On the Geometry of Null Cones in Einstein-Vacuum Spacetimes

www.elsevier.com/locate/anihpc

On the geometry of null cones in Einstein-vacuum spacetimes

Qian Wang

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA Received 6 February 2007; accepted 8 March 2008

Available online 26 April 2008

Abstract

In this paper we study the geometry of null cones in smooth Einstein vacuum spacetimes. We provide theL^∞estimate for the trace of the null second fundamental form, as well as estimates for other geometric quantities. This paper is based on the work of Klainerman and Rodnianski [S. Klainerman, I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (3) (2005) 437–529; S. Klainerman, I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16 (1) (2006) 164–229; S. Klainerman, I. Rodnianski, A geometric Littlewood–Paley theory, Geom. Funct. Anal. 16 (1) (2006) 126–163].

©2008 Elsevier Masson SAS. All rights reserved.

1. Introduction

This paper is concerned with the geometry of null cones in 3+1 smooth Einstein vacuum spacetimes, i.e. 3+1 Lorentzian manifolds(M,g)with Ricci flat metrics,

Rαβ(g)=0.

Letp∈Mbe a fixed point and letT be a fixed timelike vector atpsatisfyingT , T = −1. We choose all future null vectors Lω,ω∈S², atp such thatLω, T = −1 and Lω, Lω =0. For eachω∈S²let Γ (s, ω)denote the outgoing null geodesic parametrized by the affine parameterswith the initial dataΓ (0, ω)=pand_ds^dΓ (0, ω)=Lω. The union of all these outgoing null geodesics forms a 3-D null cone starting frompwhich is denoted byH.

We define the vector fieldLbyL:=_ds^dΓ.ObviouslyL(0, ω)=LωandLsatisfies g(L, L)=0 and DLL=0.

The parameterscan be regarded as a function onHverifyingL(s)=1 ands(p)=0. We introduce the one parameter flowΓs(ω):=Γ (s, ω). It generates a family of 2-D closed surfaces{Ss}bySs:=Γs(S²), which form the geodesic foliation ofH. It is clear that eachSs is diffeomorphic toS²for s >0 sufficiently small. By rescaling the metricg we may assume without loss of generality that for 0< s 1 each slice Ss is diffeomorphic to S². Let Ht be the portion ofHwhensvaries in(0, t]. For simplicity, we still denote byHthe portionH1. Every pointqinHcan be parametrized by the coordinates(s, ω)for whichq=Γs(ω). We then call(s, ω)the transport local coordinates.

E-mail address:[email protected].

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.03.002

LetDdenote the Levi-Civita connection of Einstein vacuum metricg. Letγ be the induced metric onS_s, and∇ its induced covariant derivative. At any pointq∈S_s⊂Hwe denote byLthe null vector conjugate toLrelative to the S_sfoliation, i.e.L, L = −2 andL, X =0 for allX∈T_q(S_s). A smooth choice of an orthonormal frame{e_a}a=1,2

inT_q(S_s)combined withL, Lforms a null frame associated to the foliation.

We introduce the null components of the curvature tensorRof the spacetime metricgrelative toLandLas follows (see [2, Section 7.3] and [4, Section 3.1.2]):

αab=R(L, ea, L, eb), βa=1

2R(ea, L, L, L), ρ=1

4R(L, L, L, L), σ=1

4 R(L, L, L, L), βa=1

2R(e_a, L, L, L), α_ab=R(L, e_a, L, e_b), (1.1)

where R_{μνγ δ}=¹_2μνλτR^λτ_{γ δ}and_μνλτare components of the volume element in(M,g). The total curvature fluxR0

is then defined by R0=

α²_L2(H)+ β²_L2(H)+ ρ²_L2(H)+ σ²_L2(H)+ β²_L2(H)

¹

2. The geometry ofHin particular depends on the null second fundamental form

χ (X, Y )= D_XL, Y

withX andY being arbitrary vector fields tangent toS_s. We will denote trχandχˆ the trace and traceless part ofχ respectively. Other important geometric quantities are the dual null second fundament form and the torsion

χ (X, Y )= D_XL, Y and ζ (X)=1

2D_XL, L.

