A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary
aa r X i v : . [ m a t h . DG ] S e p A COMPACTNESS RESULT FOR SCALAR-FLAT METRICS ONLOW DIMENSIONAL MANIFOLDS WITH UMBILICBOUNDARY
MARCO G. GHIMENTI AND ANNA MARIA MICHELETTI
Abstract.
Let ( M, g ) a compact Riemannian n -dimensional manifold withumbilic boundary. It is well know that, under certain hypothesis, in the con-formal class of g there are scalar-flat metrics that have ∂M as a constant meancurvature hypersurface. In this paper we prove that these metrics are a com-pact set in the case of low dimensional manifolds, that is n = 6 , , , providedthat the Weyl tensor is always not vanishing on the boundary. Introduction
Let ( M, g ) be a n -dimensional ( n ≥ ) compact Riemannian manifold withboundary ∂M . In [20, 21] J. Escobar investigated the question if M can be con-formally deformed to a scalar flat manifold with boundary of constant mean curva-ture hypersurface. This problem is particularly interesting because it is a higher-dimensional generalization of the well known Riemann mapping Theorem and it isequivalent to finding positive solutions to a linear equation on the interior of M with a critical nonlinear boundary condition of Neumann type:(1.1) (cid:26) L g u = 0 in MB g u + ( n − u nn − = 0 on ∂M . Here L g = ∆ g − n − n − R g where − ∆ g is the Laplace-Beltrami operator on ( M, g ) and R g the scalar curvature of M and B g = − ∂∂ν − n − h g , where ν is the outwardnormal to ∂M and h g is the mean curvature of the boundary.The existence of solutions in established by Escobar [20], Marques [24], Almaraz[3], Chen [8], Mayer and Ndiaye [23]. Once the existence of solutions of (1.1) issettled, it is natural to study the compactness of the full set of solutions. Defined Q ( M, ∂M ) := inf (cid:8) Q ( u ) : u ∈ H ( M ) , u on ∂M (cid:9) , where Q ( u ) := R M (cid:16) |∇ u | + n − n − R g u (cid:17) dv g + R ∂M n − h g u dσ g (cid:18) R ∂M | u | n − n − dσ g (cid:19) n − n − , we have that when Q ( M, ∂M ) ≤ the solution is unique up to a constant factor.The situation turns out to be delicate if Q ( M, ∂M ) > and the underlying manifoldis not the euclidean ball (in the case of the euclidean ball the set of solution isknown to be non compact). Compactness has be proven firstly by Felli and OuldAhmedou in [10] for any dimension n ≥ in the case of locally conformally flatmanifolds with umbilic boundary. If the dimension of the manifold is n ≥ andthe trace-free second fundamental form in non zero everywhere on ∂M , Almaraz Mathematics Subject Classification.
Key words and phrases.
Scalar flat metrics, Umbilic boundary, Yamabe problem, Compactness,low dimensions. in [1] proved compactness. Very recently, Kim Musso and Wei [18] showed thatcompactness continues to hold when n = 4 and when n = 6 , and the trace-freesecond fundamental form in non zero everywhere on ∂M .Compactness was proved also by the authors in [11] for manifold with umbilicboundary when n = 8 and the Weyl tensor of the boundary is always different fromzero, or if n > and the Weyl tensor of M is always different from zero on theboundary. An example of non compactness is given for n ≥ and manifolds withumbilic boundary in [2]. We recall that the boundary of M is called umbilic if thetrace-free second fundamental form of ∂M is zero everywhere.In the present work we are interested to extend the result of [11] to dimension n = 6 , , when the Weyl tensor of M is always different from zero on the boundary.Namely we want to prove compactness of the set of positive solutions to(1.2) (cid:26) L g u = 0 in MB g u + ( n − u p = 0 on ∂M where ≤ p ≤ nn − and the boundary of M is umbilic. Our main result is thefollowing. Theorem 1.
Let ( M, g ) a smooth, n -dimensional Riemannian manifold of positivetype with regular umbilic boundary ∂M . Suppose that n = 6 , , and that the Weyltensor W g is not vanishing on ∂M . Then, given ¯ p > , there exists a positiveconstant C such that, for any p ∈ h ¯ p, nn − i and for any u > solution of (1.2), itholds C − ≤ u ≤ C and k u k C ,α ( M ) ≤ C for some < α < . The constant C does not depend on u, p . Our strategy follows the argument of the seminal paper of Khuri Marques andSchoen [19]. A crucial step is to provide a sharp correction term (see Subsection2.2) for the usual approximation of a rescaled solution by a bubble around anisolated simple blow up point. This sharp correction term is a solution of a suitablelinearized equation (see (2.17)). The assumption of the umbilicity of the boundaryforces us to deal to higher order terms in the expansion of the metric tensor, andthis makes the proof of the result technically hard. Moreover, it determines theright hand side of the equation (2.17), which gives the aforementioned correctionterm.Another crucial step relies on a classical local argument with a Pohozaev typeidentity and we need a local Pohozaev sign condition which is essential for the proof.In the case of low dimensional manifolds this requires a very accurate pointwiseestimate of the correction term which seems not to have an explicit form in the caseof boundary Yamabe problem. This process is somewhat inspired to the strategyused by Kim Musso and Wei [18] to estimate the correction term on low dimensionalmanifold with non umbilic boundary.The paper is organized as follows: in Section 2 we provide some necessary pre-liminary notions; in particular in Subsection 2.1 we introduce some type of blowup points and in Subsection 2.2 we define the correction term. Section 3 containsan accurate description of the correction term, and the Pohozaev sign condition isstudied in Section 4, for the case n = 7 , , and in Section 5, for the case n = 6 . Theproof of Theorem 1 is shown in Section 6. Some technical proofs are postponed tothe Appendix. 2. Preliminaries and notations
