A compactness theorem of the fractional Yamabe problem, Part I: The non-umbilic conformal infinity

Assume that $(X, g^+)$ is an asymptotically hyperbolic manifold, $(M, [\bar{h}])$ is its conformal infinity, $\rho$ is the geodesic boundary defining function associated to $\bar{h}$ and $\bar{g} = \rho^2 g^+$. For any $\gamma \in (0,1)$, we prove that the solution set of the $\gamma$-Yamabe problem on $M$ is compact in $C^2(M)$ provided that convergence of the scalar curvature $R[g^+]$ of $(X, g^+)$ to $-n(n+1)$ is sufficiently fast as $\rho$ tends to 0 and the second fundamental form on $M$ never vanishes. Since most of the arguments in blow-up analysis performed here is irrelevant to the geometric assumption imposed on $X$, our proof also provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem.


Introduction
Given any n ∈ N, let (X n+1 , g + ) be an asymptotically hyperbolic manifold with conformal infinity (M n , [h]). According to [31,56,38,11,28], there exists a family of self-adjoint conformal covariant pseudo-differential operators P γ [g + ,h] on M in general whose principal symbols are the same as those of (−∆h) γ . If (X, g + ) is Poincaré-Einstein and γ ∈ N, the operator P γ [g + ,h] coincides with the GJMS operator constructed by Graham et al. [30] via the ambient metric; refer to Graham and Zworski [31]. In particular, P γ [g + ,h] is equal to the conformal Laplacian or the Paneitz operator for γ = 1 or 2, respectively.
Let us call Q γ [g + ,h] = P γ [g + ,h](1) the γ-scalar curvature. One natural question is if a conformal metrich ′ toh on M exists whose γ-scalar curvature Q γ [g + ,h ′ ] is constant. By virtue of the conformal covariance property of P γ , it is reduced to search a smooth solution of the equation P γ [g + ,h]u = cu p and u > 0 on (M n ,h) (1.1) for some constant c ∈ R provided n > 2γ and p = 2 * n,γ − 1 = (n + 2γ)/(n − 2γ). For γ = 1, the study on the existence of a solution to (1.1) was initiated by Yamabe [70] and completely solved through the successive works of Trudinger [68], Aubin [4] and Schoen [62]. See also Lee and Parker [43] and Bahri [5] where a unified proof based on the use of conformal normal coordinates and a proof not depending on the positive mass theorem are devised, respectively. If γ = 2 (and n ≥ 5), existence theory of (1.1) becomes considerably harder because of the lack of a maximum principle. Up to now, only partial results are available such as Qing and Raske [60], Gursky and Malchiodi [32] and Hang and Yang [34]. In [34], the authors could treat a general class of manifolds having the property that the Yamabe constant (1.4) is positive and there exists a representative of the conformal class [h] with semi-positive Q-curvature Q 2 . Meanwhile, equation (1.1) with γ = 1/2 has deep relationship with the boundary Yamabe problem (or the higher dimensional Riemann mapping theorem) formulated by Escobar [21]; refer to Remark 1.2 (1). Its existence theory has been completed due to the effort of Escobar [21,22], Marques [52,53], Almaraz [1], Chen [12] and Mayer and Ndiaye [54].
If γ / ∈ N, it is not a simple task to solve (1.1) directly, since the operator P γ [g + ,h] is nonlocal and defined in a rather abstract way. However, Chang and González [11] discovered that (1.1) can be interpreted as a Caffarelli-Silvestre type degenerate elliptic equation in [8], which is indeed local, for which a variety of well-known techniques like constraint minimization and the Moser iteration technique can be applied; see Proposition 2.1 for a more precise description. From this observation, González and Qing [28] succeeded to find solutions to (1.1) for γ ∈ (0, 1) under the hypothesis that the dimension n of the underlying manifold M is sufficiently high and M is the non-umbilic boundary of X. Their approach was further developed in the works of González and Wang [29] and the authors of this paper [39], which cover most cases when the local geometry dominates. In [39], the author also established the existence result for 2-dimensional or locally conformally flat manifolds provided that a certain technical assumption on the Green's function of P γ holds. Recently, Mayer and Ndiaye [55] and Daskalopoulos et al. [14] pursued the critical point at infinity approach and the flow approach, respectively, removing the technical condition on the Green's function.
Furthermore, Case and Chang [10] obtained an extension result for γ ∈ (1, n/2) which generalizes [11]. By utilizing it and adapting the argument in Gursky and Malchiodi [32], they also deduced that P γ [g + ,h] satisfies a strong maximum principle for γ ∈ (1, min{2, n/2}) when [h] possesses a metric whose scalar curvature is nonnegative and γ-curvature is semipositive. It is plausible that this with some further ideas in [32] may allow one to get certain existence results of (1.1) under the prescribed conditions. In many cases, (1.1) may contain higher Morse index (or energy) solutions as shown in [64,59] for γ = 1. When γ = 1 and (M,h) is the round sphere, Obata [58] found all the solutions to (1.1) and concluded that there is no uniform L ∞ (M )-bound on them. Analogously, if γ ∈ (0, 1) and (X, g + ) is the Poincaré ball whose conformal infinity is the standard sphere, the classification result [36,Theorem 1.8] of Jin et al. shows that the solution set of (1.1) is not bounded in L ∞ (M ).
In this regards, for manifolds M that are non-conformally diffeomorphic to the standard sphere, Schoen [63] raised a question on the C 2 (M )-compactness of the solution set to (1.1) with γ = 1 and suggested a general strategy towards its proof. The first affirmative answer was given by Schoen himself [65] in the locally conformally flat case. Li and Zhu [50] obtained it in n = 3 and Druet [20] did it in n ≤ 5. If n ≥ 6, the analysis is more delicate because one needs to prove that the Weyl tensor vanishes at an order greater than ⌊(n − 6)/2⌋ at a blow-up point. By solving this technical difficulty, Marques [51] and Li and Zhang [47] could deal with the situation that either n ≤ 7 holds or the Weyl tensor never vanishes on M . Assuming the validity of the positive mass theorem and performing refined blow-up analysis on the basis of the linearized problem (as in our Section 4), Li and Zhang [48] extended the result up to dimension 11, and Khuri et al. [39] finally verified it for n ≤ 24. Surprisingly, according to Brendle [6] and Brendle and Marques [7], there are C ∞ -metrics on the sphere S n with n ≥ 25 such that even though they are not conformally equivalent to the canonical metric, a blowing-up family of solutions to (1.1) does exist.
