit geometric deformations of the metric. To find these deformations, we use stereographic
projection to write the Delaunay metric as
4
uˆn−2dx2
ǫ
where dx2 is the standard Euclidean metric on Rn and u > 0 has a singularity at the origin.
Now we can deformthis metric by taking translates a7→uˆ (·+a), so we obtain a Jacobifield by
ǫ
pulling back
d n−2
uˆ (·+a)=e−t(−vˆ0,+(t)+ uˆ (t))η (θ)
da ǫ ǫ 2 ǫ j
j(cid:12)a=0
(cid:12)
whereηj isthejtheigenfun(cid:12)(cid:12)ctionof∆Sn−1 (see[KMPS]). Noticeinparticularthatthesefunctions
all decay like e−t. To find the other solution to equation (3), we first take the Kelvin transform
of uˆ (x) 7→|x|2−nuˆ (x/|x|2), translate as before and then take the Kelvin transform again. One
ǫ ǫ
canthink ofthisdeformationasatranslationatinfinity. Also,onecanshowthatthe Jacobifield
associated to this deformation grows like et (again, see [KMPS]).
At this point we introduce the indicial roots of L = ∆ +n, denoted γ (ǫ). These are the
gǫ gǫ j
exponentialgrowthratesofthesolutionstoequation(3). Fromtheabovecomputations,onesees
that γ (ǫ) = 0 and γ (ǫ) = 1 for j = 1,...,n. By the maximum principle, γ (ǫ) > 1 for j > n.
0 j j
4
Indeed, λ ≥ 2n for j ≥ n+1, so the zero order term in this ODE is (n−λ )uˆn−2 ≤ −n. It
j j ǫ
is rather remarkable that one can compute γ (ǫ) for j = 0,...,n and that they are independent
j
of ǫ, but the other indicial roots are quite hard to compute and probably depend on ǫ in some
nontrivial way.
Anotherwaytorecovertheindicialrootsistoconjugatetheoperator∆ +nbyanexponential
gǫ
function eδt and the Fourier-Laplace transform. Then one obtains a one (complex) parameter
family of operators on a fixed function space, which varies analytically with the parameter. By
the Analytic Fredholm Theorem, this family of operators has a meromorphic solution operator,
and the indicial roots turn out to be the imaginary parts of the poles of this solution operator.
In fact, they show that any solution u to
L u=0
gǫ
has an asymptotic expansion
u(t,θ)∼ (a vj,+(t)+a vj,−(t))η (θ)
j,+ ǫ j,− ǫ j
j≥0
X
5
where ηj is the jth eigenfunction of ∆Sn−1 (counting multiplicity) and vj are the particular
solution of equation (3) listed above. In particular,
|vj,±(t)|=O(e±γj(ǫ)t).
ǫ
See [MPU2] for more about this approach.
A more thorough explanation of the indicial roots in the mean curvature setting occur in
[MPPR], including an explanation of why they are called “indicial roots.”
To sum up this discussion:
• v0,+ is bounded and periodic and arises from translating the neck of the Delaunay metric
ǫ
towards the singularity
• v0,− is linearly growing and arises from changing the necksize of the Delaunay metric
ǫ
• vj,± grow/decaylike e∓t for j =1,...,n andbotharisefromtranslatingthe singularsetof
ǫ
the Delaunay metric
• in fact, any solution u to L u = 0 which is L2 orthogonal to v0,± on Sn−1-cross-sections
gǫ ǫ
hasanexpansionu(t,θ)∼ j≥1vj(t)ηj(θ)where|vj(t)|=O(e±γj(ǫ)t)withγ1(ǫ)=γ2(ǫ)=
···= γ (ǫ)=1 and 1< γ (ǫ)≤ γ (ǫ)≤···→∞ (we have to exclude the v0,± terms
n nP+1 n+2 ǫ
because one of them grows linearly).
One can find rigorous proofs of the above facts in [MPU2], [MP] and [KMPS].
4 The Approximate Solution
In this section we construct the approximate solution g˜ .