Remark . We collect here our main notations. We will use the indices ≤ i, j, k, m, p, r, s, t, τ ≤ n − and ≤ a, b, c, d ≤ n . Moreover we use the Einstein COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 3 convention on repeated indices. We denote by g the Riemannian metric, by R abcd the full Riemannian curvature tensor, by R ab the Ricci tensor and by R g the scalarcurvature of ( M, g ) ; moreover the Weyl tensor of ( M, g ) will be denoted by W g . Thebar over an object (e.g. ¯ W g ) will means the restriction to this object to the metricof ∂M . Finally, on the half space R n + = { y = ( y , . . . , y n − , y n ) ∈ R n , y n ≥ } weset B r ( y ) = { y ∈ R n , | y − y | ≤ r } and B + r ( y ) = B r ( y ) ∩{ y n > } . When y = 0 we will use simply B r = B r ( y ) and B + r = B + r ( y ) . On the half ball B + r we set ∂ ′ B + r = B + r ∩ ∂ R n + = B + r ∩ { y n = 0 } and ∂ + B + r = ∂B + r ∩ { y n > } . On R n + we willuse the following decomposition of coordinates: ( y , . . . , y n − , y n ) = (¯ y, y n ) = ( z, t ) where ¯ y, z ∈ R n − and y n , t ≥ .Fixed a point q ∈ ∂M , we denote by ψ q : B + r → M the Fermi coordinatescentered at q . We denote by B + g ( q, r ) the image of ψ q ( B + r ) . When no ambiguity ispossible, we will denote B + g ( q, r ) simply by B + r , omitting the chart ψ q .We recall that ω n − is the n − dimensional spherical element.Since the boundary ∂M of M is umbilic, it is well know the existence of aconformal metric related to g and the existence of the conformal Fermi coordinates,which will simplify the future computations.Given q ∈ ∂M there exists a conformally related metric ˜ g q = Λ q g such thatsome geometric quantities at q have a simpler form which will be summarized inthe next claim. We also know that Λ q ( q ) = 1 , ∂ Λ q ∂y k ( q ) = 0 for all k = 1 , . . . , n − . In order to simplify notations, we will omit the tilde symbol and we will omit thefermi conformal coordinates ψ q : B + r → M whenever it is not needed, so we willwrite y ∈ B + r instead of ψ q ( y ) ∈ M , instead of q = ψ q (0) , u instead of u ◦ ψ q andso on. Remark . In Fermi conformal coordinates around q ∈ ∂M , it holds (see [24])(2.1) | det g q ( y ) | = 1 + O ( | y | N ) for some N large | h ij ( y ) | = O ( | y | ) | h g ( y ) | = O ( | y | ) (2.2) g ijq ( y ) = δ ij + 13 ¯ R ikjl y k y l + R ninj y n (2.3) + 16 ¯ R ikjl,m y k y l y m + R ninj,k y n y k + 13 R ninj,n y n + (cid:18)
120 ¯ R ikjl,mp + 115 ¯ R iksl ¯ R jmsp (cid:19) y k y l y m y p + (cid:18) R ninj,kl + 13 Sym ij ( ¯ R iksl R nsnj ) (cid:19) y n y k y l + 13 R ninj,nk y n y k + 112 ( R ninj,nn + 8 R nins R nsnj ) y n + O ( | y | ) (2.4) ¯ R g q ( y ) = O ( | y | ) and ∂ ii ¯ R g q = − | ¯ W | (2.5) ∂ tt ¯ R g q = − R ninj − R ninj,ij (2.6) ¯ R kl = R nn = R nk = R nn,kk = 0 (2.7) R nn,nn = − R nins . All the quantities above are calculate in q ∈ ∂M , unless otherwise specified. COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 4
We set U ( y ) := 1[(1 + y n ) + | ¯ y | ] n − to be the standard bubble. The function U solves the problem(2.8) (cid:26) ∆ U = 0 in R n + ∂U∂y n + ( n − U nn − = 0 on ∂ R n + . Remark . Let f : R × R + → R be a smooth integrable function and fix a c ≥ .We have the following integral identities(2.9) R ninj Z ∂ R n + f ( | ¯ y | , c ) y i y j d ¯ y = 0 (2.10) ¯ R tτsp Z ∂ R n + f ( | ¯ y | , c ) y t y τ y s y p d ¯ y = 0 (2.11) R ninj ¯ R tτsp Z ∂ R n + f ( | ¯ y | , c ) y i y j y t y τ y s y p d ¯ y = 0 (2.12) ¯ R ijkl ¯ R tτsp Z ∂ R n + f ( | ¯ y | , c ) y i y j y k y l y t y τ y s y p d ¯ y = 0 R ninj R nknl Z ∂ R n + f ( | ¯ y | , c ) y i y j y k y l d ¯ y = 23 R ninj Z ∂ R n + f ( | ¯ y | , c ) y d ¯ y (2.13) = 2 n − R ninj Z ∂ R n + f ( | ¯ y | , c ) | ¯ y | d ¯ y Proof.
The first two identities follows by the symmetries of the curvature tensor.For the last formula we have, again by symmetry, R ninj R nknl Z ∂ R n + f ( | ¯ y | , c ) y i y j y k y l d ¯ y = 2 R ninj Z ∂ R n + f ( | ¯ y | , c ) y i y j d ¯ y = 2 R ninj Z ∂ R n + f ( | ¯ y | , c ) y y d ¯ y, and we can conclude by the elementary identities Z R f ( x + y ) x y dxdy = Z R f ( x + y ) x dxdy and Z ∂ R n + f ( | ¯ y | , c )¯ y d ¯ y = 3 n − Z ∂ R n + f ( | ¯ y | , c ) | ¯ y | d ¯ y. (cid:3) Remark . We collect here some result contained in [1, Lemma 9.4] and in [1,Lemma 9.5]. The proof is by direct computation. For m > k + 1 Z ∞ t k dt (1 + t ) m = k !( m − m − · · · ( m − − k ) (2.14) Z ∞ dt (1 + t ) m = 1 m − Moreover, set, for α, m ∈ N , I αm := Z ∞ s α ds (1 + s ) m COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 5 it holds I αm = 2 mα + 1 I α +2 m +1 for α + 1 < m (2.15) I αm = 2 m m − α − I αm +1 for α + 1 < mI αm = 2 m − α − α + 1 I α +2 m for α + 3 < m. Blow up points and the Khuri-Marques-Schoen scheme.
By the con-formal invariance property of the operators L g and B g it is more convenient to dealwith the conformally invariant family of problems(2.16) (cid:26) L g i u = 0 in MB g i u + ( n − f − τ i i u p i = 0 on ∂M . where p i ∈ h ¯ p, nn − i for some fixed ¯ p > , τ i = nn − − p i , f i → f in C loc for somepositive function f and g i → g in the C loc topology.First, we collect the definition of various type of blow up points. Definition 6.
We say that x ∈ ∂M is a blow up point for the sequence u i ofsolutions of (2.16) if there is a sequence x i ∈ ∂M such that(1) x i → x ;(2) x i is a local maximum point of u i | ∂M ;(3) u i ( x i ) → + ∞ . Shortly we say that x i → x is a blow up point for { u i } i .We say that x i → x is an isolated blow up point for { u i } i if x i → x is a blowup point for { u i } i and there exist two constants ρ, C > such that u i ( x ) ≤ Cd ¯ g ( x, x i ) − pi − for all x ∈ ∂M r { x i } , d ¯ g ( x, x i ) < ρ. Here ¯ g denotes the metric on the boundary induced by g and d ¯ g ( · , · ) is the geodesicdistance on the boundary between two points.Finally, given x i → x an isolated blow up point for { u i } i , and given ψ i : B + ρ (0) → M the Fermi coordinates centered at x i , we define the spherical averageof u i as ¯ u i ( r ) = 2 ω n − r n − Z ∂ + B + r u i ◦ ψ i dσ r and w i ( r ) := r − pi − ¯ u i ( r ) for < r < ρ. We say that x i → x is an isolated simple blow up point for { u i } i solutions of(2.16) if x i → x is an isolated blow up point for { u i } i and there exists ρ such that w i has exactly one critical point in the interval (0 , ρ ) .It is possible to prove the following proposition (see, for example [1, 10, 11, 19]) Proposition 7.
