Conformally Prescribed Scalar Curvature on Orbifolds

We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension 4, and an existence theorem which holds in dimensions n≥4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 4$$\end{document}. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the U(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{U}(2)$$\end{document}-invariant Leray–Schauder degree for a family of negative-mass orbifolds found by LeBrun.


Introduction
To begin, we give the definition of a Riemannian orbifold.
Definition 1.1. We say that (M, g) is a Riemannian orbifold of dimension n if M is a smooth manifold of dimension n with a smooth Riemannian metric away from a finite singular set Σ = {(q 1 , . . . , q l )}. (1.1) Near each singular point q j , there exists a neighborhood U j of q j , a nontrivial finite subgroup Γ j ⊂ O(n) acting freely on R n \ {0} and a Γ j -equivariant diffeomorphism where U j is the completion (by adding a pointq j ) of the universal cover of U j \ {q j }, and B σ j (0) is a ball of radius σ j about the origin in R 4 . Furthermore, (ϕ j ) * π * j g extends to a smooth Riemannian metric on B σ j (0), where π j : U j \ {q j } → U j \ {q j } is a universal covering map.
Our convention is that if l = 0, then Σ = ∅, and (M, g) is a smooth Riemannian manifold. But if l ≥ 1, then there must be nontrivial singular points. We next define the meaning of a C k (M ) function on a Riemannian orbifold (M, g).
, and near each singularity, there exists a coordinate system ϕ j such that the function f •π j •ϕ −1 j : B σ j (0) → R is in C k (B σ j (0)). We can also define the spaces C ∞ (M ), C k,α (M ), C k loc (M ) in a similar fashion.
Note that since linear terms are never invariant under a nontrivial orbifold group, a C 1 function necessarily has a critical point at any nontrivial orbifold singularity. In other words, Σ ⊂ Crit(f ), where Crit(f ) ≡ {x ∈ M | ∇ g f (x) = 0}. Remark 1.3. In general, an orbifold can have higher-dimensional singular sets. So our definition is restrictive in that we only allow isolated quotient singularities.
Assume (M, g) is a compact Riemannian n-orbifold with positive scalar curvature R g > 0. Let K > 0 be a positive C 2 function on M . We will study the following equation where c(n) = n−2 4(n−1) , 1 < p ≤ n+2 n−2 and ∆ g is the Laplacian operator associated with g. Let L g = ∆ g − c(n)R g denote the conformal Laplacian of the metric g. When p = n+2 n−2 , the solution of equation (1.3) corresponds to the prescribed scalar curvature problem. That is, the metric given byg = u 4 n−2 g has scalar curvature Rg = 4(n−1) n−2 K. The Yamabe problem on manifolds is well-understood, and we refer to [Aub98a,LP87,Sch84,Yam60] and references therein. The prescribed scalar curvature problem on S n and other manifolds has been studied in many works; see for example [BAA08,CGY93,CL01,Li95,Li96,Mal02,SZ96]. Prescribed scalar curvature on manifolds is studied for example in [BACCH96,MM20,May17]. More recently, the Yamabe problem on singular spaces has been of interest; see for example [AB03, AB04, ACM14, ACM19, CLV21, Mon17, Mon18, Via10,Via13].
Analogous to the generalization from manifolds to orbifolds, there is the following generalization of asymptotically flat (AF) metrics.
Definition 1.4. A complete Riemannian orbifold (X n , g) with finitely many singular points is called asymptotically locally Euclidean or ALE of order τ if it has finitely many ends and for each end there exists a finite subgroup Γ ⊂ O(n) acting freely on R n \ {0} and a diffeomorphism ψ : X \ K → (R n \ B R (0))/Γ where K is a compact subset of X, and such that under this identification, (1.4) ∂ k (ψ * g) ij = O(ρ −τ −|k| ), (1.5) for any partial derivative of order |k|, as ρ → ∞, where ρ is the distance to some fixed basepoint.
We will occasionally refer to an AF metric as an ALE metric, since AF is exactly the case of ALE with Γ = {e}. Next, we give the definition of the ADM mass on asymptotically locally Euclidean (ALE) orbifolds.
Definition 1.5. Given an n-dimensional ALE orbifold (X, g) with asymptotic coordinates {z i } and quotient group Γ near ∞, define the ADM mass as follows: where S r /Γ is the hypersurface at |z| = r, and dV z is the Euclidean volume element.
Remark 1.6. Bartnik proved that in the AF case, if τ > (n − 2)/2 then the mass is well-defined and independent of the choice of coordinates at infinity [Bar86]. A similar argument shows that the same result holds in the ALE case.