R,φ
First we choose some R=mT for some positive integer m and φ∈SO(n) and define M by
ǫ0
M =(M1\Br0e−(T0+R+1)(p0))∪(M2\Br0e−(T0′+R+1)(q0))/∼
where we identify (t ,θ ) with (τ ,φθ ) if t ≥ T +R−1 and τ ≥ T′ +R−1 and t +τ =
0 0 0 0 0 0 0 0 0 0
T +T′+2R. TheballsB (p )andB (q )areballsinthestandardroundmetric. WewillletC
0 0 r 0 r 0 R
bethecylinder{(t ,θ ):T +R−1≤t ≤T +R+1}∼{(τ ,θ ):T′+R−1≤τ ≤T′+R+1}.
0 0 0 0 0 0 0 0 0 0
We will also find it convenient in the following sections to define the extended cylinder
CˆR =(Br0e−T0(p0)\Br0e−T0−R+1(p0))∪CR∪(Br0e−T0′(q0)\Br0e−T0′−R+1(q0)),
parameterized by (t,θ) ∈ [−R,R]×Sn−1. The relationship between t and t or τ is given by
0 0
t=t −R−T for t<0 and t=−τ +R+T′ for t>0. This relationship for between t and t
0 0 0 0 0
and τ agrees with the identification of (t ,θ) with (τ ,φθ) in C listed above.
0 0 0 R
Now we will define the metric g˜ . First pick a cutoff function χ on M such that
R,φ
χ(p)= 1 p∈M1\Br0e−(T0+R−1)(p0)
(cid:26) 0 p∈M2\Br0e−(T0′+R−1)(q0).
We define the metric g˜R,φ by letting g˜R,φ = g1 on M1\Br0e−(T0+R−1)(p0), letting g˜R,φ = g2 on
M2\Br0e−(T0′+R−1)(q0) and by letting
g˜R,φ(t0,θ0)=(uǫ(T0+t0)+χ(t0,θ0)u1,0(t0,θ0)+(1−χ(t0,θ0))u2,0(T0′+T0+2R−t0,φθ0))n−42(dt20+dθ02)
on C . The analysis below will often be independent of at least one of the parameters R and φ;
R
in this case we will suppress the appropriate subscripts.
6
We denote the scalar curvature of g˜ by S˜ . Outside of C , g˜ is either g or g , and
R,φ R,φ R R,φ 1 2
so S˜ =n(n−1) is these regions. A priori, we also have
R,φ
ku k =O(e−R),
1,0 C2,α(CR)
(andasimilarestimateforu ). However,wecanadjustg andg byconformaltransformations
2,0 1 2
as follows. The term u has an asymptotic expansion near p as
1,0 0
u ∼ (a vj,+(t )+a vj,−(t ))η (θ ). (4)
1,0 j,+ ǫ 0 j,− ǫ 0 j 0
j>1
X
The functions vj,± correspond explicitly to translations of the origin or infinity once under the
ǫ
stereographicprojection which sends p to infinity. So change g by the conformalmotion which
0 1
translates the origin by (−a ,...,−a ) and then by the conformal motion which translates
1,+ n,+
infinity by (−a ,...,a ). Thishas the effectofeliminatingthe firstn termsinthe expansion
1,− n,−
(4), and so the new metric, which we will still call g , has an expansion of the form (u +
1 ǫ
u1,0)n−42(dt20+dθ02) where now
u ∼ (a vj,+(t )+a vj,−(t ))η (θ ),
1,0 j,+ ǫ 0 j,− ǫ 0 j 0
j≥n+1
X
and so
ku1,0kC2,α(CR) =O(e−γn+1(ǫ)R).
We can perform as similar adjustment to g2 so that ku2,0kC2,α(CR) = O(e−γn+1(ǫ)R). Notice we
canonlydothisadjustmentforoneendofeachoftheM . Thegeometriceffectofthisadjustment
i
istotranslatethep andq (forj ≥0)aroundsoastomakethemetricsg andg nearp andq
j j 1 2 0 0
(respectively)closerto being Delaunay metrics. The aboveestimates imply the followinglemma.