Let x i → x is an isolated blow up point for { u i } i and ρ as inDefinition 6. We set v i ( y ) = M − i ( u i ◦ ψ i )( M − p i i y ) , for y ∈ B + ρM pi − i (0) , where M i := u i ( x i ) Then, given R i → ∞ and β i → , up to subsequences, we have | v i − U | C (cid:16) B + Ri (0) (cid:17) < β i and lim i →∞ p i = nn − . Furthermore, if x i → x is an isolated simple blow up point for { u i } i , then thereexist C, ρ > such that COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 6 (1) M i u i ( ψ i ( y )) ≤ C | y | − n for all y ∈ B + ρ (0) r { } ; (2) M i u i ( ψ i ( y )) ≥ C − G i ( y ) for all y ∈ B + ρ (0) r B + r i (0) where r i := R i M − p i i and G i is the Green’s function which solves L g i G i = 0 in B + ρ (0) r { } G i = 0 on ∂ + B + ρ (0) B g i G i = 0 on ∂ ′ B + ρ (0) r { } and | y | n − G i ( y ) → as | y | → . The usual strategy to prove compactness of solutions of Yamabe problems datesback to the seminal Khuri Marques and Schoen paper [19]. Their idea is to provefirstly that only isolated simple blow up points may occur, then, to give a precisedescription of the asymptotic profile of a rescaled solution around an isolated simpleblow up points. Finally they rule out also the possibility of having isolated simpleblow up points.The key tool to accomplish these steps is a sign estimates of a Pohozaev typeformula for a blowing up sequence of solutions that we recall here.
Theorem 8 (Pohozaev Identity) . Let u a C -solution of the following problem (cid:26) L g u = 0 in B + r B g u + ( n − f − τ u p = 0 on ∂ ′ B + r for B + r = ψ − q ( B + g ( q, r )) for q ∈ ∂M , with τ = nn − − p > . Let us define ¯ P ( u, r ) := Z ∂ + B + r n − u ∂u∂r − r |∇ u | + r (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂r (cid:12)(cid:12)(cid:12)(cid:12) ! dσ r + r ( n − p + 1 Z ∂ ( ∂ ′ B + r ) f − τ u p +1 d ¯ σ g and P ( u, r ) = − Z B + r (cid:18) y a ∂ a u + n − u (cid:19) [( L g − ∆) u ] dy + n − Z ∂ ′ B + r (cid:18) ¯ y k ∂ k u + n − u (cid:19) h g ud ¯ y − τ ( n − p + 1 Z ∂ ′ B + r (cid:0) ¯ y k ∂ k f (cid:1) f − τ − u p +1 d ¯ y + (cid:18) n − p + 1 − n − (cid:19) Z ∂ ′ B + r ( n − f − τ u p +1 d ¯ y. Then ¯ P ( u, r ) = P ( u, r ) A sharp approximation of blow up points.
To describe the asymptoticprofile of a rescaled solution around an isolated simple blow up point in the case ofmanifolds with umbilic boundary we introduce the function γ q = γ which solves(2.17) ( − ∆ γ = (cid:2) ¯ R ikjl ( q ) y k y l + R ninj ( q ) y n (cid:3) ∂ ij U on R n + ∂γ∂y n = − nU n − γ on ∂ R n + . In [12] and in [11] the authors prove the following lemma.
Lemma 9.
Assume n ≥ . Given a point q ∈ ∂M , there exists a solution γ : R n + → R of the linear problem (2.17).In addition it holds (2.18) |∇ τ γ ( y ) | ≤ C (1 + | y | ) − τ − n for τ = 0 , , (2.19) Z R n + γ ∆ γdy ≤ COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 7 (2.20) Z ∂ R n + U nn − ( t, z ) γ ( t, z ) dz = 0; (2.21) γ (0) = ∂γ∂y (0) = · · · = ∂γ∂y n − (0) = 0 . Let x i → x an isolated simple blow up point for u i of solutions of (2.16) . Set v i ( y ) := δ pi − i u i ( δ i y ) for y ∈ B + Rδi (0) where δ i := u − p i i ( x i ) , we know that v i satisfies(2.22) L ˆ g i v i = 0 in B + Rδi (0) B ˆ g i v i + ( n −
2) ˆ f − τ i v p i i = 0 on ∂B + Rδi (0) where ˆ g i := ˜ g i ( δ i y ) = Λ n − x i ( δ i y ) g ( δ i y ) , ˆ f i ( y ) = f i ( δ i y ) , f i = Λ x i f → Λ x f and τ i = nn − − p i .Using the term γ we are able to give a good estimate of the rescaled solution v i around the isolated blow up point x i → x . Indeed we have (see [11, Proposition9]) Proposition 10.
Assume n ≥ . Let γ be defined in (2.17). There exist R, C > such that | v i ( y ) − U ( y ) − δ i γ x i ( y ) | ≤ Cδ i (1 + | y | ) − n (cid:12)(cid:12)(cid:12)(cid:12) ∂∂ j (cid:0) v i ( y ) − U ( y ) − δ i γ x i ( y ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ i (1 + | y | ) − n (cid:12)(cid:12)(cid:12)(cid:12) y n ∂∂ n (cid:0) v i ( y ) − U ( y ) − δ i γ x i ( y ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ i (1 + | y | ) − n (cid:12)(cid:12)(cid:12)(cid:12) ∂ ∂ j ∂ k (cid:0) v i ( y ) − U ( y ) − δ i γ x i ( y ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cδ i (1 + | y | ) − n for | y | ≤ R δ i . A characterization of function γ In this section we give a an accurate description of a solution γ of (2.17), similarlyto [18]. First we split γ = Φ + E where Φ = ˜Φ + ˜Φ is a polynomial function and ˜Φ , ˜Φ solve, respectively − ∆ ˜Φ = R ninj ( q ) y n ∂ ij U on R n + (3.1) − ∆ ˜Φ = 13 ¯ R ijkl ( q ) y k y l ∂ ij U on R n + (3.2)while E is an harmonic function solving(3.3) ( − ∆ E = 0 on R n + lim y n → ∂E∂y n = − nU n − E − q on ∂ R n + , with q = ∂ Φ ∂y n + nU n − Φ . COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 8
Lemma 11.
For n = 5 or n ≥ the function ˜Φ = 13 ¯ R ijkl ( q ) y i y j y k y l ( n − | ¯ y | + (1 + y n ) ) n + a n ( n − n + 4)( n − n −
4) 1( | ¯ y | + (1 + y n ) ) n +62 ) solves (3.2) for any a ∈ R . Lemma 12.