Remark 1.7. Note that if |Γ| = {e} is the trivial group, the formula above defines the mass for AF orbifold (X, g), which is consistent with [LP87,Definition 8.2]. Note also that if Γ = {e}, our coefficient differs from [HL16] due to the factor of |Γ|. Our convention has the advantage that it eliminates the need for writing extra factors of |Γ| in several formulas.
For any Riemannian orbifold (M, g) with positive scalar curvature, we can construct a scalar-flat ALE orbifold (X,ĝ) by the following well-known procedure.
Definition 1.8. Take g-normal coordinates {x i } centered at pointx and let r = |x| denote the distance function. Let ψx > 0 be the Green's function of L g with leading term r 2−n nearx. Then (Xx,ĝx) = (M \x, ψ 4/(n−2) x g) is a scalar flat ALE orbifold. We will refer to (Xx,ĝx) as the conformal blow-up of g at the pointx.
Of course, ifx is a smooth point of M , then (Xx,ĝx) is an AF orbifold, but ifx is a singular point, then (Xx,ĝx) is an ALE orbifold.
1.1. Positive Mass Theorem on AF orbifolds. Our first result is that the positive mass theorem does hold for AF orbifolds.
Theorem 1.9. Let (X, g X ) be an asymptotically flat (AF) n-dimensional Riemannian orbifold with finitely many isolated singular points, with R(g X ) ≥ 0, and which is of order τ X > n−2 2 . Then mass(g X ) ≥ 0. and mass(g X ) = 0 if and only if there are no nontrivial orbifold singularities, and (X, g X ) is isometric to (R n , g Euc ).
This is proved in Section 2. The basic idea is to use a certain Green's function for the Laplacian based at the orbifold points, which we use to reduce to the positive mass theorem for manifolds with concave boundary due to Hirsch-Miao [HM20] using the fundamental work of Schoen-Yau [SY79,SY81,SY17].
Remark 1.10. The positive mass theorem holds on ALE spaces only with some extra assumptions; see for example [Nak90]. In contrast, the positive mass theorem does not necessarily hold for arbitrary ALE metrics with nonnegative scalar curvature. The first examples were given in [LeB88], and many more in [HL16]. Theorem 1.9 shows that the study of the Yamabe equation on orbifolds can be substantially different from the manifold case. In dimension 4, we can define a mass function m : M → R by assigning the mass of the conformal blow-up atx ∈ M . In the manifold case, this is a smooth function [Hab00]. However, the above result shows that if (M, g) is an orbifold and the conformal blow-up has negative mass atx ∈ Σ, then the mass function m is necessarily discontinuous atx; see Corollary 2.2 below.
1.2. Existence and compactness results. There is a long history of existence and compactness results for the Yamabe problem, which is the case when K = constant. The fundamental idea for compactness was due to Schoen [Sch89b,Sch91,Sch89a]; compactness results for the Yamabe problem in low dimensions were then proved in [Dru03,KMS09,LZ05,LZ99,Mar05]. As mentioned above, existence and compactness results for variable K have been studied in great detail on S n and on manifolds. Our next result generalizes many of these results in dimension four to the case of a Riemannian orbifold.
Theorem 1.11 will be proved in Sections 3, 4, and 5. One of the main difficulties is due to the "discontinuity" of conformal normal coordinates at an orbifold point. In Section 5, we will show that if concentration of a sequence u k happens at an orbifold point, then the local maxima of u k must occur exactly at the orbifold point for k sufficiently large; see Condition 3.7 which is a generalization of the isolated simple blow-up condition. Once we prove that blow-up points satisfy Condition 3.7; we can then fix the coordinates to be centered at the orbifold point, and then there is a contribution only from the mass function at the blow-up point. Note: our argument does not need continuity of the mass function which, as pointed out above, is not true in general anyway.
Remark 1.12. We emphasize that under the assumption (1.8), we are only claiming compactness; we do not have the existence in this general case. The main reason is that the subcritical method only works under the stronger assumption (1.10). However, the compactness result does allow one to define the Leray-Schauder degree. We expect that by imposing extra assumptions on K, one should be able to prove this degree is non-zero in some cases where (1.10) does not hold, but we do not pursue this here. Results along this line in the manifold case can be found in [BACCH96,Li96,MM20,May17].
1.3. Variational methods. Next, we have another existence result in dimensions n ≥ 4 which is proved using variational methods. Roughly, this says that if K is not too large away from the orbifold points, then we need assumption (1.10) to hold at only one orbifold point.