Lemma 2 When (M,g˜ ) is defined as above, S˜ = n(n − 1) − ψ where ψ is compactly
R,φ R,φ
supported and kψkC0,α(M) =O(e−γn+1(ǫ)R).
Proof: WeonlyneedtoestimateS˜ inC . Tothisend,wefirstrewritetheconformalfactor
R,φ R
onC asu (T +t )+χu (t ,θ )+(1−χ)u (T′+T +2R−t ,φθ )=u (T +t )(1+v(t ,θ )).
R ǫ 0 0 1,0 0 0 2,0 0 0 0 0 ǫ 0 0 0 0
If we plug 1+v into equation (1), we find
n−2 ψ n+2 n(n−2) n(n−2) n+2
n−1 4(1+v)n−2 = ∆gǫ(1+v)− 4 (1+v)+ 4 (1+v)n−2
(cid:18) (cid:19)
= ∆ (v)+nv+O(kvk2 ).
gǫ C2,α
The lemma now follows from the above bounds on u and u , which imply similar bounds on
1,0 2,0
v. (cid:4)
One way to rephrase the result of this lemma is to say that one can write the metric g˜
R,φ
restricted to Cˆ (in the (t,θ) coordinates) as
R
g˜R,φ =(uǫ(t)+v(t,θ))n−42(dt2+dθ2)
where
coshγn+1(ǫ)t
|v(t,θ)|=O .
coshγn+1(ǫ)R!
Because ofthe abovebounds onS˜ we will callthe metric g˜ anapproximatesolutionto our
R,φ R,φ
problem.
7
5 Linear Analysis
In this section we will develop the necessary linear analysis to find a uniformly bounded solution
operatorfortheJacobioperatorL . Westartthesectionbyrecallingsomeofthelinearanalysis
g˜R,φ
forconstantpositivescalarcurvaturemetricsin[MPU2]andthenweconstructasolutionoperator
for L .
g˜R,φ
5.1 Linear Analysis for General Constant Scalar Curvature Metrics on
Punctured Spheres
ThegrowthpropertiesforsolutionsofL (u)=0outlinedabovemotivatetheuseofthefollowing
gǫ
function spaces.
Definition 1 On (M ,g ) we define Cl,α(M ) to be the space of functions such that the norm
i i δ i
kukCδl,α(Mi) =kukCl,α(Mic)+0≤mj≤akx1−1tˆj≥seu−pr0+1ke−δtjukCl,α((tˆj−1,tˆj+1)×Sn−1)
is finite. There is a similar definition for (M ,g ). For the approximate solution (M,g˜ ) we
2 2 R,φ
will need to adjust this definition as follows. Recall that we can write
M =Mc∪(∪k1−1B (p )\{p })∪Mc∪(∪k2−1B (q )\{q })∪Cˆ .
1 1 r0 j j 2 1 r0 j j R
Then we define Cl,α(M) to be the space of functions such that the norm
δ
kukCδl,α(M) = kukCl,α(M1c)+kukCl,α(M2c)+1≤mj≤akx1−1tˆj≥seu−pr0+1ke−δtjukCl,α((tˆj−1,tˆj+1)×Sn−1)
coshδR
+1≤mj≤akx2−1τˆj≥seu−pr0+1ke−δτjukCl,α((τˆj−1,τˆj+1)×Sn−1)+|tˆ|s≤uRp−1k coshδt ukCl,α([tˆ−1,tˆ+1]×Sn−1)
is finite. We also say (M ,g ) is unmarked nondegenerate if L u = 0 does not admit solutions
i i gi
u∈C2,α(M ) for any δ <−1.
δ i
FunctionsinCδl,α(Mi)cangrowatmostlikeeδtj ontheendEj. Weremarkthatforδ ≤1the
onlysolutionsof∆ v+nv=0withv ∈C2,α(R×Sn+1)arelinearcombinationsofvj,+ andvj,−
gǫ δ ǫ ǫ
for j =0,...,n, each of which is either bounded and periodic or unbounded on at least one end.