For γ = 2, Y. Li and Xiong [46] obtained the C 4 (M )-compactness result if 5 ≤ n ≤ 9 or M is locally conformally flat, and the positive mass theorem holds for the Paneitz operator P 2 ; see also the previous works [35,60,44]. On the other hand, Wei and Zhao [69] established a non-compactness result for n ≥ 25. While it is believed that C 4 (M )-compactness holds in general up to dimension 24, a rigorous proof is not known yet.
For the boundary Yamabe problem, corresponding to the case γ = 1/2, compactness results were deduced when X n+1 is locally conformally flat [26] or n + 1 = 3 [27]. Compactness of the solution set also follows under the assumption that the second fundamental form on M vanishes nowhere [1]; refer to [33,19] for more results. Almaraz [3] showed that a blow-up phenomenon still happens if n + 1 ≥ 25.
If γ is non-integer value, only a few has been revealed up to now. As far as we know, the only article that investigates compactness of the solution set to (1.1) for γ ∈ [1, n/2) is [61] due to Qing and Raske, which concerns with locally conformally flat manifolds M with the positive Yamabe constant and the Poincaré exponent less than (n − 2γ)/2. For the non-compactness, the authors of this paper [41] constructed asymptotic hyperbolic manifolds which are small perturbations of the Poincaré ball and exhibit a blow-up phenomenon for n ≥ 24 if γ ∈ (0, γ * ) and n ≥ 25 if γ ∈ [γ * , 1) where γ * is a number close to 0.940197. However, the compactness issue on (1.1) for γ ∈ (0, 1) has not been discussed in the literature so far, unless the underlying manifold is the Poincaré ball; see [36].
In this paper, we are concerned with the compactness of the solution set to the γ-Yamabe problem (1.1) provided γ ∈ (0, 1) and c > 0. As can be predicted from the representation theorem of Palais-Smale sequences associated with fractional Yamabe-type equations in [25], the conformal covariance property of P γ makes us perform local analysis even though it is a nonlocal operator.
We will state the main theorem under a slightly more general setting; more precisely, we will allow p to be subcritical. Since we always assume that the metric g + in X is fixed, we write P γ h = P γ [g + ,h] and Q γ h = Q γ [g + ,h]. Theorem 1.1. Let γ ∈ (0, 1), and (X n+1 , g + ) be an asymptotically hyperbolic manifold with conformal infinity (M n , [h]). Denote by ρ a geodesic defining function associated to M , i.e., a unique defining function splitting the metricḡ = ρ 2 g + as dρ 2 + h ρ near M where {h ρ } ρ is a family of metrics on M such that h 0 =h. Assume that the first L 2 (X)-eigenvalue λ 1 (−∆ g + ) of the Laplace-Beltrami operator −∆ g + satisfies the inequality is positive and where R[g + ] is the scalar curvature of (X, g + ).
If the second fundamental form π of (M,h) ⊂ (X,ḡ) never vanishes, then for any ε 0 > 0 small, there exists a constant C > 1 depending only on X n+1 , g + ,h, γ and ε 0 such that  (2) Hypothesis (1.5) implies that the mean curvature H is identically 0 on M ; see [42,Lemma 2.3]. As a particular consequence, π = π − Hḡ on M , the latter tensor being the trace-free second fundamental form. Thus our theorem generalizes the result of Almaraz [2, Theorem 1.2] on the boundary Yamabe problem (corresponding to the case γ = 1/2) under the further assumption that H = 0 on M .
(3) The standard transversality argument implies that the set of Riemannian metrics on M whose second fundamental form is nonzero everywhere is open and dense in the space of all Riemannian metrics on M . On the other hand, there exists an asymptotically hyperbolic manifold X n+1 that can be realized as a small perturbation of the Poincaré half-space, for which the solution set of the γ-Yamabe problem is non-compact provided that n ≥ 24 or 25 according to the magnitude of γ ∈ (0, 1); refer to [41]. In this example, the conformal infinity M is the totally geodesic (in particular, umbilic) boundary of X.
(4) It is remarkable that the dimension restriction (1.2) is exactly same as the one appeared in the existence result [42, Corollary 2.7] for equation (1.1) on non-umbilic conformal infinities.
(5) We believe that the same type of a compactness result as Theorem 1.1 can be achieved for any umbilic non-locally conformally flat conformal infinity whose Weyl tensor never vanishes, whenever n ≥ 7 for γ ∈ [1/2, 1) and n ≥ 8 for γ ∈ (0, 1). The condition on the dimension is suggested by an existence result [42,Corollary 3.4]. Also, if a suitable condition on the Green's function on P γ h is assumed, a compactness result may be obtained provided that M is either locally conformally flat or 2-dimensional as in Felli and Ould Ahmedou [26,27]. An interesting question is to confirm whether the dimension assumption in [41] is optimal to form a blow-up phenomenon of the solution set to (1.1).
One can establish Theorem 1.1 from the next theorem and elliptic regularity theory; see Subsection 6.2 and Appendix A. Theorem 1.3 (Vanishing theorem of the second fundamental form). For γ ∈ (0, 1) and n ∈ N satisfying (1.2), let (X n+1 , g + ) be an asymptotically hyperbolic manifold with conformal infinity (M n , [h]) such that (1.3) is valid. Moreover assume that ρ is a geodesic defining function associated to M ,ḡ = ρ 2 g + , Λ γ (M, [h]) > 0, and (1.5) holds. If {(u m , y m )} m∈N is a sequence of pairs in C ∞ (M ) × M such that each u m is a solution of (1.1), y m is a local maximum point of u m satisfying u m (y m ) → ∞ and y m → y 0 ∈ M as m → ∞, then the second fundamental form π at y 0 vanishes.