Theorem 1.13. Let (M, g) be a compact Riemannian n-dimensional orbifold with singularities Σ Γ = {(q 1 , Γ 1 ), · · · , (q l , Γ l )} and positive scalar curvature. Let K be a positive smooth function on M . Assume that whereĝ q i 0 is as in Definition 1.8. Then there exists a positive smooth solution of equation (1.3) with p = n+2 n−2 . This will be proved in Section 6, which is closely related to [ES86, Theorem 2.1], [Aku12, Theorem 3.1] and [MM21, Proposition 5.1]. The analogue on manifolds is that (1.12) has to hold at one maximum point of K, which is only possible in dimension n = 4. Hence the theorem is quite special to the orbifold case in dimensions n ≥ 5. We next apply this to some examples.
Consider S n ⊂ R n+1 , and let Γ ⊂ SO(n) be a finite subgroup acting freely on S n−1 ⊂ S n ∩ {x n+1 = 0}. Then S n /Γ with the spherical metric g S is a Riemannian orbifold with 2 orbifold points q 1 and q 2 .
We note that for |Γ| = 2, this result also follows from [LZ14, Theorem 1.12]. Another non-trivial example to which Theorem 1.13 applies is to the Calabi metric g CAL(n) , which is a U (n)-invariant Ricci-flat Kähler ALE metric on the total space X n of the line bundle O(−n) → P n−1 . For details on the Calabi metric, see for example [MV20].
Corollary 1.15 (Calabi orbifold). For n ≥ 2, take a U (n)-invariant conformal compactification (X n ,ǧ CAL(n) ) of the Calabi metric (X n , g CAL(n) ), such that the infinity of g CAL(n) is compactified to an orbifold pointq, with quotient group Z/nZ. Let K be a positive smooth function onX n . Assume that then there exists a positive smooth solution of equation (1.3) with g =ǧ CAL(n) and p = n+1 n−1 . Remark 1.16. In this case, from Theorem 1.3 and Remark 4.7 in [Via10], we know that there is no solution with K = constant > 0, which certainly does not contradict with the condition on the Laplacian at the orbifold point. However, Corollary 1.15 implies that any sufficiently small perturbation of a constant satisfying ∆ǧ CAL(n) K(q) > 0 does admit a solution. So the case of K = constant is right on the boundary of existence. Furthermore, if we take a sequence of functions K k satisfying (1.16), but converging to a positive constant in C 2 norm, then this gives an example where bubbling must occur as k → ∞.
For n = 2, (X 2 , g CAL(2) ) is also known as the Eguchi-Hanson metric, which is included in the family of LeBrun metrics, which we discuss next.
1.4. LeBrun metrics. In this subsection, we will use the previous results to analyze the prescribed scalar curvature problem on the family of orbifolds mentioned above: the LeBrun negative mass metrics on O P 1 (−n). Note: these are 4-dimensional manifolds, so in this subsection the integer n will instead be used to denote −c 1 , where c 1 is the first Chern class of this line bundle. These metrics possess an isometric U(2)-action. As we will see below, we have some results in the general case, but our most complete results are under a U(2)-symmetry assumption.
(1.18) For any n ∈ N * , g LEB(n) is scalar-flat. Note that g LEB(1) is conformal to the Fubini-Study metric on P 2 , and is known as the Burns metric, which is AF and has positive mass. As pointed out above, g LEB(2) is the Eguchi-Hanson metric, which is Ricci-flat ALE and has zero mass. For any n ≥ 3, g LEB(n) has negative mass. More details and properties of LeBrun metrics can be found in [DL16,LeB88,Via10]. We next choose an orbifold compactification by defininǧ g LEB(n) = 1 (n +r 2 ) 2 · g LEB(n) .
Theorem 1.18. For any n ∈ N * , there exists a constant C, depending only on (Ǒ P 1 (−n),ǧ LEB(n) ) such that we have the following conclusions.
(2) For all K ∈ X n,− and Λ > C, we have deg U(2) (F 3,K,n , Ω Λ,n , 0) = 0. (1.28) Remark 1.19. We note that vanishing of the Leray-Schauder degree does not give any information regarding the existence of a solution. However, we can moreover show that there is no solution at all for a large class of functions in X n,− ; see Theorem 7.3. In particular, there is no solution for K = constant when n ≥ 2. For n = 2 and K = constant, nonexistence of any solution (symmetric or non-symmetric) was proved in [Via10]. However, it is still an open question whether the case K = constant possibly admits some non-symmetric solution when n > 2.
Remark 1.20. For any n ∈ N * , the set X n,0 can be viewed as a "wall" in the space of positive radial functions X n , and the Leray-Schauder degree jumps by 1 upon crossing this wall, which is a phenomenon observed in many other geometric PDE problems.