For n = 5 or n ≥ the function ˜Φ = R ninj ( q ) y i y j ( | ¯ y | + (1 + y n ) ) n − + n −
26 1 + y n − y n ( | ¯ y | + (1 + y n ) ) n + a n ( n − n − n − " ( n + 4) (1 + y n )( | ¯ y | + (1 + y n ) ) n +62 − | ¯ y | + (1 + y n ) ) n +42 + a ′ " n ( n − n − | ¯ y | + (1 + y n ) ) n +22 − n ( n + 2) 1( | ¯ y | + (1 + y n ) ) n +42 + a ′ n ( n − " | ¯ y | + (1 + y n ) ) n +22 . solves (3.1) for any a , a ′ , a ′ ∈ R . The proof of these two results is postponed in the appendix.For our purpose will be sufficient to fix a = a ′ = 0 . This allows also to extendthe previous results for n = 6 , as we summarize hereafter. Corollary 13.
For n ≥ the functions ˜Φ := R ninj ( q ) y i y j ( | ¯ y | + (1 + y n ) ) n − + n −
26 1 + y n − y n ( | ¯ y | + (1 + y n ) ) n ) ˜Φ := 13 ¯ R ijkl ( q ) y i y j y k y l (cid:26) n − | ¯ y | + (1 + y n ) ) n (cid:27) solve respectively (3.1) and (3.2).Proof. For n = 5 and n ≥ the result is proved in the appendix, in the proofs ofLemmas 11 and 12. For n = 6 , notice that both functions ˜Φ , ˜Φ are well definedThen the claim follows by direct computation. (cid:3) Case n = 7 , In [11] it is proved that, if x i → x is isolated simple blow-up point for u i , then,for n ≥ it holds P ( u i , r ) ≥ R ( U, U ) + R ( U, δ i γ ) + R ( δ i γ, U ) + O ( δ n − ) (4.1) ≥ δ i ( n − ω n − I nn ( n − n − n − n − (cid:20) ( n − | ¯ W ( x i ) | + 4( n − n − R nlnj ( x i ) (cid:21) − δ i Z R n + γ x i ∆ γ x i dy + o ( δ i ) . where(4.2) R ( u, v ) := − Z B + r/δi (cid:18) y b ∂ b u + n − u (cid:19) [( L ˆ g i − ∆) v ] dy. and ˆ g i := Λ n − x i ( δ i y ) g ( δ i y ) . COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 9
This, for n = 7 , becomes P ( u i , r ) ≥ δ i ω I (cid:20) | ¯ W ( x i ) | − R i j ( x i ) (cid:21) − δ i Z R n + γ ∆ γdy + o ( δ i ) , (4.3)and, for n = 8 , P ( u i , r ) ≥ δ i ω I | ¯ W ( x i ) | − δ i Z R n + γ ∆ γdy + o ( δ i ) , (4.4)The proof of (4.1) can be found in [11, Prop. 14].In this section we will prove the following result Lemma 14.
Let x i → x is an isolated simple blow-up point for u i solution of(2.16) then it holds P ( u i , r ) ≥ δ i ω I (cid:20) | ¯ W ( x i ) | + 754 R i j ( x i ) (cid:21) + o ( δ i ) for n = 7; (4.5) P ( u i , r ) ≥ δ i ω I (cid:20) | ¯ W ( x i ) | + 108934020 R i j ( x i ) (cid:21) + o ( δ i ) for n = 8 . (4.6)4.1. A crucial estimate.
To prove Theorem 1 it will be necessary to estimate thevalue of − R R n + γ ∆ γd ¯ ydy n in order to obtain that the right hand sides of (4.3) andof (4.4) are positive. By the description of γ in terms of E and Φ , we can simplifythis integral term as following. Lemma 15.
We have − Z R n + γ ∆ γd ¯ ydy n = Z ∂ R n + q Φ d ¯ y + Z ∂ R n + qEd ¯ y − Z R n + Φ∆Φ d ¯ ydy. Proof.
We get, since E is harmonic, and integrating by parts, that − Z R n + γ ∆ γd ¯ ydy n = − Z R n + ( E + Φ)∆Φ d ¯ ydy n = Z R n + ∇ ( E + Φ) ∇ Φ d ¯ ydy n + Z ∂ R n + ( E + Φ) ∂ n Φ d ¯ y = − Z R n + (∆Φ)Φ d ¯ ydy n − Z ∂ R n + ∂ n ( E + Φ)Φ d ¯ y + Z ∂ R n + ( E + Φ) ∂ n Φ d ¯ y = − Z R n + (∆Φ)Φ d ¯ ydy n − Z ∂ R n + ∂ n E Φ d ¯ y + Z ∂ R n + E∂ n Φ d ¯ y. Now, keeping in mind that q = ∂ Φ ∂y n + nU n − Φ and equation (3.3) we have − Z ∂ R n + ∂ n E Φ d ¯ y + Z ∂ R n + E∂ n Φ d ¯ y = Z ∂ R n + ( nU n − E + q )Φ d ¯ y + Z ∂ R n + E ( q − nU n − Φ) d ¯ y and we get the result. (cid:3) Lemma 16. If n > we have Z ∂ R n + qEd ¯ y = Z R n + |∇ E | d ¯ ydy n − n Z ∂ R n + U n − E d ¯ y ≥ . Proof.
First of all, by (3.3), integrating by parts we have Z R n + − E ∆ Ed ¯ ydy n = Z R n + |∇ E | d ¯ ydy n − n Z ∂ R n + U n − E d ¯ y − Z ∂ R n + qEd ¯ y COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 10 which proves the first equality. Notice that E ∈ D , ( R n + ) by difference, since γ, Φ ∈ D , ( R n + ) if n > .To conclude we argue as in [18, Lemma 4.6]. Firstly, observe that, since q = ∂ (˜Φ +˜Φ ) ∂y n + nU n − ( ˜Φ + ˜Φ ) , by Lemma 11, Lemma 12, and in light of identities(2.9), (2.10) we immediately get Z ∂ R n + qU d ¯ y = 0 . Now, we use E and U as test functions respectively in equation (2.8) and in equation(3.3), obtaining ( n − Z ∂ R n + U nn − Ed ¯ y = Z R n + ∇ U ∇ Ed ¯ ydy n = n Z ∂ R n + U nn − Ed ¯ y + Z ∂ R n + qU d ¯ y = n Z ∂ R n + U nn − Ed ¯ y, thus R ∂ R n + U nn − Ed ¯ y = 0 . At this point we can conclude the proof of the Lemma.In fact, it is well known that the function U is minimizer for J ( u ) = 12 Z R n + |∇ u | dy − ( n − n − Z | u | n − n − dy on the Nehari manifold M := (cid:26) u ∈ D , ( R n + ) r , : k u k D , = ( n − | u | n − n − n − n − (cid:27) . Since E ∈ D , ( R n + ) and R ∂ R n + U nn − Ed ¯ y = 0 we have that E ∈ T U M and we cancompute ≤ d dt J ( U + tE ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z R n + |∇ E | d ¯ ydy n − n Z ∂ R n + U n − E d ¯ y which ends the proof. (cid:3) We can further simplify the estimate for − R R n + γ ∆ γd ¯ ydy n . Lemma 17. If n > we have − Z R n + γ ∆ γd ¯ ydy n ≥ Z ∂ R n + ∂ ˜Φ ∂y n ˜Φ d ¯ y + Z ∂ R n + nU n − ˜Φ d ¯ y − Z R n + ˜Φ ∆ ˜Φ d ¯ ydy. Proof.