As a corollary of Theorem 1.1, we can compute the Leray-Schauder degree of all solutions to equation (1.1) if every hypothesis imposed in the theorem holds. Since P γ h is a self-adjoint operator as shown in [31], any L p+1 (M )-normalized minimizer of for which it holds that F p (u) = 0 if and only if u is a solution of (1.1). Elliptic estimate in Lemma A.2 below implies that F p is the sum of the identity and a compact map. Besides we infer from Lemma 6.6, a consequence of Theorem 1.1, that 0 / ∈ F p (∂D Λ ) for all 1 ≤ p ≤ 2 * n,γ −1 if Λ is sufficiently large. Therefore the Leray-Schauder degree deg(F p , D Λ , 0) of the map F p in the domain D Λ with respect to the point 0 ∈ L ∞ (M ) is well-defined.
In particular, the fractional Yamabe equation (1.1) possesses a solution.
Theorem 1.4 gives a new proof on the existence of a solution to (1.1) under rather restrictive assumptions. Compare it with [28,42]. We also expect that there exists the strong Morse inequality in our framework; refer to [39,Theorem 1.4].
Although certain parts of the proof can be achieved from minor modification of the classical arguments, there are still plenty of technical difficulties which demand new ideas. We will pay attention to, for instance, the following features.
-In our setting, the freedom of conformal transform is limited on the boundary M and one cannot use the standard conformal normal coordinate on the whole manifold X.
To handle the vertical direction to M , we use the geometric assumption (1.5) and examine the first-order partial differential equation satisfied by the geodesic defining function; -We largely depend on the extension result of Chang and González [11] to analyze solutions. Because of the degeneracy of the extended problem (2.4), it is not simple to study the asymptotic behavior of the Green's function near its singularity; see Appendix B.1 where some of its qualitative properties are obtained. Hence, in order to show the decay property of rescaled solutions, we do not use potential analysis, but iteratively apply the rescaling argument based on the maximum principle; -Regularity theory which we require is technically more difficult to deduce than ones for the classical local problems, or even nonlocal problems on the Euclidean space; -Because the bubbles have no explicit expression in general (see Subsection 2.2), we have to put some extra efforts compute integrals involving them. To reduce overlaps, we will omit the proofs of several intermediate results which closely follow the standard arguments, leaving appropriate references. Our main concern is to clarify the novelty of the nonlocal problems defined on general conformal infinities.
The paper is organized as follows: In Section 2, we recall some analytic and geometric tools necessary to investigate the fractional Yamabe problem (1.1). In Section 3, we introduce some concepts regarding a blowing-up sequence {u m } m∈N of solutions to (1.1) and perform an asymptotic analysis near each blow-up point of {u m } m∈N . Section 4 is devoted to deducing sharp pointwise estimate of u m near each isolated simple blow-up point. This allows one to establish the vanishing theorem of the second fundamental form at any isolated simple blowup point, which is discussed in Section 5. Finally, the main theorems are proved in Section 6 with the aid of a local Pohozaev sign condition which guarantees that every blow-up point is isolated simple. In the appendices, we provide technical results needed in the main body of the proof as well as their proofs. Firstly, in Appendix A, we present several elliptic regularity results. Then we study the asymptotic behavior of the Green's functions near its singularity in Appendix B.1. We also derive a fractional Bôcher's theorem in Appendix B.2. Finally, a number of integrals involving the standard bubble W 1,0 , whose precise definition is given in Subsection 2.2, will be computed in Appendix C. Notations.
-The Einstein convention is adopted throughout the paper. We shall use the indices 1 ≤ i, j, k, l ≤ n.
-For a function f on R N + , we often write -For anyx ∈ R n , x ∈ R N + and r > 0, B n (x, r) signifies the n-dimensional ball whose center and radius arex and r, respectively. Similarly, we define B N + (x, r) by the N -dimensional upper half-ball centered at x having radius r. We often identify B n (x, respectively. The natural function space W 1,2 (X; ρ 1−2γ ) for the fractional Yamabe problem (2.4) is analogously defined.
-Assume that (M,h) and (X,ḡ) are compact Riemannian manifolds. Then Bh(y, r) ⊂ (M,h) stands for the geodesic ball centered at y ∈ M of radius r > 0. Besides, dvḡ is the volume form of (X,ḡ) and dσ represents a surface measure.
-C > 0 denotes a generic constant possibly depending on the dimension n of an underlying manifold M , the order γ of the conformal fractional Laplacian P γ and so on. It may vary from line to line. Moreover, a notation C(α, β, · · · ) means that the constant C depends on α, β, · · · .  ) ֒→ L q (Ω) is compact for any q ∈ [1, 2 * n,γ ) and a smooth bounded domain Ω ⊂ R n .

Geometric background.
We recall the extension result involving the conformal fractional Laplacian P γ obtained by Chang and González [11]; see also [8,28].
Proposition 2.1. Suppose that γ ∈ (0, 1), n > 2γ, (X, g + ) is an asymptotically hyperbolic manifold with conformal infinity (M, [h]). Also, assume that ρ is a geodesic defining function associated to M ,ḡ = ρ 2 g + and the mean curvature H is 0 on M . Set s = n/2 + γ and Then we have and R[g + ] are the scalar curvatures of (X,ḡ) and (X, g + ), respectively, and |ḡ| is the determinant ofḡ. In addition, where ν denotes the outward unit normal vector with respect to X and Γ(z) is the Gamma function.
Note that the condition Λ γ (M, [h]) > 0 (see (1.4)) implies that the functional . See [16, Lemma 2.5] for its proof. Therefore, given any u ∈ H γ (M ), the standard minimization argument guarantees the existence and uniqueness of the extension U ∈ W 1,2 (X; ρ 1−2γ ) of u which satisfies (2.2). Furthermore, testing u − in (2.2), we easily observe that if u ≥ 0 on M , then U ≥ 0 in X. If it holds that u > 0 on M , then the strong maximum principle for (non-degenerate) elliptic operators gives U > 0 on X.