All of the above results regarding the LeBrun metrics are proved in Section 7.
1.5. Acknowledgements. The authors would like to thank Richard Schoen for suggesting to consider the Green's function on an AF orbifold to prove Theorem 1.9; this greatly simplified our original proof which was based on a resolution of singularities argument.

Properties of the mass
In this section, we will prove Theorem 1.9. For simplicity, let us assume that there is exactly 1 orbifold point, which we denote as q, with orbifold group Γ ⊂ O(n) (the argument below easily generalizes to the case of multiple orbifold singularities). Let r denote a positive smooth function which is the Euclidean distance in the AF coordinate system, and near q is the distance to q. We will first prove a lemma showing existence of a certain harmonic function on X \ {q}.
Lemma 2.1. There exists a unique harmonic function H : X \ {q} → R which satisfies H > 1 and admits the expansion for ǫ > 0 sufficiently small, for some constant A > 0.
for some constant A. We then define H = 1 + h, which is harmonic. We have that lim r→∞ H = 1, and lim r→0 H = +∞. If H were not strictly larger than 1, then it would have an interior minimum. The strong maximum principle would then imply that H is constant, which is impossible. So H > 1, which clearly implies that A > 0. Obviously, H is unique.
Proof of Theorem 1.9. For any constant δ > 0, we define which satisfies ∆ g H δ = 0, H δ > 1, and admits the expansion for ǫ > 0 sufficiently small, for the fixed constant A from Lemma 2.1.
Next, we consider the metric (X q , g δ ) = (X \ {q}, H 4 n−2 δ g X ). Near r ∼ ∞, g δ has a single AF end of order min{τ X , n − 2}. Since q is an orbifold point, near q, g δ has a single ALE end of order τ = 2 − ǫ. To see this, choose Riemannian normal coordinates {x i } for g X around q, then we have the expansions (2.14) A computation then shows that as |y| → ∞, so g δ is indeed ALE of order τ = 2 − ǫ. Note also that the scalar curvature of g δ is given by (2.16) Given δ > 0, we can choose a very large distance sphere Σ δ in the ALE end of g δ which is strictly concave with respect to the normal pointing to the AF end. Let X Σ δ be the manifold with boundary obtained by removing the ALE end outside of Σ, which is a manifold with strictly concave boundary with a single AF end. From [HM20, Theorem 1.5 and Remark 1.7], we conclude that m(X Σ δ , g δ ) ≥ 0. (2.17) But an easy computation shows that m(X Σ δ , g δ ) = m(X, g X ) + b(n)δA, (2.18) where b(n) > 0 is a dimensional constant. Since this is true for any constant δ > 0, and A is a fixed constant, we conclude that Note that if m(X, g X ) = 0, then we cannot conclude that m(X Σ δ , g δ ) = 0, since A > 0. Therefore we cannot directly use the equality case in [HM20, Theorem 1.5]. So to finish the proof, if mass(X, g X ) = 0 then we instead argue as in [LP87,Lemma 10.7] to conclude that g X is Ricci-flat (this argument is valid in our orbifold setting). Since g X is asymptotically flat, we have asymptotic equality in Bishop's volume inequality (which holds for orbifolds; see [Bor92]). This implies that g X is flat, which clearly implies that there can be no nontrivial orbifold singularities, and (X, g X ) is isometric to (R n , g Euc ).
As mentioned above, in dimension 4, there are examples of ALE spaces which have negative mass [LeB88,HL16]. So we note the following corollary of Theorem 1.9.
Corollary 2.2. If (X, g) is a 4-dimensional Riemannian orbifold with negative mass at an orbifold point q, then the mass function m : X → R is necessarily discontinuous at q.

2.1.
Odd-dimensional cases. If the dimension n is odd, we have the following.
Proposition 2.3. Let (X, g) is a compact Riemannian orbifold with isolated singularities and odddimensional. Then any nontrivial orbifold point must have Γ = Z/2Z. Furthermore, (X, g) is a good orbifold. That is, there is a Z/2Z action on a compact manifoldX with finitely many fixed points such that X =X/(Z/2Z). Letting π :X → X denote the quotient mapping, then π * g is a smooth Riemannian metric onX.