We will also use trχandχˆ to denote the trace and traceless part ofχ.

The mass aspect functionμis defined by μ= −divζ+1

2χˆ· ˆχ−ρ+ |ζ|². (1.2)

We are now ready to state the main theorem in this paper.

Theorem 1.1. Consider an outgoing null hypersurface Hin a smooth 3+1 Einstein vacuum spacetime (M,g), initiating from a point p and foliated by the geodesic foliation associated to the affine parameter s with s|p=0.

Assume that the total curvature fluxR0is sufficiently small. Then we have trχ−2

t

L^∞_t L^∞_ω

R0

and 1 0

| ˆχ|²dt L∞ω

+ 1

0

|ζ|²dt L∞ω

R0, sup

t1

|t∇trχ| L²_ω

+ sup

t1

t³²|μ| L²_ω

+ μL²(H)R0, N1(χ )ˆ +N1(ζ )+N1

trχ−2

t

R0,

∇trχ_B⁰+ t^1/2μ_B⁰+ μ_P⁰R0, sup

t1

t¹² trχ+2

t

L²_ω

+ sup

t1

t¹²| ˆχ| L²_ω

R0,

trχ+2 t

L²_tL²_ω

+ ˆχ_L²

tL²_ωR0, ∇L

trχ+2

t

L²(H)

R0, t^−1/2

trχ+2

t

B⁰+ t^−1/2χˆ_B⁰R0, t⁻¹

trχ+2

t

P⁰+ t⁻¹χˆ_P⁰R0.

The various norms appearing in the statement will be defined in Section 3 (see (3.3)–(3.6) and (3.20), (3.21)).

Throughout this paper we will use the notationABto meanAC·Bfor some appropriate universal constantsC. In [9,11,12] Klainerman and Rodnianski developed systematic methods to prove that, on truncated null hypersur- faces initiating from a 2-D surface diffeomorphic toS², within the radius of injectivity, trχ can be controlled by appropriate norms of the small initial data and small total curvature fluxR0, which is one of their steps toward the answer of the minimal local regularity of the initial data that guarantees the existence and uniqueness of local devel- opments for Einstein vacuum equation. See [7] and [8] for the best known regularity result. For the background of the initial data problem of Einstein vacuum equations and related results, please refer to [1,3,21,5,10]. In this work, we extend their result to null cones in smooth Einstein vacuum spacetimes. Our result shows that trχ−²_s can be bounded only by small total curvature flux before the formation of caustics or geodesic loops. This result is used in [13] to provide the uniform lower bound on the radius of injectivity of null boundaries in Einstein vacuum spacetimes. Such lower bound is essential in understanding the causal structure and propagation properties of solutions to the Einstein equations, and is important in construction of a Kirchoff–Sobolev type parametric for wave equations onM(see [14]), which is used in [15] to prove a large data break-down criterion for solutions of the Einstein vacuum equations.

We will follow the framework of [9] to prove the main theorem by the bootstrap principle. Since our null hyper- surfaceHinitiates from a point, many quantities behave likes^−afor some numbera >0 ass→0, we have to keep track the weights^ain each step. Note that the Besov norm estimates (see [9, Proposition 5.11])

∇ ·D⁻¹F_P⁰F_P⁰

of the 0-order Hodge operator ∇ ·D⁻¹ were used in [9, Section 6] to control the terms such as the commutator [∇L,∇D⁻²] ˇR, where P⁰ is a certain Besov norm. However, these estimates hold true only when some additional terms involving the L⁴_s⁺L²_ω norm of D⁻¹F is added,¹ due to the limited regularity of Gauss curvatureK of each slice S_s. The corrected versions we will present on theP⁰ estimates of the 0-order Hodge operators and on some product estimates add much complexity to the commutator estimates.