On the other hand, without loss of generality, we can always assume that the constant c > 0 in equation (1.1) is exactly 1. As a result, (1.1) is equivalent to the degenerate elliptic problem (2.4) Next, choose any y ∈ M and identify it with 0 ∈ R n . Also, let x = (x, x N ) ∈ R N + be Fermi coordinates on X around y, i.e.,x = (x 1 , · · · , x n ) normal coordinates on M at y and x N = ρ. In [21, Lemma 3.1], the following expansion of the metricḡ near y is given. Lemma 2.2. In terms of Fermi coordinates x on X around y ∈ M , and Here -δ ij is the Kronecker delta; -π 2 =h ikhjl π ij π kl is the square of the norm of the second fundamental form π; is a component of the Riemannian curvature tensor on M and R iN jN [ḡ] is that of the Riemannian curvature tensor on X; Every tensor in the expansion is evaluated at y = 0 and commas denote partial differentiation.
If H = 0 on M , which is the case when (1.5) holds, all the coefficients of the terms x N , x i x N , x i x j x N , etc. in the expansion of |ḡ| are 0. In particular, condition (A.5) holds.
As pointed out in papers on the fractional Yamabe problem [28,29,42], only the boundary metric can be controlled through conformal changes. It is one of the main differences compared to the boundary Yamabe problem treated in e.g. [21,52,12,55,2]. The following lemma is a reformulation of Lemmas 2.4 and 3.2 in [42].
(2.5) We also introduce the γ-harmonic extension W λ,σ of w λ,σ , namely, the unique solution of Then it is known that where κ γ > 0 is the number appeared in (2.3) and ν is the outward unit normal vector with respect to R N + .

Lemma 2.4. (1) [Symmetry]
The value of W 1,0 (x, x N ) for (x, x N ) ∈ R N + is governed by |x| and x N . In particular, (2) [Decay] There exists a constant C > 0 depending only on n, γ and ℓ such that for all x ∈ R N + and ℓ ∈ N ∪ {0}.
Moreover the sharp decay estimate [42, Section A] for W 1,0 gives (2). Assertions (3) and (4) It turns out that it is convenient to deal with the following form of the equation Suppose thath m = w wheref m = w m f m , which is the same form as that of (2.10).

2.4.
Pohozaev's identity. Pick a small number r 1 ∈ (0, r 0 ) (see (2.1)) such thatḡ m -Fermi coordinate centered at y ∈ M is well-defined in the closed geodesic half-ball B N + (y, r 1 ) ⊂ X for every m ∈ N and y ∈ M .
In this subsection, we provide a local version of Pohozaev's identity for where κ γ > 0 is the constant in (2.3). Then we have for all r ∈ (0, r 1 ).
Proof. The proof is similar to that of [36,Proposition 4.7].

Basic properties of blow-up points
3.1. Various types of blow-up points. We start this section by recalling the notion of blow-up, isolated blow-up and isolated simple blow-up. Our definition is a slight modification of the one introduced in [2, Section 4] (cf. [36,39,46]). Definition 3.1. As before, let (X, g + ) be an asymptotically hyperbolic manifold with conformal infinity (M, [h]). Here we use the notations in Subsection 2.3 and the small number For simplicity, we will often say that y m → y 0 ∈ M is a blow-up point of {U m } m∈N .
for some C > 0, r 2 ∈ (0, r 1 ] whereh m =ḡ m | T M and dh m is the distance function in the metric h m . (3) Define a weighted spherical averagē of u m . We say that an isolated blow-up point y 0 of {U m } m∈N is simple if there exists a number r 3 ∈ (0, r 2 ] such thatū m possesses exactly one critical point in the interval (0, r 3 ) for large m ∈ N.
Roughly speaking, Item (2) (or (3), respectively) in the above definition depicts the situation when clustering of bubbles (or bubble towers, respectively) is excluded among various blow-up scenarios.
Hereafter, we always assume that Also, denote byh m a representative of the class [h m ] satisfying properties (1) and (2) in Lemma 2.3 with y = y m , and by U m a solution to (2.12). We shall often use x ∈ R N + to denoteg m -Fermi coordinates on X around y m so that U m can be regarded as a function in R N + near the origin.

3.2.
Blow-up analysis. We study asymptotic behavior of a sequence of solutions {U m } m∈N to (2.10) near blow-up points.
(3) It holds that Proof. The validity of this proposition comes from the Liouville type theorem [ We have a remark on (3.3): According to Proposition A.8, only the C β -convergence is guaranteed on the closed half-ball B N + (0, 2R). However, we have the C 2+β -convergence on its bottom B n (0, 2R).
Lemma A.2 and the standard rescaling argument readily give the annular Harnack inequality around an isolated blow-up point. Lemma 3.3. Suppose that y m → y 0 ∈ M is an isolated blow-up point of a sequence of solutions {U m } m∈N to equation (2.10). If we are inḡ m -Fermi coordinate system centered at y m , then there exists C > 0 independent of m ∈ N and r > 0 such that Proof. The proof is similar to that of [36,Lemma 4.3].
If y m → y 0 ∈ M is an isolate blow-up point of solutions {U m } m∈N to (2.10), Proposition 3.2 can be extended in the following manner.
for all m ∈ N.
In order to prove it, we first need the following type of the Hopf lemma.
). Assume also that there exist a small number r > 0 and Proof. Our proof is in the spirit of those in [28, Theorem 3.5] and [10, Proposition 7.1]. Let with C 1 , C 2 > 0 sufficiently large. Then there exists a small number δ > 0 such that On the other hand, Lemma 3.5 implies . Because of the classical maximum principle, V m does not attain its infimum in the interior of B N + (0, r) for any r ∈ (0, 1], unless it is a constant function. However, it cannot be constant, otherwise we get an absurd relation . Moreover, the infimum of V m is not achieved on the bottom B n (0, r), because the existence of a minimum pointx m ∈ B n (0, r) of V m and the Hopf lemma produce a contradictory relation where C > 0 is a constant independent of m ∈ N. Accordingly, by making use of (3. and V m (·, 0) → w 1,0 in C 2+β loc (R n ) passing to a subsequence. The assertion of the lemma is true.