Proof. For n odd, any element A ∈ O(n) must have ±1 as an eigenvalue, so the only possibility for a nontrivial orbifold point is Γ = Z/2Z. Near any singular point q, a small distance sphere is homeomorphic to RP n−1 , which is non-orientable if n is odd. So if there is any nontrivial orbifold point, then X contains a non-orientable 2-sided hypersurface, which implies that X \ Σ is nonorientable, where Σ is the finite set of singular points. Let π : X ′ → X \ Σ denote the orientable double cover. ConsiderX = X ′ ∪ {q 1 , . . . ,q j } whereq j are points, and extend π :X → X by letting π(q j ) = q j . We extend to Z/2Z action toX with fixed points atq j , and endowX with the quotient topology. It is then straightforward to show thatX is a smooth manifold and π * g extends as a smooth Riemannian metric toX.
Corollary 2.4. Let (X, g) be a compact Riemannian orbifold with isolated singularities and odddimensional. Then the mass function m : X → R satisfies m > 0 everywhere, unless (X, g) is conformal to (S n , g round ) or a "football" metric S n /(Z/2Z) with exactly 2 singular points.
Proof. At any smooth point of X, the mass of the Green's function metric is positive by Theorem 1.9. If the mass at a smooth point were zero, then the Green's function metric would be Euclidean space, which would imply that (X, g) is conformal to (S n , g round ). At a singular point q, by uniqueness of the Green's function metric, the Green's function metric at q must be the Z/2Z quotient of the Green's function of (X, π * g) atq. SinceX is a manifold, by the usual positive mass theorem, the Green's function metric upstairs must have non-negative mass, so the Green's function downstairs must also. If the mass at an orbifold point was 0, then the Green's function metric upstairs would have to be Euclidean space. This implies that the Green's function metric downstairs is a Z/2Z-quotient of Euclidean space, which implies that (X, g) is conformal to a "football" metric S n /(Z/2Z).
The above remarks show that the odd-dimensional case of the orbifold Yamabe problem is equivalent to the Z/2Z-equivariant Yamabe problem on the manifoldX.

Compactness preliminaries
Remarkable work in analyzing the blow-up points of equation (1.3) has been done in [Mar05], [LZ05], [LZ99], [KMS09]. In this section, we are going to quote some of their definitions and local results on manifolds, which also appear to hold on orbifolds by modification of proofs.
In the following context, we will write R n /Γ or B r (x)/Γ. If Γ = {e} is the trivial group, it denotes the Euclidean space or a smooth ball; if Γ = {e} is a finite nontrivial group in O(n), it denotes the Euclidean cone or a quotient of a ball centered at a singular pointx.
3.1. Conformal scalar curvature equation. Instead of dealing with equation (1.3), we will study the following conformal scalar curvature equation. Let Ω ⊂ R n /Γ be an open neighborhood of the origin, and suppose g is a Riemannian metric in Ω. Suppose also f is a positive C 1 function defined in Ω. Consider a positive C 2 function u satisfying where K is a positive C 2 function, 1 < p ≤ n+2 n−2 and τ = n+2 n−2 − p. We note that (3.1) is scale invariant. To see this, let u be a solution to equation (3.1). For any s > 0, define the rescaled solution v(y) = s 2 p−1 u(sy). Then L h v +Kf −τ v p = 0, wherẽ K(y) = K(sy),f (y) = f (sy) and the components in metric h in normal coordinates are given by h ij (y) = g ij (sy). Note that v satisfies an equation of the same type as equation (3.1).
We also note that (3.1) is conformally invariant. To see this, supposeg = φ 4 n−2 g is a metric conformal to g and let u be a solution to equation (3.1).
These two properties, will be used later to take a rescaling of the coordinates and conformally map g to some conformal normal metric, without changing the type of equation (3.1).

Isolated and isolated simple blow-up points.
Let Ω = B σ (x)/Γ be (a quotient of) an open ball centered at pointx. Suppose {g k } is a sequence of Riemannian metrics in Ω converging, in the C 2 loc topology, to a metric g. Let R k denote the scalar curvature of g k and R g denote the scalar curvature of the limit metric g. Suppose {f k } is a sequence of positive C 1 functions converging in the C 1 loc topology to a positive function f . Also suppose {K k } is a sequence of positive C 2 functions converging in the C 2 loc topology to a positive function K ∞ . Consider a sequence of positive C 2 functions u k satisfying where 1 + ǫ 0 < p k ≤ n+2 n−2 for some ǫ 0 > 0 and τ k = n+2 n−2 − p k . Definition 3.2. Suppose u k is a sequence of positive functions satisfying equation (3.5). If Γ = {e}, definex to be an isolated blow-up point for u k if there exists a sequence x k ∈ Ω, converging tox, so that: Remark 3.3. From now on, for each k, assume that we work in g k -normal coordinates {x i } centered at point x k . Then, we will simply write u k (x) instead of u k (exp x k (x)) and |x| instead of d g k (x, x k ).