Combining Lemma 15 and Lemma 16 we have that − Z R n + γ ∆ γd ¯ ydy n ≥ Z ∂ R n + q Φ d ¯ y − Z R n + Φ∆Φ d ¯ ydy = Z ∂ R n + (cid:18) ∂ Φ ∂y n Φ + nU n − Φ (cid:19) d ¯ y − Z R n + Φ∆Φ d ¯ ydy. At this point we can prove immediately by (2.11) that Z ∂ R n + ∂ ˜Φ ∂y n ˜Φ d ¯ y = Z ∂ R n + ∂ ˜Φ ∂y n ˜Φ d ¯ y = Z ∂ R n + nU n − ˜Φ ˜Φ d ¯ y = 0 and by (2.12) that Z ∂ R n + ∂ ˜Φ ∂y n ˜Φ d ¯ y = Z ∂ R n + nU n − ˜Φ d ¯ y = 0 . COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 11
Now, taking in account equation (3.2), we have − Z R n + Φ∆ ˜Φ d ¯ ydy = 13 Z R n + Φ ¯ R ijkl y k y l ∂ ij U = n ( n − Z R n + Φ ¯ R ijkl y k y l y i y j ( | ¯ y | + (1 + y n ) ) − n = 0 again by (2.11) and (2.12). Similarly we prove that − R R n + ˜Φ ∆ ˜Φ d ¯ ydy = 0 and weconclude the proof. (cid:3) Case n = 7 . In this case we can take a = a ′ = a ′ = 0 in the expression of ˜Φ given in Lemma 12, so we set ˜Φ = R ninj y i y j A ( | ¯ y | , y n ) , where A ( | ¯ y | , y n ) := 112( | ¯ y | + (1 + y n ) ) n − + n −
26 1 + y n − y n ( | ¯ y | + (1 + y n ) ) n and we have the final result of this subsection Lemma 18. If n ≥ we have (4.7) − Z R n + γ ∆ γd ¯ ydy n ≥ n − R ninj "Z ∂ R n + A ( | ¯ y | , ∂∂y n A ( | ¯ y | , y n ) (cid:12)(cid:12)(cid:12)(cid:12) y n =0 | ¯ y | d ¯ y + n Z ∂ R n + A ( | ¯ y | , | ¯ y | + 1 | ¯ y | d ¯ y + n ( n − Z R n + A ( | ¯ y | , y n )( | ¯ y | + (1 + y n ) ) n +22 | ¯ y | y n d ¯ ydy in addition for n = 7 − Z R n + γ ∆ γdy ≥ ω I R i j . Proof.
We have, by (2.13) Z ∂ R n + ∂ ˜Φ ∂y n ˜Φ d ¯ y = Z ∂ R n + R ninj y i y j A ( | ¯ y | , R nlnk y l y k ∂∂y n A ( | ¯ y | , y n ) (cid:12)(cid:12)(cid:12)(cid:12) y n =0 d ¯ y = 2 n − R ninj Z ∂ R n + A ( | ¯ y | , ∂∂y n A ( | ¯ y | , y n ) (cid:12)(cid:12)(cid:12)(cid:12) y n =0 | ¯ y | d ¯ y. Similarly we have n Z ∂ R n + U n − ˜Φ d ¯ y = n Z ∂ R n + A ( | ¯ y | , | ¯ y | + 1 R ninj y i y j R nlnk y l y k d ¯ y = 2 nn − R ninj Z ∂ R n + A ( | ¯ y | , | ¯ y | + 1 | ¯ y | d ¯ y. COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 12
Finally, using (3.1) and (2.13) we have − Z R n + ˜Φ ∆ ˜Φ d ¯ ydy = Z R n + A ( | ¯ y | , y n ) R ninj y i y j R nknl y n ∂ kl U d ¯ ydy = R ninj R nknl Z R n + A ( | ¯ y | , y n ) y i y j y n ∂ kl U d ¯ ydy = n ( n − R ninj R nknl Z R n + A ( | ¯ y | , y n )( | ¯ y | + (1 + y n ) ) n +22 y i y j y k y l y n d ¯ ydy = 2 n ( n − n − R ninj Z R n + A ( | ¯ y | , y n )( | ¯ y | + (1 + y n ) ) n +22 | ¯ y | y n d ¯ ydy which proves the first claim.To conclude the proof we will have to estimate several integral quantities involv-ing the functions A ( | ¯ y | , y n ) and its derivative ∂∂y n A ( | ¯ y | , y n ) (cid:12)(cid:12)(cid:12)(cid:12) y n =0 = − (cid:0) | ¯ y | (cid:1) − − (cid:0) | ¯ y | (cid:1) − which we compute below. Notice also that, by change of variables, we have Z ∂ R | ¯ y | d ¯ y (1 + | ¯ y | ) α = ω I α and Z R | ¯ y | y βn d ¯ ydy n ((1 + y ) + | ¯ y | ) α = ω I α Z ∞ t β dt (1 + t ) α − . Keeping in mind (2.15) we have(4.8) Z ∂ R A ∂∂y n A | ¯ y | d ¯ y = − ω I . and(4.9) Z ∂ R A | ¯ y | (1 + | ¯ y | ) d ¯ y = ω I . Finally, in light of (2.14), we have(4.10) Z R A | ¯ y | y n ((1 + y n ) + | ¯ y | ) d ¯ ydy n = ω I . By (4.8), (4.9) and (4.10) we get the proof (cid:3)
Proof of first claim of Lemma 14.
By (4.3), (2.15), and by Lemma 18 we immedi-ately get (4.5). (cid:3)
Case n = 8 . For n = 8 we want to repeat the same strategy used for n = 7 .Unfortunately, taking all the coefficients equal to zero in ˜Φ does not prove the signcondition. For this case thus we consider ˜Φ = R ninj y i y j A ( | ¯ y | , y n , b ) , where A ( | ¯ y | , y , b ) := 112( | ¯ y | + (1 + y ) ) + 1 + y n − y n ( | ¯ y | + (1 + y ) ) + b ( | ¯ y | + (1 + y ) ) . Lemma 19.
For n = 8 and b = − we have − Z R γ ∆ γdy ≥ ω I R i j COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 13
Proof.