So the Delaunay metrics are unmarked nondegenerate. We also remark that the function space
Cl,α(M) is the same space of functions as if we had not weighted the middle cylinder, but it has
δ
a different norm. This difference in norms will become important later when we want uniform
bounds on a solution operator. We further remark that the notion of unmarked nondegenerate
is weaker than the notion of marked nondegenerate, which requires that L u=0 does not have
gi
any solutions where u∈C2,α(M ) with δ <0.
δ i
In order to find a function space on which L has suitable mapping properties we will need
gi
the following definition.
Definition 2 The deficiency space W of (M ,g ) is the span of all the functions χ vi,±, where
gi i i i ǫj
1≤j ≤k (so the sum runs over all the ends of M ) and 0≤i≤n.
i i
Notice that W is a vector space of dimension k (2n+2) and it has a basis {χ vi,±} which only
gi i i ǫj
dependsonthemetricg andthechoiceofcutofffunctionχ . WewillusethisbasistogiveW the
i i gi
Euclidean norm. We also remark that this is the proper deficiency space to use to parameterize
the unmarked moduli space, meaning that one fixes the cardinality but not the position of the
singular set. For the marked moduli space (fixing both the cardinality and the position of the
singular set) one should work with a smaller deficiency space which only incorporatesthe Jacobi
fields arising from translating the necks of the Delaunay metrics along their axes and changing
their necksizes.
8
Remark 1 LetE beanendofM correspondingtothepuncturepointp . Itturnsoutthewecan
j 1 j
use particular conformal Killing fields on the sphere to show that for any end E there is always
j
a Jacobi field of g which is asymptotic to v0,+ along E . To see this, consider stereographic
i ǫj j
projection sending p to ∞ composed with a dilation about the origin. This provides a one-
j
parameter family of scalar curvature n(n−1) metrics on M which translate the Delaunay neck
1
on E . Taking the infinitesimal generator of this family we obtain a Jacobi field asymptotic to
j
v0,+. Similar Jacobi fields exist on M . These Jacobi fields are also in C2,α(M ). This seems
ǫ 2 1 i
to be a special property of spheres, as one cannot in general find such conformal Killing fields
on arbitrary compact manifolds with positive scalar curvature. In the mean curvature case the
corresponding Jacobi fields arise from global translations of the surface.
A similar Linear Decomposition result to the one stated below appears as Lemma 4.18 of
[MPU2], as stated for weighted Sobolev spaces and exactly constant scalar curvature metrics.
The result below is essentially the next term in the asymptotic expansion; see Proposition 4.15
of[MPU2]. The prooffor weightedHo¨lder spacesis nearlyidentical andreallyonly requiresthat
the ends are asymptotically Delaunay.
Proposition 3 (Mazzeo, Pollack, Uhlenbeck, 1996) Let δ ∈ (1,infγ (ǫ )). If u ∈ C2,α(M ),
n+1 j δ i
f ∈C0,α(M ) and L u=f then u∈W ⊕C2,α(M ).
−δ i gi gi −δ i
Suppose g is a unmarked nondegenerate metric on M . Then for δ ∈(1,inf(γ (ǫ ))
i i n+1 j
L :C2,α(M )→C0,α(M )
gi −δ i −δ i
is injective, which in turn implies
L :C2,α(M )→C0,α(M )
gi δ i δ i
is surjective. If we combine this with the Linear Decomposition result in proposition 3 then we
see that
L :W ⊕C2,α(M )→C0,α(M )
gi gi −δ i −δ i
is surjective. We will call the kernel of this map B , the bounded null space of L . Mazzeo,
gi gi
Pollackand Uhlenbeck ([MPU2]) show that if M has k ends and g is unmarked nondegenerate
i i i
then B is k (n+1)-dimensional(in generalB couldcontaina spaceof exponentially decaying
gi i gi
functions of some unknown dimension). From this reasoning one can see (using the Implicit
FunctionTheorem)thatnearanunmarkednondegeneratepointthemodulispaceofsuchmetrics
has the structure of a real analytic manifold of dimension k (n+1).
i
5.2 Solvability of the Linear Problem
To construct the deficiency space W we take cutoffs of the Jacobi fields vi,±from the model
g˜R,φ ǫ
Delaunay metrics arising from p ...p and q ,...q . Notice we do not include p and q .