Moreover, by [LP87, Theorem 5.1], there exists a conformal factor φ k , such thatg k = φ 4 n−2 k g k is the conformal normal metric with conformal normal coordinates {x i } centered at x k . Letũ k = φ −1 k u k . By the property that equation (3.1) is conformally invariant as stated in Section 3.1, u k andũ k satisfies the same type of conformal scalar curvature equation (3.1). Hence we may assume g k is already the conformal normal metric and {x i } is already the conformal normal coordiantes centered at x k .
Consider the change of variables and define the rescaled metric and functions where r is as in Definition 3.2. Using the property that equation (3.1) is rescale invariant as stated in Section 3.1, the rescaled functions satisfy (3.10) The following property holds for an isolated blow-up point.

Proposition 3.4 ([Mar05] Proposition 4.3).
Assume u k is a sequence of positive functions satisfying equation (3.5) and x k →x is an isolated blow-up point. Moreover, if Γ = {e}, we require x k =x for all large k's. Assume p k → n+2 n−2 , then there exist R ′ k → ∞ and ǫ k → 0, such that after passing to a subsequence, where p = lim k→∞ p k and ∆ denotes the Euclidean Laplacian. By [CGS89], we must have p = n+2 n−2 , and v(y) = U K∞(x) (y). The proof of (3.12) is the same as in [Mar05].
Remark 3.5. Under the same assumptions as Proposition 3.4, we also have where we are using g k -normal coordinates and integrating with respect to the Euclidean volume form. We say x k →x is an isolated simple blow-up point if there exists a real number 0 < ρ < r such that the functionsû have exactly one critical point in the interval (0, ρ), for k large.
We say thatx is an isolated simple blow-up point for u k if π * (x) is an isolated simple blow-up point for π * (u k ) in the lifting-up space.
Next, we give the special blow-up condition that we will work on.
Condition 3.7. Assume u k is a sequence of positive functions satisfying equation (3.5) and x k →x is an isolated simple blow-up point. Moreover, if Γx = {e}, then we require that x k =x for all k sufficiently large.
Remark 3.8. In Condition 3.7, the main reason why we require x k =x for all k sufficiently large in the singular point case Γ = {e} is the following. By assuming so, for each k, the geodesic ball centered at x k will be B r (x)/Γ. Then, when we later analyze some local integrals and let k → ∞, we will not run into the case that integrals over smooth balls converge to an integral over a quotient of a smooth ball. We also note that eventually, we will prove in Corollary 5.4 that if blow-up occurs at a singular point, then Condition 3.7 must hold.
From now on, assume we are working in dimension n = 4. In the following context, we will use C to denote various constants which only depend on the limit metric g, inf K ∞ , K ∞ C 2 and possibly the chosen small radius ρ 1 , δ and σ. The dependency is implied in the proof. Fix δ > 0, and define Proposition 3.9 ([LZ99]). Assuming Condition 3.7, for sufficiently small δ > 0, there exists constants 0 < ρ 1 < ρ and C > 0 such that for every x satisfying for every y satisfying Proof. The proof is the same as [LZ99, Lemma 7.3]. That proof was for n = 3, but directly generalizes to higher dimensions.
Proposition 3.10. Assuming Condition 3.7, then there exists a constant C such that Proof. The proof is very similar to [LZ99, Lemma 7.8], we provide here only a brief outline. Recall that {x i } is the g k -normal coordinates centered at point x k . For some fixed positive small σ, let η be a smooth cutoff function such that η(x) = 1 for |x| ≤ σ/2 and η(x) = 0 for |x| ≥ σ. Multipying equation (3.5) by η(∂u k /∂x j ), integrating by parts on {|x| ≤ σ}/Γ, we get (3.24) Then, using Proposition 3.4 in the ball |x| ≤ R ′ k M − p k −1 2 k and Proposition 3.9 in the annuli together with the assumption that K k converges to K ∞ in the C 0 loc norm, for large k's we have (3.25) where C 1 depends on inf K ∞ , K ∞ C 0 and σ. Next, the power series expansion where C 2 depends on K ∞ C 2 . Multiplying by u p k +1 k , integrating over {|x| ≤ σ}/Γ and using inequality (3.25), we get (3.28) The integral on the left limits to the volume of the bubble which is a finite constant, the integral on the right can be estimated similarly as above using Proposition 3.4 and Proposition 3.9. Therefore, we have proved where C is a constant depending on inf K ∞ , K ∞ C 2 , g and σ.