This paper is organized as follows. In Section 2, we recall the structure equations on various geometric quantities onHand provide the results on the initial data. In Section 3 we present the complete set of bootstrap assumptions, introduce some important norms and establish some preliminary estimates. In Section 4, we provide theL² type estimates and theP⁰type estimates of 0-order Hodge operators∇ ·D⁻¹. The result onP⁰estimate, which has special significance to Section 6, is a correction of [9, Proposition 5.11]. The proof is based on the unpublished notes of Klainerman and Rodnianski [6]. In Section 5, we prove some important product estimates. In Section 6, we use the results in Sections 4 and 5 to fulfill the decomposition of the commutators. Finally, in Section 7 we use the results in previous sections to complete the proof of the main theorem.

2. Structure equations and initial data

As the starting point we state the results on the behaviors of the main geometric quantities near the vertex of the null cone which can be obtained by local analysis, see [16] or [22, Appendix] for the proofs.

Proposition 2.1.Near the vertex of the null coneHthere hold

1 In this paper we will usea+to represent a numberq > a, anda−to represent a numberq < a.

strχ=2+O(s³) and strχ→ −2 ass→0, (2.1) ˆ

χ= −1

3sU+O(s²) and χˆ→0 ass→0, (2.2)

ζ = −1

6sη+O(s²) ass→ ∞ (2.3)

and

s∇trχ , s∇ ˆχ , sdivζ, s∇ζ, sμ→0 ass→0 (2.4)

whereUis a symmetric traceless 2-tensor andηis a1-form, both of which are finite at the vertex, depending on the curvature tensor in(M,g).

Let γ⁽⁰⁾ be the canonical metric on the standard 2-sphere S² and letγ^◦ =s⁻²γs. Set as =√

|γs|/ |γ⁽⁰⁾|and r=r(s)= (4π )⁻¹|Ss|with|Ss|being the area ofSs. Then

γ◦=γ⁽⁰⁾+O(s²), s⁻²as→1 and r

s →1 ass→0. (2.5)

We callr:=r(s)the radius of each leafS_s.

We also state the structure equations of the geodesic foliation (see [2] or [9, Section 2.12] for the derivations) d

dstrχ= −1

2(trχ )²− | ˆχ|², (2.6)

∇Lχˆ= −trχ· ˆχ−α, (2.7)

∇L(∇trχ )= −3

2trχ· ∇trχ− ˆχ· ∇trχ−2χˆ· ∇ ˆχ , (2.8)

∇Lζ = −trχ ζ−2χˆ·ζ −β, (2.9)

d

dstrχ= −1

2trχtrχ−2 divζ − ˆχ· ˆχ+2|ζ|²+2ρ, (2.10)

∇Lχˆ= −∇⊗ζ−1

2(trχ· ˆχ+trχ· ˆχ )+ζ⊗ζ, (2.11)

divχˆ =1

2∇trχ−1

2trχ·ζ− ˆχ·ζ−β, (2.12)

curlζ= −1

2χˆ∧ ˆχ+σ, (2.13)

divχˆ =1

2∇trχ−1

2trχ ζ+ζ · ˆχ+β, (2.14)

K= −1

4trχtrχ+1

2χˆ· ˆχ−ρ (2.15)

and the renormalized null Bianchi identities

∇Lβ=divα+ζ·α, (2.16)

L(ρ)ˇ +3

2trχ· ˇρ=divβ−ζ·β+1 2χˆ·

∇⊗ζ+1

2trχ· ˆχ−ζ⊗ζ

, (2.17)

L(σ )ˇ +3

2trχ· ˇσ= −curlβ+ζ∧β+1

2χˆ∧(∇⊗ζ−ζ⊗ζ ), (2.18)

∇Lβˇ= −∇ρ+(∇σ ) −2(∇⊗ζ )·ζ+3

ζ·ρ−ζ σ

−trχ β, +2ζ·

−1

2trχ· ˆχ−1

2trχ· ˆχ+ζ⊗ζ

−4χ· ˆχ·ζ (2.19)

whereKdenotes the Gauss curvature of each leafS:=S_s, and ˇ

ρ=ρ−1

2χˆ· ˆχ , σˇ =σ−1

2χˆ∧ ˆχ , βˇ=β+2χˆ·ζ,

moreover,∇Lπ, for anyS-tangent tensor fieldπ, is defined as in [9, Definition 2.9], i.e. the projection ofD_Lπ on each leafS_s.