Keeping in mind that our proof is not affected by the act of picking a subsequence of {U ℓ } ℓ∈N , we always select {R m } m∈N first and then {U ℓm } m∈N satisfying (3.4) and R mǫℓ m → 0. From now on, we write {U m } m∈N to denote {U ℓm } m∈N and so on to simplify notations.
The next result is a simple consequence of the previous lemma under the choice τ m = w 1,0 (R m )/2.
Proof. Choose numbers l > 0 and L > 0 such that l ≤ w −1 m =ũ m /u m ≤ L on M for all m ∈ N. If we use the normal coordinate on M at y m , it follows from (3.4) that Hence there exists a sufficiently large number for any y ∈ M \ {ỹ m }, dh m (y,ỹ m ) < r 2 where the magnitude of r 2 > 0 may be reduced if necessary. As a result, an application of the proof of Lemma 3.4 toũ m shows that for R ′ ≫ R large enough,ũ m is C 2 (Bh m (ỹ m , R ′ǫ m ))close to a suitable rescaling of the standard bubble w 1,0 so that it has the unique critical point on Bh m (ỹ m , R ′ǫ m ), i.e., the local maximumỹ m . However, by (2.11), y m ∈ Bh m (ỹ m , R ′ǫ m ) is already a critical point ofũ m , and so it should be equal toỹ m . This completes the proof of (1). The verification of (2) is plain.
The objective of this subsection is to show that the behavior of each U m in the geodesic half-ball Bg m (y m , r) ∩ X can be controlled whenever r > 0 is chosen to be sufficiently small. We will useg m -Fermi coordinate system centered at y m , so Bg m (y m , r) ∩ X is identified with Proposition 3.7. Assume n > 2 + 2γ and y m → y 0 ∈ M is an isolated simple blow-up point of a sequence of solutions {U m } m∈N to (2.10). Then one can choose C > 0 large and where M m = u m (y m ) and U m is the function constructed in Subsection 3.1.
Proof of Proposition 3.7. Its proof consists of six steps. Let o(1) denote any sequence tending to 0 as m → ∞.
Step 1 (Rough upper decay estimate of { U m } m∈N ). For any fixed sufficiently small It can be proved as in [26,Lemma 2.7] and [36,Lemma 4.6]. However, since this is one of the places where hypothesis (1.5) is used, we sketch the proof.
Thanks to Lemma 3.3 and Corollary 3.6 (2), it turns out that Assume that 0 ≤ µ ≤ n − 2γ. Then one can calculate From (3.15), (3.16) and the computation , with a suitable choice of L 0 , L m > 0 large and ζ, η > 0 small, satisfies Consequently, the generalized maximum principle (Lemma A.5) leads us to conclude U m ≤ Φ 1m in Γ 2 . From this and the assumption that y m → y 0 is an isolated simple blow-up point of {U m } m∈N , we infer that (3.13) or (3.12) holds.
Step 2 (Lower decay estimate of { U m } m∈N ). We claim that there is a large constant where the magnitude of r ′ 4 is reduced if necessary. Let G m be the Green's function that solves (B.1) providedḡ =ḡ m and B R = B N + (0, r ′ 4 ). By (2.10), (3.4

) and (B.3), we find that
We apply the standard rescaling argument depicted in, e.g., the proof of [27, Lemma 2.6]. Given any m ∈ N and R ∈ [2R mǫm , r ′ 4 /2], set Inequalities (3.18) will be valid if there exists C > 0 independent of m and R such that Because of relatively poor regularity property of degenerate elliptic equations, especially when γ ∈ (0, 1) is small, derivation of (3.20) is rather technical. In particular, as we will see shortly, it requires the lower estimate (3.17) of M λm m U m in contrast to the local case γ = 1. In light of (3.12), it holds that U R (x) ≤ C in Γ 3 . Applying the Hölder estimate (A.3), a bootstrap argument with the Schauder estimate (A.17), and the derivative estimate (A.6) to (3.19), we obtain on ∂Γ ′ 3 . This together with (A.17) and (3.21) gives that for all proper compact subsets K ′ K of Γ ′ 3 and exponents 1 ≤ β < β ′ ≤ min{2, β + 2γ}. Step 4 (Estimate of δ m ). For δ m = (2 * n,γ − 1) − p m ≥ 0, it holds that The proof makes use of Pohozaev's identity in Lemma 2.5 and is analogous to that in [36,Lemma 4.8]. Hence we omit it. Step for any sufficiently small r 4 ∈ (0, r ′ 4 ]. Suppose that it does not hold. Then there exists a sequence {z m } m∈N of points on the half-sphere {x ∈ R N Let us write Eḡ m (x N ) = x 1−2γ N A m so that {A m } m∈N is a family of functions whose C 2 -norm is uniformly bounded. We divide the cases according to the sign of A m .
In this case, one can argue as in [51,Proposition 4.5] or [2,Proposition 4.3] to reach a contradiction. The proof is omitted.