Proposition 3.11 ([Mar05] Proposition 4.5). Assuming Condition 3.7, then there exists a constant C > 0 and 0 < ρ 1 < ρ such that Proof. The proof is very similar to the proof of [Mar05, Proposition 4.5]. That proof was assuming K k = constant. For variable K k , every step in the proof remains valid, except for Claim 2 of [Mar05, Proposition 4.5], which says that there exists C > 0 such that This estimate does not hold in our setting, however, a modification of his arguments shows that there exists a constant C > 0 such that (3.32) To verify this, note that when K k is a variable function, there will be an extra term on the left hand side of [Mar05,inequality (4.18)]. That extra term is (3.33) For small ρ 1 and large k, when |x| ≤ ρ 1 2 , by power series expansion and Proposition 3.10, we have (3.34) Then for |y| ≤ ρ 1 2 M p k −1 2 k and large k, (3.35) On the other hand, by Proposition 3.9, we know (3.36) Therefore, Corollary 3.12 ([Mar05] Corollary 4.6). Assuming Condition 3.7, after maybe passing to a subsequence, we have where M k is as defined in Definition 3.2 and hx = aG(·,x) is a constant multiple of the standard Green function, i.e. L g (G(·,x)) = δx is the Dirac delta function at pointx. (Here, g stands for the limit metric.) Proof. The proof of Marques remains valid for variable K k .
Then, we have the following: Proposition 3.13. Assuming Condition 3.7, then
(4.16) 4.2. Main Estimates. By [CGS89], it is sufficient to consider the blow-up case as p k → 3, consequently τ k = 3 − p k → 0. Then we know lim k→∞ M τ k k = 1 from (3.32). We can estimate terms in the Pohozaev identity through the following lemmas. We will use C, C 1 to denote various positive constants independent of k and σ. For notational simplicity, we will omit dx, dy, dσ(x) and dσ(y) terms in integrals. Proof. Using (3.5), By Corollary 3.12, M k u k → h in the C 2 loc norm, hence for small σ,  Proof. Because U K k (x k ) (y) is a radial function and {y i } is a rescale of the conformal normal coordinates, we know Then, we have For σ ≤ 1, we have |y| ≤ σM By (4.6) and Proposition 3.13, for large k, we have and (4.24) Thus, for large k, (4.25) Lemma 4.3. There exist constants C > 0 and 0 < δ < 1 such that when σ < δ, Proof. Due to (4.1), there exist C > 0 and 0 < δ < 1 such that when σ < δ, on |x| = σ, by power On the other hand, on |y| = σM (4.27) Then, we have the estimate (4.28) Lemma 4.4. There exist constants C > 0 and 0 < δ < 1 such that when σ < δ, (4.29) Proof. By (4.1), there exist C > 0 and 0 < δ < 1 such that when σ < δ, in the ball |x| ≤ σ, by power series expansion, we have (4.30) The second inequality implies (4.31) It follows that in the ball |y| ≤ σM On the other hand, similar to (4.27), for |y| ≤ σM p k −1 2 k and large k, v 2 k (y) ≤ C(1 + |y| 2 ) −2 . Then, for large k, we have the estimate (4.33) Next, we estimate the most important term M 2 k I k,5 . Lemma 4.5. There exists a constant 0 < δ < 1 such that when σ < δ, uniformly for |x| ≤ σ. It follows There exists a constant 0 < δ < 1 such that when σ < δ, by power series expansion and Proposition 3.10, we have for |x| ≤ σ, where (K k ) ,ij (0) denotes the second order partial derivatives of K k in coordinates {x i } at the point x k . It is not hard to verify that (1 + |y|) 2 (4.41) and for |y| ≤ σM p k −1 2 k and large k. On the other hand, by power series expansion and Proposition 3.13, (4.43) Together with estimate (4.41), for large k, we have (4.45) Using estimate (4.42), for large k, we have (4.46) Therefore, (4.47) Since U K k (x k ) is a radial function, (4.48) Thus we obtain lim k→∞ M 2 k I k,5 = lim (4.50) Finally, we have achieved the following proposition.
Proposition 4.6. Assuming Condition 3.7, we have the following inequality Moreover, if τ k = 0 for all k, we have the equality (4.52) Here, ∆ g is the Laplacian with respect to the limit conformal normal metric g.