The transport equation for the mass aspect functionμdefined by (1.2) takes the form d

dsμ+3

2trχ μ= ˆχ·(∇⊗ζ )+1

2trχρˇ+2ζ· ∇trχ−4χˆ·ζ ·ζ+trχ|ζ|²−1

4trχ| ˆχ|². (2.20) 3. The bootstrap assumptions

3.1. A preliminary bootstrap assumption

As a preliminary bootstrap assumption, we require that there exists a sufficiently small positive constant 0< Δ0<

1/2 such that

VL^∞(H)Δ0, (3.1)

whereV (s, ω)=trχ−²_s. We also setV (s, ω)=trχ+²_s,which will be used later.

In the following we will provide some preliminary estimates under (3.1). Recall that for the induced metricγ:=γs

onS:=Ss, we have _ds^dγ =2χ. Thus _ds^das=trχ as. In view of (2.5), we then gett⁻²at =exp(t

0Vsds). Therefore for 0< t1 there holds|t⁻²a(t, ω)−1|t Δ0. Thus for smallΔ0we have

1

2t²a(t, ω)2t². (3.2)

In view of [9, Lemma 2.26] which says ^dr_ds=^r₂trχ, it is easy to check d

dslogr s =1

2V_s.

Using (2.5) and integrating the above equation along any null geodesic yields ^r_s =exp(s·O(Δ₀)). Therefore we get Proposition 3.1.Under the bootstrap assumption(3.1), the radiusr(s)of each leafS_s and the affine parametersare always comparable in the sense that|¹_r −¹_s|Δ₀for0< s1.

In view of (3.2), we have for anyS-tangent tensorF onH 1

2

|ω|=1

F (s, ω)^ps²dωF^p_L^p_(Ss)2

|ω|=1

F (s, ω)^ps²dω, where|F|denotes the norm ofF under the induced metricγon each leafS_s.

ForS-tangent tensor fieldsF onH, we introduce the following norms. For 1p, q∞we define theL^q_tL^p_x norm

F_L^q_tL^p_x :=

1 0 |ω|=1

|F (s, ω)|^ps²dω _p^q

ds _q¹

(3.3)

and theL^pxL^∞_t norm F_L^p_xL∞

t :=

|ω|=1

sup

s

s^2pF (s, ω)p_p¹

. (3.4)

We also define the norms N1(F ):= t⁻¹F_L²

tL²_x + ∇LF_L²

tL²_x+ ∇F_L²

tL²_x (3.5)

and

N2(F ):= t⁻²F_L²

tL²_x +t⁻¹∇LF

L²_tL²_x+ t⁻¹∇F_L²

tL²_x + ∇∇LF_L²

tL²_x+ ∇²F_L²

tL²_x. (3.6)

On each sliceS=S_s, we have the following Sobolev inequalities for scalar functionsf and tensor fieldsF (see [12]

for the proofs):

fL²(S)∇fL¹(S)+ s⁻¹fL¹(S), (3.7)

fL^∞(S)∇²fL¹(S)+ ∇fL²(S)+ s⁻²fL¹(S), (3.8)

FL^p(S)∇F¹_L⁻2(S)^p2 · F_L^p22(S)+s^p2−1F

L²(S), 2p <∞, (3.9)

FL^∞(S)∇²F_L^p12(S)

∇F_L^p−2p2(S)F_L^p12(S)+s^p2−1F_L^p−1p2(S)

+ ∇F_L²_(S). (3.10)

The following preliminary estimates will be used routinely.

Lemma 3.1.Let F be an arbitraryS-tangent tensor field onH. Then t⁻¹²F

L^∞_t L^∞_ω + t⁻¹F_L²

tL^∞_ω N2(F ), FL⁴_xL^∞_t + F_L⁶

tL⁶_x+t^−1/2F

L²_xL^∞_t N1(F ), (3.11)

t^{−q1−p2+12}F

L^q_tL^px N1(F ) with2p <∞,2< q 2p

p−4. (3.12)

Proof. The proof of the first inequality can be found in [2]. In the following we show the second inequality in (3.11).