1 k1−1 1 k2−1 0 0
Again, we will use the basis formed by {χ vi,±,χ vi,±}, which induces the Euclidean norm on
1 ǫj 2 ǫ′j
W .
g˜R,φ
Recall that the Jacobi operator L is given by
g˜+R,φ
n−2 ψ
L =∆ +n+ ,
g˜R,φ g˜R,φ n−1 4
(cid:18) (cid:19)
whichisa perturbationof∆ +nwhere the perturbationiscompactlysupportedandglobally
g˜R,φ
of size O(e−γn+1(ǫ)R).
9
Proposition 4 Suppose both g are unmarked nondegenerate and there exists a one-parameter
i
family of scalar curvature n(n−1) metrics g on M such that the asymptotic necksize of the
t 1
end at p with respect to to g is ǫ+t. Then for δ ∈ (1,inf{γ (ǫ ),γ (ǫ′)}) there exists an
0 t n+1 j n+1 j
R >0 such that for R≥0 one can find an operator
0
G :C0,α(M)→W ⊕C2,α(M)
R,φ −δ g˜R,φ −δ
such that u = G (f) solves the equation L (u) = f and kuk ≤ ckfk
R,φ g˜R,φ Wg˜R,φ⊕C−2,δα(M) C−0,δα(M)
where c is independent of R and φ.
The idea behind this proof was communicated to me by F. Pacard.
Proof: We wish to solve the equation
L (u)=f.
g˜R,φ
To this end, first let u +v ∈W ⊕C2,α(M ) solve
1 1 g1 −δ 1
L (u +v )=χf.
g1 1 1
Such a solution exists because g is unmarked nondegenerate. Moreover,we have the estimate
1
ku k +kv k ≤c kχfk . (5)
1 Wg1 1 C−2,δα(M1) 1 C−0,δα(M1)
In B (p ) (the standard spherical ball), u (t ,θ ) ∼ α vi,±(t ). Now choose Φ ∈ B
r0 0 1 0 0 i,± i,± ǫ 0 1 g1
such that |Φ1(t0,θ0)+ αi,±vǫi,±(t0)| = O(e−γn+1(ǫ)tP0). We can choose such a Φ1 because of
the existence of the Jacobi fields in remark 1 and because of the assumption that there is a one-
P
parameter family of metrics g on M such that g = g and the asymptotic necksize of the end
t 1 0 1
at p with respect to g is ǫ+t. Thus L (u +v +Φ )=χf and
0 t g1 1 1 1
|u1(t0,θ0)+v1(t0,θ0)+Φ1(t0,θ0)|≤2c1kχfkC−0,δα(M1)e−δt0 (6)
for (e−t0,θ)∈Br0(p0). We also have the estimate
kΦ k ≤c kχfk . (7)
1 Wg1⊕C−2,δα(M1) 1 C−0,δα(M1)
Similarly we let u +v ∈W ⊕C2,α(M ) solve
2 2 g2 −δ 2
L (u +v )=(1−χ)f
g2 2 2
with the estimate
ku k +kv k ≤c kfk . (8)
2 Wg2 2 C−2,δα(M2) 2 C−0,δα(M2)
This time we cannot cancel the nondecaying part of u +v on E . Instead, let β be such
2 2 2 i,±
that |u2(τ0,θ0)− βi,±vǫi,±(τ0)| = O(e−γn+1(ǫ)τ0) and let Φ2 ∈ Bg1 be such that |Φ2(t0,θ0)−
βi,±vǫi,±(t0)|=PO(e−γn+1(ǫ)t0) (recallthat the relationshipbetweent0 andτ0 in CˆR is givenby
t +τ =T +T′+2R). This time the salient estimates are
P0 0 0 0
kΦ k ≤c k(1−χ)fk (9)
2 Wg1⊕C−2,δα(M1) 2 C−0,δα(M2)
and
kL (Φ +u )k ≤c kfk e−δR. (10)
g˜R,φ 2 2 C0,α(CR) 3 C−0,δα(M)
10