We can recast (4.7) for n = 8 obtaining(4.11) − Z R γ ∆ γd ¯ ydy ≥ R i j "Z ∂ R A ( | ¯ y | , , b ) ∂∂y A ( | ¯ y | , y , b ) (cid:12)(cid:12)(cid:12)(cid:12) y =0 | ¯ y | d ¯ y + 8 Z ∂ R A ( | ¯ y | , , b ) | ¯ y | | ¯ y | + 1 d ¯ y +48 Z R A ( | ¯ y | , y , b ) | ¯ y | y ( | ¯ y | + (1 + y ) ) d ¯ ydy . We have(4.12) ω Z ∂ R A ∂A∂y (cid:12)(cid:12)(cid:12)(cid:12) y =0 | ¯ y | d ¯ y = (cid:20) − − b − b (cid:21) I , (4.13) ω Z ∂ R A | ¯ y | | ¯ y | + 1 d ¯ y = I (cid:20) b + 716 b (cid:21) and(4.14) ω Z R A ( | ¯ y | , y , b ) | ¯ y | y ( | ¯ y | + (1 + y ) ) d ¯ ydy = I (cid:20)
56 + b (cid:21) . So by (4.12), (4.13) and (4.14), the inequality (4.11) becomes − Z R γ ∆ γd ¯ ydy ≥ R i j ω I (cid:20) − − b − b (cid:21) which for b = − gives the claim. (cid:3) Proof of second claim of Lemma 14.
By (4.3), (2.15), and by Lemma 19 we imme-diately get (4.6). (cid:3) Case n = 6 When dealing with low dimensions, often it is convenient to work in cylindricalsets D + r := [0 , r ] × B r ⊂ R instead of spheres B + r = B r ∩ R . In the limit r → ∞ the difference between thetwo approaches is of higher order, but the boundary of D + r is easier to manage. So,we compute the Pohozaev identity on cylindrical sets. Again, as in [11, Proposition14] we have that, if x i → x is isolated simple blow-up point for u i , then(5.1) P ( u i , r ) ≥ R ( U, U ) + R ( U, δ i γ ) + R ( δ i γ, U ) + O ( δ i ) where R ( u, v ) in this case is R ( u, v ) := − Z D + r/δ (cid:18) y b ∂ b u + n − u (cid:19) [( L ˆ g i − ∆) v ] dy. Throughout this section we will the following lemma.
Lemma 20.
Let x i → x is an isolated simple blow-up point for u i solution of(2.16) then it holds P ( u i , r ) ≥ R ( U, U ) + R ( U, δ i γ x i ) + R ( δ i γ x i , U ) + O ( δ i )= ω I δ i log (cid:18) δ i (cid:19) (cid:20) | ¯ W ( x i ) | + 815 R i s ( x i ) (cid:21) + O ( δ i ) (5.2) COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 14
Remark . We recall the following elementary identity, obtained by change ofvariables(5.3) Z r Z B n − r | ¯ y | β dy [(1 + y n ) + | ¯ y | ] α = Z r (1 + y n ) β + n − dy n (1 + y n ) α Z B n − r | ¯ y | β d ¯ y [1 + | ¯ y | ] α and, finally, that Z r/δ y αn (1 + y n ) α +1 = log (cid:18) δ (cid:19) + O (1) , and Z r/δ y α +2 n − y αn (1 + y n ) α +3 = log (cid:18) δ (cid:19) + O (1) for α = 0 , , With these premises we have the following result (in order to simplify notation,we denote δ for δ i and q for x i ). Lemma 22.
We have R ( U, U ) = ω I δ log (cid:16) rδ (cid:17) (cid:20) | ¯ W ( q ) | − R nins (cid:21) + O ( δ ) . Proof.
The proof is similar to [11, Lemma 15]. We focus here on the main differ-ences, omitting the standard calculations. By definition of L ˆ g i we have R ( U, U ) = ( n − Z D + rδ − | y | − y n ) + | ¯ y | ] n +1 ny i y j (cid:0) g ij ( δy ) − δ ij (cid:1) dy − ( n − Z D + rδ − | y | − y n ) + | ¯ y | ] n (cid:0) g jj ( δy ) − (cid:1) dy − ( n − Z D + rδ − | y | − y n ) + | ¯ y | ] n δ∂ i g ij ( δy ) y j dy − ( n − n − Z D + rδ − | y | − y n ) + | ¯ y | ] n − δ R g ( δy ) dy + O ( δ )=: A + A + A + A + O ( δ ) . Using the symmetries of the curvature tensor and the expansion of the metric wehave that, for n = 6 , A = δ R nins Z D + rδ − ( | y | − | ¯ y | y n [(1 + y n ) + | ¯ y | ] dy (5.4) + δ R ninj,ji Z D + rδ − ( | y | − | ¯ y | y n [(1 + y n ) + | ¯ y | ] dy + O ( δ ) .A + A = − δ R nins Z D + rδ − ( | y | − y n [(1 + y n ) + | ¯ y | ] dy (5.5) − δ R ninj,ij Z D + rδ − ( | y | − | ¯ y | y n [(1 + y n ) + | ¯ y | ] dy + O ( δ ) . and A = δ | ¯ W ( q ) | Z D + rδ − ( | y | − | ¯ y | [(1 + y n ) + | ¯ y | ] dy (5.6) + δ R nins Z D + rδ − ( | y | − y n [(1 + y n ) + | ¯ y | ] dy + δ R ninj,ij Z D + rδ − ( | y | − y n [(1 + y n ) + | ¯ y | ] dy + O ( δ ) . COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 15
Now, using (5.3) we have Z D + rδ − ( | y | − | ¯ y | y n [(1 + y n ) + | ¯ y | ] = Z r/δ y n dy n Z B r/δ | ¯ y | d ¯ y [(1 + y n ) + | ¯ y | ] + Z r/δ y n ( y n − dy n Z B r/δ | ¯ y | d ¯ y [(1 + y n ) + | ¯ y | ] = Z r/δ y n (1 + y n ) dy n Z B r/δ | ¯ y | d ¯ y [1 + | ¯ y | ] + Z r/δ y n − y n (1 + y n ) dy n Z B r/δ | ¯ y | d ¯ y [1 + | ¯ y | ] = ω Z r/δ y n (1 + y n ) dy n Z r/δ ρ d ¯ y [1 + ρ ] + ω Z r/δ y n − y n (1 + y n ) dy n Z B r/δ ρ d ¯ y [1 + ρ ] = log(1 /δ ))( I + I ) + O (1) . In a similar way we proceed for all the terms in (5.4), (5.5), (5.6), obtaining A = ω δ log (cid:18) δ (cid:19) (cid:20) R nins ( I + I ) + 4835 R ninj,ji ( I + I ) (cid:21) + O ( δ ) , (5.7) A + A = − ω δ log (cid:18) δ (cid:19) (cid:20) R nins ( I + I ) + 85 R ninj,ij ( I + I ) (cid:21) + O ( δ ) , (5.8)and A = ω δ log (cid:18) δ (cid:19) | ¯ W ( q ) | ( I + I ) (5.9) + ω δ log (cid:18) δ (cid:19) (cid:0) R nins + R ninj,ij (cid:1) ( I + I ) . Finally, in light of (2.15), by (5.7), (5.8) and (5.9) we get the claim. (cid:3)
Lemma 23.