Note that it suffices to prove it for scalar functionsf. First

t²f (t, ω)⁴ t

0

sf (s, ω⁴ds+ t

0

|f|³·∇Lf (s, ω)s²ds

t 0

f (s, ω)⁶s²ds ¹

2t

0

s⁻²f (s, ω)²s²ds+ t

0

∇Lf (s, ω)²s²ds ¹

2

. Hence, integrating on|ω| =1, we obtain

fL⁴_xL^∞_t f_L³⁴6(H)

s⁻¹fL²(H)+ ∇LfL²(H)

¹

4. (3.13)

On the other hand, by (3.7) we have on eachSs that

S_s

|f|⁶dAsf³²_L2

x

S_s

|∇f| +s⁻¹|f|

|f|²dAs

2

S_s

|∇f|²+s⁻²|f|² dA_s·

S_s

|f|⁴dA_s.

Then integrating insyields fL⁶(H)f_L²³∞

t L⁴_x

s⁻¹fL²(H)+ ∇fL²(H)

¹₃

. (3.14)

Finally we note that t^−1/2f

L²_xL^∞_t 1 t

d ds

s|f|² ds

1 2

L²_ω

+f (1)

L²_ω.

The first term is bounded by N1(f ) by using Hölder inequality. The second term can be estimated by taking a test function 0θ 1 supported in [1/2,1] with θ (1)=1 and vanishing identically at 0< t 1/2, and sup_ts1|θ(s)|<₁₋¹_t. Since

f (1)²

L²_ω=

|ω|=1

1 1/2

d ds

sθ|f|² ds

dω∇Lf²_L2(H)+ s⁻¹f²_L2(H)N1(f )²

we concludet^−1/2fL²_xL^∞_t N1(f ). Combining (3.13) with (3.14), we conclude that the estimates in the second line of (3.11) hold true. We can prove (3.12) similarly. 2

3.2. The full set of bootstrap assumptions

In order to provide the full set of bootstrap assumptions, we introduce the following conventions.

• Rdenotes the full collection of null curvature componentsα,β,ρ,σ,β.

• R0denotes the collection of the null curvature componentsβ,ρ,σ,β.

• ˇRdenotes the collection of the renormalized null curvature components(ρ,ˇ − ˇσ ),β.ˇ

• Adenotes the collectionV,χˆ,ζ,V,χˆ.

• Adenotes the collection ofV,χˆ,ζ.

• Mdenotes the collection∇trχ,μ.

• ∇Adenotes the collection∇trχ,∇ ˆχ,∇ζ.

The bootstrap assumptions we will rely on in this paper are trχ−2/tL^∞_t L^∞_ω,A_L∞

ωL²_t,N1(A),∇trχL²_xL^∞_t Δ0, (BA1)

and

t^−1/2AL²_xL^∞_t Δ₀, ∇LAL²(H)Δ₀ (BA2)

where 0<R0Δ0<¹₂is a sufficiently small constant. Note that the preliminary bootstrap assumption (3.1) is a part of (BA1).

In order to complete the proof of Theorem 1.1, by the bootstrap principle it suffices to show, under (BA1) and (BA2), that all the inequalities in them still hold true withΔ₀ replaced byΔ₀/2 when 0<R0Δ₀is sufficiently small. This will be done in Section 7 after the preparations given in the next three sections.

Lemma 3.2.Under the bootstrap assumption(BA1), the metricγ^◦_ij(s)on eachS_sverifies weakly spherical conditions, i.e. relative to the transport local coordinates(s, ω1, ω2)the metric componentsγ^◦_ij(s)satisfy

γ^◦_ij(s)−γ_ij⁽⁰⁾

L^∞_ω Δ₀, (3.15)

∂_kγ^◦_ij(s)−∂_kγ_ij⁽⁰⁾

L²_ωΔ₀, (3.16)

whereΔ₀is a small constant.

Proof. Recall that relative to the transport local coordinates(s, ω1, ω2)onH, Proposition 2.1 says

slim→0

γ◦_ij=γ_ij⁽⁰⁾ and lim

s→0∂_kγ^◦_ij=∂_kγ_ij⁽⁰⁾ (3.17)

wherei, j, k=1,2. Recall also that the metricγ verifies _ds^dγ_ij =2χijwithi, j=1,2. Consequently, d

ds

γ◦_ij=γ^◦_ijV +2s⁻²χˆ_ij.