Case 2. Assume the condition that
To handle this situation, we will recover positivity of A m by employing conformal change of the metricḡ m on M . Owing to (1.5) (or the condition H = 0 on M ), (2.1) and Lemma 2.2, it holds that where π m is the second fundamental form of (M,h m ) ⊂ (X,ḡ m ). Also, inspecting the proof of [42, Lemma 2.4], we see We want to find a representativeȟ m of the conformal class [h m ] and a small number whereǧ m is the metric on X defined via the geodesic boundary defining function associated toȟ m . To this end, it suffices to confirm that given a fixed small number ε > 0, Hereπ m is the second fundamental form of (M,ȟ m ) ⊂ (X,ǧ m ). Set f m (x) = −K|x| 2 in B n (0, r ′ 4 ) for some large K > 0 and then extend it to M suitably so that f m ∈ C ∞ (M ). If we letȟ m = e 2fmh m , then the transformation law of the scalar curvature and the umbilic tensor under a conformal change (see (1.1) of [21] and (2.2) of [42]) gives which establishes (3.25). Verifying (3.26) requires a little more work. We extend the function f m on M to its collar neighborhood M × [0, r ′ 4 ) by solving a first-order partial differential equation Since it is non-characteristic, a solution exists and is unique provided r ′ 4 small. Locally, it is written as We easily see thatǧ m = e 2fmḡ m on M × [0, r ′ 4 ). Hence, by the assumption H = 0 on M and (3.27), it is sufficient to find a small number r ′′ 4 = r ′′ 4 (K) > 0 such that so as to ensure the validity of (3.26). Given an arbitrary pointx ∈ B n (0, 2r ′′ 4 ), the characteristic equation of (3.27) is the system of 2N + 1 ordinary differential equations of functions Here the dot notation stands for the differentiation with respect to s and the domain of the functions (p, z, x)(·;x) is assumed to be the interval [0, 2r ′′ 4 ). Then asymptotic analysis on the system indicates and for any fixed r ′′ 4 ∈ (0, K −2 ), thereby establishing the desired inequalities. Now, with the fact that x 1−2γ NǍ m ≥ 0 in B N + (0, r ′′ 4 ), we may consider the family {Ǔ m } m∈N of solutions to (2.10) in which the tildes are replaced with the checks. Notice that since H = 0 is an intrinsic condition that comes from (1.5), the new metricsǧ m on X still satisfy necessary conditions for the regularity results in Appendices A and B.2. Hence our situation is reduced to Case 1 and we get the same contradiction.

4.2.
Refined blow-up analysis. As before, let y m → y 0 ∈ M be an isolated simple blow-up point of {U m } m∈N . In view of Corollary 3.6 (1) and (2.11), y m → y 0 is an isolated blow-up point of { U m } m∈N and M m = U m (y m ) (= U m (0) ifg m -Fermi coordinate system around y m is used). Also, Proposition 3.7 ensures the validity of the pointwise estimate (3.11) for { U m } m∈N near y 0 . The objective of this subsection is to refine it by analyzing the ǫ m -order terms. Recall the functions W 1,0 and Ψ m defined in (2.6) and constructed in Proposition 4.1 (where the tensor π m is replaced by the second fundamental formπ m (y m ) at y m of (M,h m ) ⊂ (X,g m )), respectively.
Step 2 (Estimate of δ m ). We assert Its proof can be done as in [2, Lemma 6.2] with minor modifications, so is omitted.
The remaining inequalities, i.e., (4.6) for ℓ = 1, 2 and (4.7) are derived as in the justification of (3.18). Indeed, a tedious but straightforward calculation shows that the second-order derivatives of the functions B m (x) and Q 2m (x) have the required decay rate as |x| → ∞. The proof is now completed.  Particularly,π 0 (y 0 ) = 0.

Vanishing theorem of the second fundamental form
Proof. In the proof, we will use Lemma 2.3 withg =g m ,h =h m and y = y m , and think as if U m is a function in R N + near the origin by appealingg m -Fermi coordinates on X around y m . Denotingĝ m =g m (ǫ m ·) andf m =f m (ǫ m ·), we set Also, let E m be the functions introduced in (4.10) so that V m in (4.5) is a solution of For a fixed number r ∈ (0, r 5 ], let Owing to (4.6) and (4.7), we are led to We first estimate F m (W 1,0 , W 1,0 ). This amounts to calculate the integrals where Z 0 1,0 = x · ∇W 1,0 + (n − 2γ)/2 · W 1,0 is the function described in (2.9). For the value F 1m , we discover from Lemmas 2.4 (1), 2.2 and 2.3 that where H m = tr(π m )/n. Similarly, we have Moreover, the Gauss-Codazzi equation and Lemma 2.3 yield Hence we find Consequently, by combining (5.5)-(5.8) and employing (C.1), we obtain that for some constant C 0 > 0 relying only on n and γ. We note that the coefficient of ǫ 2 m (orǫ 2 m ) is positive if and only if n satisfies (1.2) for each γ ∈ (0, 1).
On the other hand, it holds that whose verification is deferred to the end of the proof, and By plugging (5.9)-(5.11) into (5.4), we reach at Using (3.11), (3.23) and (4.5), we deduce if r > 0 is selected to be small enough. As a result, estimate (5.1) follows.
Derivation of (5.10). Sinceπ m →π 0 in C 1 (M ), the norm |π m | ∞ (see the statement of Proposition 4.1) is uniformly bounded in m ∈ N. Thus, by virtue of (5.2), (2.8), (4.2), Lemma 2.2 and integration by parts, we observe provided that n > 2 + 2γ. On the other hand, applying another integration by parts and testing Ψ m in (4.1) lead to It is well-known that the Morse index of w 1,0 ∈ H γ (R n ) is 1 due to the contribution of w 1,0 itself. Hence we see from (4.4) that I ≥ 0; see the proof of [16,Lemma 4.5] for more explanation. This completes the proof.
6. Proof of the main theorems 6.1. Exclusion of bubble accumulation. Set which is a part of the function P defined in (2.14).
Lemma 6.1. Assume that γ ∈ (0, 1) and the dimension condition (1.2) holds. Let y m → y 0 be an isolated simple blow-up point of {U m } m∈N , and { U m } m∈N be the sequence of the functions constructed in Subsection 3.1. Suppose further thatπ 0 (y 0 ) = 0. Then, given m ∈ N large and r > 0 small, there exist universal constants C 1 , · · · , C 4 > 0 such that inḡ m -Fermi coordinates centered in y m . Here, η > 0 is an arbitrarily small number.
We shall use the following Liouville type lemma to prove Lemma 6.3.
then it should be a nonnegative constant.
Proof. According to Hölder estimate (A.3) and the asymptotic condition in (6.3), U is contained in C β loc (R N + ) and bounded from below. Let us denote m 0 = inf R N + U and U m 0 = U − m 0 ≥ 0. By the Harnack inequality (A.4) and scaling invariance of (6.3), we have where C > 0 is independent of r. Letting r → ∞ in (6.4), we obtain that U m 0 = 0 or U = m 0 in R N + . One more application of the asymptotic condition on U forces m 0 ≥ 0. The proof is concluded.
up to a subsequence; see [2, Proposition 8.1]. It follows from (6.2) that and as m → ∞. Hence, taking the limit on (6.6) and employing (6.7), (6.8), (6.5) and (6.1), we obtain which is a contradiction. The assertion in the statement must be true.
