On the Positivity of Scattering Operatorsfor Poincar´e-Einstein Manifolds

Fang Wang

School of Mathematical Sciences,Shanghai Jiao Tong University,800Dongchuan Rd,Shanghai200240,China.

Abstract.In thispaper,wemainly study thescattering operatorsfor a Poincar´e-Einstein manifold(Xn+1,g+),which def ine the fractional GJMSoperators P2γof order 2γfor 0<γ

Key words:Scattering operators,fractional GJMS,positivity,Poinca´e-Einstein.

1 Introduction

We call(Xn+1,g+)a Poincar´e-Einstein manifold with conformal inf inity(M,[g]),ifg+is a smooth Riemannian metric in the interiorXwhich satisf ies

Herewerequirethatx2g+can beCk,αextended to theboundary for somek≥2,0<α<1.By the boundary regularity theorem given in[4],without loss of generality,we will assumek=∞fornodd andk≥n−1 forneven in thispaper.A straightforward calculation shows that all the sectional curvatures of(Xn+1,g+)converge to−1 when approaching to the boundary.A standard example is the hyperbolic space Hn+1in the ball model:

Thespectrum and resolvent for the Laplacian-Beltramioperator of(Xn+1,g+)is studied by Mazzeo-Melrose[13],Mazzeo[14]and Guillarmou[7].Actually the authors dealt with more general asymptotically hyperbolic manifolds.They showed that Spec(△+)=σpp(△+)∪σac(△+),whereσpp(△+)is theL2-eigenvalue set andσac(△+)is the absolute spectrum,and

We def ine the scattering operatorS(s)by

S(s):C∞(M)−→C∞(M),S(s)f=G|M.

HereP2kis the GJMSoperator of order 2kon(M,g)withg=x2g+|TM.In particular,P2is the conformal Laplacian andP4is the Paneitz operator on(M,g).See[9][10]for more details.

For simplicity,we def inethe renormalised scattering operators on(M,g)by

ThenP2γat regular pointsareconformally covariantγpowersof the Laplacian and hence also called thefractional GJMSoperators.Similarly,the fractional Q-curvatures are def ined by

And thefractional Yamabe constants are def ined by

Whenγ=1,у2(M,[g])is the classical Yamabe constant.

From the def inition ofP2γ,whenγis not an integer,P2γwould also depend on the interior metric(Xn+1,g+),not only on(M,[g]).A special case is(Sn,[gc]),the conformal inf inity of hyperbolic space Hn+1,wheregc=dθ2is thecanonical spherical metric.In this case,rigidity theorems given in[18][5]and[12]tell us that a Poincar´e-Einstein manifold(Xn+1,g+)with conformal inf inity(Sn,[gc])has to be the hyperbolic space Hn+1.So the fractional GJMSoperatorsP2γareuniquely determined,which aregiven in thefollowing:

Then the fractional Q-curvature can be calculated immediately:

In this paper,we are mainly interested in the positivity of renormalised scattering operatorsP2γ.Forγ∈(0,1),it was studied by Guillarmou-Qing in[8].

Theorem 1.1(Guillarmou-Qing).Suppose(Xn+1,g+)(n≥3)isa Poincar´e-Einstein manifold with conformal inf inity(M,[g]).Fix a smooth representative g for the conformal inf inity and assumethescalar curvature Rg ispositiveon(M,g).Then forγ∈(0,1),

(a)Q2γ>0on M;

(b)Thef irst eigenvalueof P2γispositive;

(c)The Green function of P2γispositive;

(d)Thef irst eigenspaceof P2γisspanned by apositivefunction.

Based on thepositivity result and theidentityS(n−s)S(s)=Id,they also showed that

Forγ∈(1,2),we prove the following positivity result forP2γ.

Theorem 1.3.Supposethe Poincar´e-Einstein manifold(Xn+1,g+)(n≥5)isahyperbolic manifold with conformal inf inity(M,[g]).Fix a smooth representative g for the conformal inf inity.Assumethescalar curvature Rg is apositiveconstant and Q4≥0on(M,g).Then forγ∈(1,2),

(a)Q2γ>0on M;

(b)Thef irst eigenvalueof P2γispositive;

(c)The Green function of P2γis positive;

for any smooth 1-formωon(M,g).We also denote

△g+u−s(n−s)u=0,xs−nu|M=f.

andG=S(s)f+O(x2).Here

T2(n−s)=L2(n−s),

T4(n−s)=L2(n−s+2)L2(n−s)−2(2s−n−2)L4(n−s).