Integrating this equation along any null geodesic onHand using (3.17) we derive

i,j

γ^◦_ij−γ_ij⁽⁰⁾

L^∞_t L^∞_ω

i,j

sup

t

t

0

γ◦_ijV +s⁻²χˆ_ijds L^∞_ω

VL^∞_t L^∞_ω + ˆχ_L∞ωL²_t

ij

γ^◦_ij−γ_ij⁽⁰⁾

L^∞_t L^∞_ω +1

. This gives (3.15), by using (BA1) and the smallness ofΔ₀.

The proof of (3.16) is similar by noting that d

ds∂k

γ◦_ij=∂k

γ◦_ijV +γ^◦_ij∂kV +2s⁻²∂kχˆij, wherei, j, k=1,2. 2

On eachS:=S_s we will use the geometric Littlewood–Paley (GLP) projectionsP_kintroduced in [12] which take the form

P_kF :=

∞

0

m_k(τ )U (τ )F dτ

for any tensor fieldF, wherem_k(τ ):=2^2km(2^2kτ )for some smooth functionmon[0,∞)vanishing sufficiently fast and verifying the vanishing moment property

∞

0

τ^k1∂^k2m(τ ) dτ=0, k₁+k₂N,

andU (τ )F is defined by the heat flow on(S_s,γ )^◦

∂

∂τU (τ )F−Δ◦

γU (τ )F=0, U (0)F=F. (3.18)

One may refer to [12] for various properties of GLP projections, such as the finite band property and the Bernstein inequalities, etc, which will be frequently used in this paper.

We will also use the notations F_n:=P_nF, F₀:=

k0

P_kF and F_>0:=

k>0

P_kF for anyS-tangent tensor fieldF.

Let 0θ <1, we define the Besov normB_2,1^θ for tensor fieldsF on 2-D surfaceSby F_B^θ

2,1=

k>0

2^kt⁻¹θ

P_kF

L²_x+ t^−θFL²_x. (3.19)

We also define the BesovB^θ andP^θ norms forS-tangent tensor fieldsF onHas follows:

F_B^θ =

k>0

2^kt⁻¹θ

P_kF

L^∞_t L²_x + t^−θFL^∞_t L²_x, (3.20)

F_P^θ=

k>0

2^kt⁻¹θ

P_kF

L²_tL²_x+ t^−θF_L²

tL²_x. (3.21)

By using the heat flow (3.18), we can define the operatorΛ^awitha0 such that for anyS-tangent tensor fieldsF Λ^aF := s^−a

(−a/2) ∞

0

τ^−a2−1e^−τU (τ )F dτ.

The definition ofΛ^aextends to the rangea >0 by defining for 0< a2mthat Λ^aF=Λ^a−2m·

s⁻²Id−Δγ

m

F.

We record the basic properties ofΛ^ain the following result (see [12,20]).

Proposition 3.2.

(i) Λ⁰=IdandΛ^a·Λ^b=Λ^a+bfor anya, b∈R.

(ii) For anyS-tangent tensor fieldF and anya0 s^aΛ^aFL²(S)FL²(S).

(iii) For anyS-tangent tensor fieldF and anyba0

s^aΛ^aFL²(S)s^bΛ^bFL²(S) and Λ^aFL²(S)Λ^bF_L^ab2(S)F¹_L⁻2(S)^ab . (iv) For anyS-tangent tensor fieldsF andGand any0a <1

Λ^a(F·G)

L²(S)ΛFL²(S)Λ^aGL²(S)+ Λ^aFL²(S)ΛGL²(S). (v) For anyS-tangent tensor fieldF there holds with2< p <∞anda >1−_p²

FL^p(S)Λ^aFL²(S).

(vi) For anya∈Rand anyS-tangent tensor fieldF F²_H^a_(S):= Λ^aF²_L2(S)≈

k0

2^2kas^−2aP_kF²_L2(S)+s^−2aP₀F²_L2(S).