We have R ( U, δ γ ) + R ( δ γ, U ) = − δ Z D + rδ − γ ∆ γdy + O ( δ ) . Proof.
Again, we follow the main lines of [11, Lemma 16]. We have by definitionof R ( u, v ) , by (2.3) and (2.18), that R ( U, δ γ ) + R ( δ γ, U ) = − δ Z D + rδ − ( y b ∂ b U + 2 U ) (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j γdy − δ Z D + rδ − y b ∂ b γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dy − δ Z D + rδ − γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dy + O ( δ )=: − δ ( A + A + A ) + O ( δ ) . By (2.17), immediately we have(5.10) A = 2 Z D + rδ − γ ∆ γ. COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 16
Integrating by parts, and recalling that the index b = 1 , . . . , n while i, j, k, l, s =1 , . . . , n − , we have A =6 Z D + rδ − γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dy + Z D + rδ − y b γ∂ b (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dy + Z D + rδ − y b γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ b ∂ i ∂ j U dy − Z r/δ Z ∂B r/δ y s ν s γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dσdy n − Z B r/δ y n γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U (cid:12)(cid:12)(cid:12)(cid:12) y n = rδ d ¯ y. (5.11)Now, we estimate the boundary terms. On ∂B r/δ we have y s ν s = | ¯ y | = r/δ . Takingin account (2.18) we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z r/δ Z ∂B r/δ y s ν s γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dσdy n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z r/δ Z ∂B r/δ (cid:16) rδ (cid:17) − dσdy n = O (1) . Similarly we obtain (cid:12)(cid:12)(cid:12)(cid:12)R B r/δ y n γ q (cid:2) ¯ R ikjl y k y l + R ninj y n (cid:3) ∂ i ∂ j U (cid:12)(cid:12) y n = rδ d ¯ y (cid:12)(cid:12)(cid:12)(cid:12) = O (1) .Moreover, Z D + rδ − y b γ∂ b (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dy = Z D + rδ − y s γ∂ s (cid:20)
13 ¯ R ikjl y k y l (cid:21) ∂ i ∂ j U dy + Z R n + y n γ∂ n (cid:2) R ninj y n (cid:3) ∂ i ∂ j U dy = 2 Z D + rδ − γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ i ∂ j U dy = − Z D + rδ − γ ∆ γ, and, using (2.17) for the first term of (5.11) we have A = − Z D + rδ − γ ∆ γdy + Z D + rδ − y b γ (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ b ∂ i ∂ j U dy + O (1) . For the term A we integrate by parts twice. As before, all the boundary terms areestimated by a constant number. So, using the symmetries of the curvature tensorwe have, after the first integration, A = Z D + rδ − ( ∂ i U + y b ∂ i ∂ b U + 2 ∂ i U ) (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ j γdy + Z D + rδ − ( y b ∂ b U + 2 U ) ∂ i (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ j γdy + O (1)= Z D + rδ − (3 ∂ i U + y b ∂ i ∂ b U ) (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) ∂ j γdy + O (1) COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 17
And, integrating again, A = − Z D + rδ − (4 ∂ j ∂ i U + y b ∂ j ∂ i ∂ b U ) (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) γdy + O (1)=4 Z D + rδ − γ ∆ γdy − Z D + rδ − y b ∂ j ∂ i ∂ b U (cid:20)
13 ¯ R ikjl y k y l + R ninj y n (cid:21) γdy + O (1) . Adding A , A and A we get the proof. (cid:3) Lemma 24.
We have Z D + rδ − γ ∆ γdy = Z D + rδ − ˜Φ ∆ ˜Φ dy + O (1) . Proof.
We have, integrating by parts, and since E is harmonic(5.12) Z D + rδ − γ ∆ γdy = Z D + rδ − (Φ + E )∆(Φ + E ) dy = Z D + rδ − (Φ + E )∆Φ dy = − Z D + rδ − ∇ (Φ + E ) ∇ Φ dy + Z r/δ Z ∂B r/δ (Φ + E ) ∇ Φ · νdσdy n . By the decay of γ in (2.18) and by the explicit expression of Φ in Corollary 13, weobtain |∇ τ E ( y ) | ≤ C (1 + | y | ) − τ − n for τ = 0 , , and the same holds for Φ . At this point we can easily see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z r/δ Z ∂B r/δ (Φ + E ) ∇ Φ · νdσdy n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (1) . We can integrate again by parts in (5.12) and, keeping in mind that again theboundary term is estimate by a constant and that E is harmonic, we get Z D + rδ − γ ∆ γdy = − Z D + rδ − ∇ (Φ + E ) ∇ Φ dy + O (1)= Z D + rδ − ∆(Φ + E )Φ dy + O (1) = Z D + rδ − Φ∆Φ dy + O (1) . Now we proceed in Lemma 17, using (2.11) and (2.12) to prove that Z D + rδ − Φ∆Φ dy = Z D + rδ − ˜Φ ∆ ˜Φ dy and concluding the proof. (cid:3) Lemma 25.
We have R ( U, δ γ ) + R ( δ γ, U ) = ω I δ log (cid:16) rδ (cid:17) R nins + O ( δ ) . Proof.
By Lemma 23 and Lemma 24, taking in account (2.13), we have R ( U, δ γ ) + R ( δ γ, U ) = − δ Z D + rδ − ˜Φ ∆ ˜Φ dy + O ( δ )= 835 δ R nins "Z D + rδ − | ¯ y | y n dy ( | ¯ y | + (1 + y n ) ) + 8 Z D + rδ − | ¯ y | ( y n + y n − y n ) dy ( | ¯ y | + (1 + y n ) ) + O ( δ )=: 835 δ R nins [ B + B ] + O ( δ ) . COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 18
By (5.3) we have B = Z r/δ y n dy n (1 + y n ) Z B r/δ | ¯ y | d ¯ y (1 + | ¯ y | ) = log (cid:18) δ (cid:19) I B =8 Z r/δ ( y n + y n − y n ) dy n (1 + y n ) Z B r/δ | ¯ y | d ¯ y (1 + | ¯ y | ) = 8 log (cid:18) δ (cid:19) I and, by (5) we get the proof. (cid:3) Proof of Lemma 20.
Lemma 22 and Lemma 25 lead us to (5.2). (cid:3) Proof of the main result
We start proving the following Weyl vanishing property.
Proposition 26.
Let n ≥ and let x i → x be an isolated simple blow up pointfor u i solution of (2.16) Then W ( x ) = 0 . Proof.