We rule out bubble accumulation by applying Lemma 6.3.
Proof. By appealing to Propositions 3. Proof of Theorem 1.1. We first claim that u ≤ C on M . If it does not hold, then by Proposition 2.1, there is a sequence {U m } m∈N ⊂ W 1,2 (X; ρ 1−2γ ) of solutions to (2.4) which blows-up at a point y 0 ∈ M . By applying Theorem 1.3, we conclude that π(y 0 ) = 0. However, it is contradictory to the assumption that π never vanishes on M , so our claim should be true. A combination of (1.4), Lemma A.2 and Proposition A.8 now yields the other estimates in (1.6), that is to say, the lower and C 2+β -estimates of u on M .
At this stage, only Theorem 1.4 is remained to be verified. To define the Leray-Schauder degree deg(F p , D Λ , 0) for all 1 ≤ p ≤ 2 * n,γ − 1 and apply its homotopy invariance property, we need the following result. Lemma 6.6. Assume the hypotheses of Theorem 1.1. Then one can choose a constant C = C(X n+1 , g + ,h, γ) > 1 such that Also, in [28, Section 4] and [9, Section 1], it was proved that the first eigenspace of P γ h is one-dimensional and spanned by a positive function on M . By the L 2 (M )-orthogonality, the other eigenfunctions must change their signs. Using these characterizations, one can follow the argument in [65], up to minor modifications, to derive deg(F 1 , D Λ , 0) = −1. The proof of Theorem 1.4 is completed.
A.2. Derivative estimates. If the functions A, Q, a and q have classical derivatives in the tangential direction, weak solutions to (A.2) have higher differentiability in the same direction. The following result is a huge improvement of [40,Lemma 5.3] in that a much milder condition onḡ is imposed. The reader is advised to see carefully why handling (A.2) becomes more difficult ifḡ is non-Euclidean and how it is resolved in the proof.

Then it weakly solves
where U h (x, x N ) = U (x + h, x N ), The most problematic term in analyzing (A.7) turns out to be div(x 1−2γ N F * ), especially, its subterm ∂ N (x 1−2γ N D h |ḡ| · ∂ N U h ). Let us concern it in depth. If we fix a small number ε > 0 and write B 3R/4,ε = B 3R/4 ∩ {x N > ε} and ∂B ′ 3R/4,ε = B 3R/4 ∩ {x N = ε}, then an integration by parts shows for any function Φ ∈ C 1 (B 3R/4 ) such that Φ = φ on ∂B ′ 3R/4 and Φ = 0 on ∂B ′′ 3R/4 . On the other hand, we obtain from (A.2) that x N ) and so on. Consequently, after substituting for Φ in the above identity, combining the result with (A.8) and then taking ε → 0, we get We have two remarks on (A.9): First, Ξ has the same regularity as that of Φ and vanishes on ∂B ′′ 3R/4 . Second, by virtue of (g2), there exists a constant C > 0 such that Furthermore, as pointed out in the proof of [40,Lemma 5.3], an application of the rescaling argument gives where C > 0 depends only n, R,ḡ and A L ∞ (B R ) . Therefore no terms in the right-hand side of (A.9) are harmful. Now, we introduce a number if it is nonzero, any positive number otherwise.
In the latter case, we send k → 0 at the last stage. For a fixed K > 0 and m ≥ 0, we define denotes a suitable cut-off function. Employing (A.9), Hölder's inequality, Young's inequality, the weighted Sobolev inequality and the weighted Sobolev trace inequality (see (1.8)) and then taking K → ∞, we derive for some C > 0 depending only on n, γ, R,ḡ, A, a L q 3 (∂B ′ R ) and U L ∞ (B R ) , and η > 1 depending only on n and γ; refer to the proofs of [24,Proposition 1] and [40,Lemma 5.3] which provide more detailed descriptions. Combining (A.10) with the corresponding inequality for (D h U ) − + k and letting h → 0, we see .
Similarly, one can obtain the weak Harnack inequality as well as the Hölder estimate for ∇xU . The cases ℓ 0 = 2 or 3 can be also treated. We omit the details.
In the following lemma, we take into account Hölder regularity of the weighted derivative Lemma A.4. Suppose that the metricḡ satisfies (g1) and U ∈ W 1,2 (B R ; x 1−2γ N ) is a weak solution to (A.2) such that U, ∇xU, ∇ 2 x U ∈ C β (B R ) for some β ∈ (0, 1). Furthermore, assume that the following conditions hold: for C > 0 depending only on n, γ, R,ḡ, A C β (B R ) and a C β (∂B ′ R ) . Proof. Refer to [40,Lemma 5.5].
A.3. Two maximum principles. In this part, we list two maximum principles which are used throughout the paper.
The following lemma describes the generalized maximum principle for degenerate elliptic equations.
then U ≥ 0 on B R . Furthermore, the same conclusion holds if B R and ∂B ′ R are substituted by R N + and R n , respectively, and the third inequality in (A.12) is replaced with the condition that |U (x)|/V (x) → 0 uniformly as |x| → ∞.
The next remark concerns on the weak maximum principle when the size of the domain is sufficiently small.
Remark A.6. For any fixed R > 0, we introduce the space is always attained, and so λ 1 (R) > 0. Moreover, we see from the dilation symmetry that Therefore if |A| ≤ M for some constant M > 0, then there exists R ′ 0 = R ′ 0 (M,ḡ) > 0 such that is a norm equivalent to the W 1,2 0 (B R ; x 1−2γ N )-norm for R ∈ (0, R ′ 0 ). In particular, we have a weak maximum principle: Given any R ∈ (0, R ′ 0 ), suppose that which is a special case of (A.2).
Proof. We shall follow closely the argument in the proof of [36, Theorem 2.14].