Under (BA1) and (BA2) we can also derive

Proposition 3.3.Under(BA1)and(BA2), if0<R0< Δ₀are sufficiently small, then for all¹₂< a <1there holds K_a:=Λ^−a

K−s⁻²

L^∞_t L²_x Δ₀.

The proof of Proposition 3.3 is a little involved. Noting that our definition ofΛ^−ainvolvess⁻², by keeping track the powers of s, the argument in the proof of Proposition 4.13 in [9] still goes through. For details please refer to [22, Chapter 4.3].

Sometimes it is convenient to work with the Besov norms defined by the classical Littlewood–Paley (LP) projec- tionsE_k. Recall that (see [17–19]) for any scalar functionf onR²we can define

E_kf = 1 (2π )²

R²

η

ξ /2^kf (ξ )eˆ ^ixξdξ,

whereηis a smooth function with support in the dyadic shell{¹₂|ξ|2}and satisfying

k∈Zη(2^−kξ )=1 when ξ=0.

Now for any scalar functionf onH, we define for any 0a <1 itsB˜^aandP˜^anorms by f_B˜a:=

k>0

2^kt⁻¹a

E_kf

L^∞_t L²_x + t^−af_L∞

t L²_x, (3.22)

f_P˜a =

k>0

2^kt⁻¹a

Ekf

L²_tL²_x+ t^−af_L²_tL²

x. (3.23)

It is worthy to say a few words about this definition. Recall that the geodesic flowΓ_s:S²→S_sfor eachs >0 is a diffeomorphism. Let(U_i, η⁽ⁱ⁾)be a finite atlas onS²with chartsη⁽ⁱ⁾mappingU_i into the unit disc inR², and let{φ} be a subordinated partition of unity onS². Then{φ◦Γ_s⁻¹}is a partition of unity on the sliceS_s for 0< s1 which

will be denoted as φ_s. Let η_s⁽ⁱ⁾:=η⁽ⁱ⁾◦Γ_s⁻¹. TheE_kf in the above definition is defined asE_k((φ_sf )◦η_s⁽ⁱ⁾⁻¹)on eachS_sand theL²_xnorms are understood to be theL²norm onR².

Using Lemma 3.2, (BA1) and (BA2), we can adapt [11, Proposition 3.28] to obtain the following lemma.

Lemma 3.3.Under the bootstrap assumptions(BA1)and(BA2), there exists a finite number of vector fields{X_i}^l_i=1 verifying the conditions

X, t∇0XL^∞_t L^∞_ω 1, t∇(∇0X)L²_xL^∞_t 1, (∇ − ∇0)X

L²_xL^∞_t Δ₀, ∇LX=0,

where ∇0 represents the covariant derivative induced by the metric s²γ⁽⁰⁾. For appropriateS-tangent tensorF ∈ L^∞_t L²_x,F ∈B^aif and only ifF ·X_i∈B^a, and

C⁻¹

i

F·X_iB^a F_B^aC

i

F·X_iB^a, with0a <1,

whereCis a universal constant. The same results hold for the spacesP^α. Moreover N1(F⊗X)+ F⊗X_L∞ωL²_t N1(F )+ F_L∞ωL²_t,

where⊗stands for either a tensor product or a contraction.

Lemma 3.3 allows us to define Besov norms for arbitrary S-tangent tensor fields F onH by the classical LP projections.

Definition 3.1.LetF be an(m, n) S-tangent tensor field onHand letF_i^j1j2...jm

1i₂...i_n be the local components ofF relative to{X_i}^l_i=1. We define theB˜^aandP˜^anorms ofF by

F_B˜a =

F_i^j1j2...jm

1i₂...i_{n B˜}a and F_P˜a =

F_i^j1j2...jm

1i₂...i_{n P˜}a, where the summation is taken over all possible(i1. . . in;j1. . . jm).

Finally we state the following equivalence results betweenB^a,P^anorms andB˜^a,P˜^anorms, whose proof can be found in [22].

Proposition 3.4.Under the bootstrap assumptions(BA1)and (BA2), for arbitraryS-tangent tensor fieldsF onH there hold

F_B˜a ≈ F_B^a and F_P˜a≈ F_P^a with0a <1.