By [11, Propositions 5 and 18] we have P ( u i , r ) ≤ Cδ n − i . This, combined with (4.5), (4.6) and with (5.2) gives | ¯ W ( x i ) | + R ninj ( x i ) ≤ Cδ i for n = 8 Cδ i for n = 7 − C (log( δ i )) − for n = 6 , which gives the result, since W ( x ) = 0 if and only if both ¯ W and R nins vanish at x . (cid:3) Now we give a series of results whose proofs are very similar to the ones containedin [11], so we will omit them.First, we can rule out the possibility to have isolated blow up points which arenot simple. As in the previous proposition, for the proof it is crucial that P ( u i , r ) is strictly positive when | W ( x ) | 6 = 0 , which we have proved in equations (4.5) and(5.2). Proposition 27.
Assume n ≥ . Let x i → x be an isolated simple blow up pointfor u i solution of (2.16). Assume | W ( x ) | 6 = 0 . Then x is isolated simple. Next, we can prove a splitting lemma.
Proposition 28.
Assume n ≥ . Given β > and R > there exist two constants C , C > (depending on β , R and ( M, g ) ) such that, if u is a solution of (6.1) (cid:26) L g u = 0 in MB g u + ( n − f − τ u p = 0 on ∂M and max ∂M u > C , then τ := nn − − p < β and there exist q , . . . , q N ∈ ∂M , with N = N ( u ) ≥ with the following properties: for j = 1 , . . . , N (1) Set r j := Ru ( q j ) − p , then (cid:8) B r j ∩ ∂M (cid:9) j are a disjoint collection; (2) we have (cid:12)(cid:12) u ( q j ) − u ( ψ j ( y )) − U ( u ( q j ) p − y ) (cid:12)(cid:12) C ( B +2 rj ) < β (here ψ j are theFermi coordinates at point q j ; COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 19 (3) we have u ( x ) d ¯ g ( x, { q , . . . , q n } ) p − ≤ C for all x ∈ ∂Mu ( q j ) d ¯ g ( q j , q k ) p − ≥ C for any j = k. Here d ¯ g is the geodesic distance on ∂M .Assume also W ( x ) = 0 for any x ∈ ∂M . Then there exists d = d ( β, R ) such that,for any u solution of (6.1) with max ∂M u > C , we have min i = j ≤ i, j ≤ N ( u ) d ¯ g ( q i ( u ) , q j ( u )) ≥ d. Now we can prove our main result.
Proof of Theorem 1.
We proceed by contradiction, supposing that there exists asequence of solutions { u i } i of problems (2.16) and that x i → x is a blow up pointfor u i . Let q ( u i ) , . . . q N ( u i ) ( u i ) the sequence of points given by proposition 28. Wecan prove that d ¯ g ( x i , q k i ( u i )) → for some sequence of k i . So q k i → x is a blowup point for u i . Now by propositions 28 and 27 we have that q k i → x is an isolatedsimple blow up point for u i . Then, by 26, this should imply that | W ( x ) | = 0 whichcontradicts our hypotheses, proving the theorem. (cid:3) Appendix: proofs of Lemma 11 and Lemma 12
We recall a result contained in [18].
Lemma 29.
Suppose n = 5 or n ≥ . We have (1) The function Φ := 14( n −
6) 1( | ¯ y | + (1 + y n ) ) n − + a ( | ¯ y | + (1 + y n ) ) n − + a , for a , a ∈ R satisfies(7.1) − ∆Φ := 1( | ¯ y | + (1 + y n ) ) n − (2) The function Φ := n − y n +1( | ¯ y | +(1+ y n ) ) n − + a y n +1( | ¯ y | +(1+ y n ) ) n = − (cid:16) n − (cid:17) ∂ n Φ , for a ∈ R satisfies(7.2) − ∆Φ := y n + 1( | ¯ y | + (1 + y n ) ) n − (3) The function Φ := n −
4) 1( | ¯ y | +(1+ y n ) ) n − + a ( | ¯ y | +(1+ y n ) ) n − + a ′ , for a , a ′ ∈ R satisfies(7.3) − ∆Φ := 1( | ¯ y | + (1 + y n ) ) n − Proof.
The first claim is proved in [18, Lemma A.1] (in particular in formula (A.2)).The second claim is proved again in [18, Lemma A.1], while the last claim corre-sponds to [18, Lemma A.2]. (cid:3)
Lemma 30.
Let n ≥ . The function ˜Φ = ( n −
8) 1( | ¯ y | +(1+ y n ) ) n − + a | ¯ y | +(1+ y n ) ) n − + a for n = 8 − log( | ¯ y | + (1 + y n ) ) + a | ¯ y | +(1+ y n ) ) + a for n = 8 COMPACTNESS RESULT FOR LOW DIMENSIONAL MANIFOLDS 20 for a , a ∈ R satisfies (7.4) − ∆ ˜Φ := 1( | ¯ y | + (1 + y n ) ) n − . Proof.
By change of variables we have that(7.5) − ∆ ˜Φ (¯ y, y n −
1) = 1( | ¯ y | + y n ) n − = 1 r n − , where r := p | ¯ y | + y n . So, in spherical coordinates, set ϕ ( r ) = ˜Φ (¯ y, y n − , (7.5)becomes(7.6) − ϕ ′′ − n − r ϕ ′ = 1 r n − and one can check that ϕ ( r ) = (cid:26) n −
8) 1 r n − + a r n − + a for n = 8 − log r + a r + a for n = 8 solves (7.6). (cid:3) Lemma 31.
Let n = 5 or n ≥ . Set β kl := ∂ kl ˜Φ ( n − n − + Φ ( n − δ kl . Then (7.7) − ∆ β kl = y k y l U. Proof.
By (7.4) we have − ∆ ∂ kl ˜Φ = ( n − n − y k y l ( | ¯ y | +(1+ y n ) ) n − − ( n − δ kl ( | ¯ y | +(1+ y n ) ) n − and by(7.1) we get the result. (cid:3) Now we can achieve the prove of the two lemmas.
Proof of Lemma 11.
Since ¯ R ijkk = 0 have ˜Φ := ¯ R ijkl ( q ) ∂ ij (cid:16) ∂ kl ˜Φ ( n − n − (cid:17) = ¯ R ijkl ( q ) ∂ ij β kl . Thus, by (7.7), we have − ∆ ˜Φ = 13 ¯ R ijkl ( q ) ∂ ij ( − ∆ β kl ) = 13 ¯ R ijkl ( q ) ∂ ij ( y k y l U ) 13 ¯ R ijkl ( q ) y k y l ∂ ij U using the symmetry of the curvature tensor. (cid:3) Proof of Lemma 12.
By Lemma 11 and Lemma 30 we have that − ∆ " ∂ nn ˜Φ ( n − n −
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E-mail address : [email protected] A. M. Micheletti,Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5, 56126 Pisa,Italy
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