By straightforward calculation,

3 The positivity of P2γforγ∈(1,2)

Weprove Theorem 1.3 in this section.The positivity of fractional GJMSoperatorsP2γforγ∈(1,2)was studied carefully by Case-Chang in Section 7 of[3].Combining their results with the spectrum theorem in[11],we f irst know that

Proposition 3.1(Lee,Case-Chang).Let(Xn+1,g+)(n≥4)be a Poinar´e-Einstein manifold with conformal inf inity(M,[g]).Fix a representative g for theconformal inf inity.Assumethe scalar curvature Rg>0and Q2γ>0for someγ∈(1,2).Then

(a)Thereisno L2-eigenvaluefor△+,i.e.spec(△+)=[n2/4,∞);

(b)Thef irst eigenvalueof P2γsatisf iesλ1(P2γ)≥minM Q2γ>0;

(c)P2γsatisf iesthestrong maximum principle,i.e.,if P2γf≥0for some f∈C∞(M),then f>0or f≡0;

(d)The Green’sfunction of P2γispositive.

Therefore,to prove Theorem 1.3,weonly need to prove itspart(a)and part(d).Here we work out a higher order comparison theorem similar as Guillarmou-Qing did in[8]to prove part(a).

Thenuis positive onXand near theboundaryuhas an asymptotical expansion:

where

IfJ>0 andQ4≥0 for(M,g),then it is easy to see that

u2<0,u4<0,onM.

Second,we def ine a test functionψby

Hereλ>max{s+2,n}will be f ixed later and

IfJ>0,then by the maximum principle,w>0 andv>0.Henceψis well def ined inX.

Lemma 3.1.Assume J>0,Q4≥0on M andλ>max{s+2,n}.Near theboundary,ψdef ined in(3.2)has an asymptotical expansion

which sastisf ies

ψ2=u2,ψ4>u4,on M.

Proof.Herev,whave asymptotical expansions as follows

By straightforward calculation,

This implies that

Soψ2=u2and

Sinceλ>s+2,λ>n>sandJ>0,Q4≥0,we haveψ4−u4>0.

Lemma 3.2.Assume J>0and Q4≥0on M andλ>max{s+2,n}.Then

Proof.A straightforward calculation gives

where

WhenJ>0,Q4≥0 andλ>max{s+2,n},it follows immediately thatK<0.

Next,takeλ=n+2 in the def inition ofψto simplify the calculations.In this case,v−2=O(x4).Denote

where

IfJ>0 andQ4≥0 onM,then Lemma 3.2 shows that

I|M<0.

We also denote

According to the asymptotical expansion(or uniformly degenerate property)of△+and the regularity ofInear boundary,it is obvious that

II|M=0.

Lemma 3.3.For Idef ined in(3.3),wehave

Here the”:”denotesthe covariant derivative with respect tog+.Hence

Direct computation shows that

Combining above,w e conclude the lemma.

Lemma 3.5.Assumethesameas Theorem1.3.Takeλ=n+2in(3.2).Thenψsatisf ies

△+ψ−s(n−s)ψ<0,in X.

Proof.SinceJis a positive constant,w0is a positive constant.Forλ=n+2,wsatisf ies△+w=0 in(3.2),which implies thatw≡w0all overX.Hence∇w=0,∇2w=0.By Lemma 3.3,

where

Vijk=v:ijk−vi[g+]jk−vj[g+]ik−2vk[g+]ij.

Notice thatII|M=0.Therefore,by the maximum principle

II=△+I≤0, inX.

which together withI|M<0 by Lemma 3.2 implies that

I<0, inX.

We f inish the proof.

Remark 3.1.The idea of constructing the test functionψwas originally from Lee[11].However,here we make the choiceλ=n+2 only for simplicity.We expect to have some intuitive explanation for it by future study,as well as some geometric interpretation of the tensorVijk.

Now we are ready to apply the comparison argument foruandψ.

Proposition 3.2.Suppose that the Poinar´e-Einstein manifold(Xn+1,g+)(n≥5)is hyperbolic with conformal inf inity(M,[g]).Fix asmooth representativeg for theconformal inf inity.Assume thescalar curvature Rg isapositiveconstant and Q4≥0on M.Then for allγ∈(1,2),Q2γ>0.Proof.Similar as Guillarmou-Qing’s proof in[8],we compare the two functionsuandψ,which are def ined in(3.1)and(3.2)withλ=n+2.Firstu/ψsatisf ies the equation:

Notice thatu/ψ>0,which has positive minimum.Applying Lemma 3.5 and the maximum principle,we see thatu/ψcan not attain an interior positive minimum.Henceu/ψ≥1,i.e.u≥ψall overX.Near theboundary,this means

1+x2u2+x2γu2γ+x4u4+O(x5)≥1+x2ψ2+x4ψ4+O(x5).

Sinceψ2=u2andψ4>u4by Lemma 3.1,we haveu2γ>0.HenceQ2γ>0 onM.

Acknowlegement

Theauthor wantsto thank Professor Sun-Yung Alice Chang and Professor Ruobing Zhang for helpful discussions.This research is supported in part by National Natural Science Foundation of China Grant No.11871331 and Shanghai Pujiang Program Grant No.14PJ1405400.