4. Elliptic estimates of Hodge operators onH

In view of the structure equations given in Section 2, it is important to consider the following Hodge operators on 2-surfaceSdiffeomorphic to the standard sphereS²:

• The operatorD1takes any 1-formF into the pair of functions(divF,curlF ).

• The operatorD2takes any symmetric traceless 2-tensorF onSinto the 1-form divF.

• The operator D1takes the pairs of scalar functions(ρ, σ )into the 1-forms−∇ρ+(∇σ ) onS.

• The operator D2takes 1-formsF onSinto the 2-covariant, symmetric, traceless tensors−¹₂LFγ, where LFγ

ab= ∇bF_a+ ∇aF_b−(divF )γ_ab.

For various properties of these operators please refer to [2,9].

4.1. L²estimates for Hodge operators

In this subsection we will give theL² estimates for the Hodge operators onHunder the bootstrap assumptions (BA1), (BA2) and the smallness conditions onR0andΔ₀.

Proposition 4.1.The following estimates hold on each leafS=Ss⊂H:

(i) The operatorD1is invertible on its range and its inverseD₁⁻¹takes pair of functionf=(ρ, σ )in the range of D1intoS-tangent1-formsF withdivF =ρ,curlF =σ. Moreover

∇D⁻₁¹f

L²(S)+s⁻¹D⁻₁¹f

L²(S)fL²(S).

(ii) The operatorD2is invertible on its range and its inverseD⁻₂¹takesS-tangent1-formsF (in the range ofD2) intoS-tangent symmetric, traceless,2-tenorfieldsZwithdivZ=F. Moreover

∇D⁻₂¹F

L²(S)+s⁻¹D⁻₂¹F

L²(S)FL²(S).

(iii) The operator(−)is invertible on its range and its inverse(−)⁻¹verifies the estimate ∇²(−)⁻¹f

L²(S)+s⁻¹∇(−)⁻¹f

L²(S)fL²(S).

(iv) The operator D1is invertible as an operator defined for pairs ofH¹functions with mean zero(i.e. the quotient ofH¹by the kernel of D1)and its inverse D⁻₁¹takesS-tangentL²1-formsF(i.e. the full range of D1)into pair of functions(ρ, σ )with mean zero, such that−∇ρ+(∇σ ) =F, verifies the estimate

∇ D⁻₁¹F

L²(S)FL²(S).

(v) The operator D2is invertible as an operator defined on the quotient ofH¹-vector fields by the kernel of D2. Its inverse D₂⁻¹ takesS-tangent2-formsZ which is inL²space intoStangent1-formsF (orthogonal to the kernel ofD2), such that D2F =Z, verifies the estimate

∇ · D⁻₂¹Z

L²(S)ZL²(S).

As a consequence of (i)–(v), letD⁻¹be one of the operatorsD⁻₁¹,D⁻₂¹, D₁⁻¹or D₂⁻¹. By dual argument, we have the following estimate for appropriate²tensor fieldsF,

D⁻¹divFL²(S)FL²(S).

The proof of this result is essentially the same as the proof of [9, Proposition 4.22]. Note that in our situation, 0< s1, which is different from [9] wheres≈1. Therefore we must keep the weights⁻¹in some of the estimates, which will be crucial for later applications.

Using the formula (2.15) for the Gauss curvatureKofS_s and the bootstrap assumptions (BA1) and (BA2) we can easily obtain

Proposition 4.2.ForK:=K−s⁻²there holdsK_L²_(H)Δ₀. For later applications, we will use the renormalized Gauss curvature

Kˇ =K−r⁻² (4.1)

which, in view of Proposition 3.1, Proposition 3.3 and Proposition 4.2, verifies

ˇKL²(H)Δ0 and Kˇα:= Λ^−aKˇL²_x Δ0 (4.2)

for any¹₂< a <1.

Using Proposition 3.3 and Proposition 4.2 and following the similar argument in [9] we can obtain (see [22])

2 By “appropriate”, we mean the tensorFsuch that divFis in the space whereD⁻¹is well-defined.