Considering a finite open cover of B R which consists of balls and half-balls with small diameters, we may assume that R > 0 is so small that the * -norm in (A.14) is equivalent to the standard W 1,2 (B R ; x 1−2γ N )-norm. For simplicity, we set Assume that β ∈ (0, 1). For m ∈ N, let W m be the unique solution in W 1,2 (B R/2 m ; Then an application of the weak maximum principle (Remark A.6) to the equation of the function for every m ∈ N. Define h m = W m+1 − W m . Thanks to Lemmas A.2 and A.3, we have so that u ∈ C β ′ (∂B ′ R/2 ) and (A.17) holds. Suppose β ∈ (1, 2). In this case, we modify W m by replacing the second equation of (A.18) with ∂ γ ν W m = q(0) + ∇xq(0) ·x − q(x) on ∂B ′ R/2 m . Then we adopt the above argument to conclude that u ∈ C β ′ (∂B ′ R/2 ) and (A.17) holds. The case β > 2 can be similarly treated. This finishes the proof.
whereḡ is the metric satisfying (g1) and δ 0 is the Dirac measure centered at 0 ∈ R N . Our argument is based on elliptic regularity theory and does not rely on parametrices.
We start with deriving an auxiliary lemma.
Proof. One can follow the lines in the proof of [13,Lemma 3.3]. The main difference is that we need to apply the Sobolev inequality here instead of the Sobolev traced inequality as used in the reference; see (1.8).
Appealing to the previous lemma, we prove the main result in this subsection. |x| n−2γ G(x) → g n,γ uniformly as |x| → 0 where g n,γ = Γ n−2γ for any fixed R ∈ (0, R 0 ).
Proof. The proof is divided into four steps.
Step 1 (Existence). By using (1.5) and (2.1), we rewrite the first equation of (B.1) as where A ∈ C 2 (B R ). In view of Remark A.6, there exists R 0 > 0 such that · * in (A.14) serves as a norm equivalent to the standard W 1,2 (B R ; x 1−2γ N )-norm for all R ∈ (0, R 0 ). Then a duality argument in the proof of [42,Lemma 4.2] shows that the desired function G exists and is contained in ) for any 1 < q < (n−2γ+2)/(n−2γ+1).
Step 2 (Regularity). Recall that By a direct calculation, we see and for all 1 < q < (n − 2γ + 2)/(n − 2γ + 1). We claim that To justify it, we consider the formal adjoint of (B.5) where Q is an arbitrary function of class C 1 (B R ). Then → 0 as ε → 0 thanks to the boundary condition and ∆xU ∈ C 0 (B R ) in light of Lemma A.3, we observe and in particular U ∈ C 1 (B R ). Thus one may use H and U as a test function for (B.9) and (B.5), respectively. As a consequence, it holds that for any Q ∈ C 1 (B R ) and small η > 0. Here η ′ and η ′′ are small positive numbers depending only on η. Also, the last inequality is due to (B.7) and Lemma B.1, and the assumption n ≥ 2 + 2γ is required to ensure that q 1 = (n − 2γ + 2)/3 + η ′′ satisfies condition (B.2). By duality, the assertion follows.
Step 3 (Blow-up rate). We use the rescaling argument in the proof of [ Step 4 (Uniqueness). The uniqueness of G follows from Bôcher's theorem stated in Proposition B.4.
Proof. The result follows immediately from (B.10) and the rescaling argument.
for all ϑ ∈ (0, R), then for any R ′ ∈ (0, R) Here c 1 is a nonnegative constant, G is the Green's function that satisfies (B.1) (where R is replaced with R ′ ) and E ∈ W 1,2 (B R ; The numbers R 0 and M were chosen in Proposition B.2 and Remark A.6. To prove the proposition, we will use the strategy from [50, Section 9] and [57, Section 3]. As a preliminary step, we derive two results. Lemma B.5. Assume that U satisfies all the conditions in Proposition B.4. If U (x) = o(|x| −(n−2γ) ) as |x| → 0, then 0 is a removable singularity of U and there exists β ∈ (0, 1) such that U ∈ W 1,2 (B R ′ ; x 1−2γ N ) ∩ C β (B R ′ ) for any R ′ ∈ (0, R).
Proof. We argue as in [57,Lemma 3.6] with minor modifications. The maximum principle in Remark A.6 combined with the asymptotic behavior (B.3) of G near the origin shows that U is bounded in B R ′ . Owing to the regularity hypotheses onḡ and A, we can apply the scaling method with Lemmas A.3 and A.4, deducing U ∈ C β (B R ′ ) for some β ∈ (0, 1) and This in turn implies that U ∈ W 1,2 (B R ′ ; x 1−2γ N ).

Appendix C. Computation of the integrals involving the standard bubble
We obtain the values of several integrals involving the standard bubble W 1,0 and its derivatives, which are needed in the proof of the vanishing theorem in Section 5.
Proposition C.1. Suppose γ ∈ (0, 1) and n > 2 + 2γ. Then there exists a constant C 0 > 0 depending only on n and γ such that (C.1) Here W 1,0 and Z 0 1,0 are the functions given in (2.6) and (2.9). Its proof is based on the Fourier transform technique which was introduced by González and Qing [28] and soon improved by González and Wang [29] and Kim et al. [41,42], where the authors studied the existence and non-compactness characteristics of the solution set of (1.1). We first need to remind a lemma obtained in [28,Section 7] and [41,Subsection 4.3].
With the help of the previous lemma, we evaluate the following nine integrals. Lemma C.3. Assume that γ ∈ (0, 1) and n > 2 + 2γ. If we set C 0 = |S n−1 |A 3 B 2 , then C 0 , N W 1,0 (∆xW 1,0 ) dx = −C 0 , Proof. The quantities I 1 , I 6 , I 7 were computed in [28,Lemma 7.2]. Besides, [42,Lemma B.4] provides the value of I 8 , and its proof suggests a way to calculate I 2 , I 3 . Accordingly we only take into account the others. Throughout the proof, we agree that the variable ofŵ 1,0 andŵ ′ 1,0 is |ξ| and that of ϕ and ϕ ′ is |ξ|x N . Also, ′ is used to represent the differentiation in the radial variable |ξ|.
It follows from Parseval's theorem that . Therefore Lemma C.2 (2) gives the value in the statement. On the other hand, we observe by applying the integration by parts that Also one can verify finishing the proof.