4.3. Green Function. Assuming Condition 3.7, Corollary 3.12 tells us that M k u k → hx = aG(·,x) in C 2 loc ((B r (x) − {x})/Γ). We are going to determine the constant a and the regular part of G(·,x) evaluated atx, where G(·,x) is the standard Green function for L g = ∆ g − 1 6 R g at the pointx, and g is the limit metric, i.e. for any C 2 function φ and small σ > 0, (4.56) Hence Corollary 3.12 implies that (4.57) Lemma 4.7. The constant a in Corollary 3.12 is . (4.58) Proof. Recall Remark 3.3 and the change of variables in (3.8), (3.9). For any C 2 function φ, definẽ Integrating by parts, we obtain (4.60) where the last equality is by Proposition 3.4. Therefore the constant a is . (4.61) On the other hand, recall that we assume for each k, g k is the conformal normal metric with conformal normal coordinates centered at point x k , hence the limit metric g is the conformal normal metric with conformal normal coordinates centered at pointx. Thus we have the following proposition. where r = |x|, ψx is from Definition 1.8, Ax is a constant, and . (4.63) Proof. The proof is given by [LP87, Definition 6.2 and Lemma 6.4]. Note that our notation is different from [LP87]. Our pointx is their point P ; our operator L g is equal to −1/6 multiplied with their box operator ; our G(·,x) is equal to −6 multiplied with their Γ P , our ψx is their G. And the Γ in our equation (4.63) is the quotient group near pointx.
Then we can relate the Pohozaev identity with the constant term Ax.
Lemma 4.9. We have (4.64) Proof. We will write G instead of G(·,x). Using hx = aG, we have (4.65) By Proposition 4.8, on |x| = σ for small σ, (4.66) (4.68) Letting σ → 0, using Lemma 4.7 and Proposition 4.8, we have (4.69) Proposition 4.10. Assuming Condition 3.7, we have the following inequality Moreover, if τ k = 0 for all k, we have the equality (4.71) Here, ∆ g is the Laplacian with respect to the limit conformal normal metric g.
Proof. Assuming Condition 3.7, by Proposition 4.6, we have (4.72) By Lemma 4.10, we have (4.74) The case that τ k = 0 for all k follows similarly.
Moreover, by relating Ax with the mass, we can remove the assumption "conformal normal metric", as follows.
Proposition 4.11. Assuming Condition 3.7, let g = lim k→∞ g k be the limit metric, but do not necessarily assume that g k , g are conformal normal metrics. Letĝx = ψ 2 x g be the conformal blow-up of g at the pointx, as in Definition 1.8. We have the following inequality Moreover, if τ k = 0 for all k, we have the equality (4.76) Proof. For each k, let r k denote the g k -distance function from the point x k . Assumeg k = φ 2 k g k is the conformal normal metric with conformal normal coordinates {x i } centered at point x k . By [LP87, Theorem 5.6], after applying a dilation and a translation to the coordinates {x i }, we may assume φ k (x k ) = 1 and ∇φ k (x k ) = 0, in other words, for small r k , φ k = 1 + O(r 2 k ). When k → 0, we have thatg = φ 2 g, where g,g, φ are the limit of g k ,g k , φ k as k → ∞. Moreover, φ = 1 + O(r 2 ), where r is the g-distance function from the pointx. Next, recall that the conformal transformation law of the Laplacian forg = e 2ϕ g is ∆g = e −2ϕ ∆ g + (n − 2)e −2ϕ g ij ∂ϕ ∂x j ∂ ∂x i . (4.77) Thus, for any f ∈ C 2 , we have ∆gf (x) = ∆ g f (x). Let Ax be the regular part corresponding to conformal blow-up ofg at pointx. Letĝx be the conformal blow-up of g at pointx. By [LP87,Lemma 9.7], we have m(ĝx) = 12Ax. (4.78) Therefore, we know which implies that this proposition is equivalent to Proposition 4.10.

Blow-up points must be isolated and simple
In [LZ99], they proved that on a 3-dimensional compact manifold, all blow-up points for (3.5) must be isolated simple blow-up points. The same result in higher dimensions is proved by [KMS09], [LZ05], but only for constant prescribed scalar curvature. Here, we will modify their proofs to show that the same result holds on 4-dimensional compact orbifolds, for a sequence of variable prescribed scalar curvatures.
Let (M, g) be a compact Riemannian 4-dimensional orbifold with singularities Σ Γ = {(q 1 , Γ 1 ), · · · , (q l , Γ l )} (5.1) and positive scalar curvature R g . Assume u is a positive C 2 solution of equation (3.1) on M , where K is a positive C 2 function and f is a positive C 1 function. For any pointx ∈ M , define Ωx ,σ in the following: a) ifx is a smooth point, define Ωx ,σ = B σ (x) for some σ > 0 such that its closureΩx ,σ doesn't include any singular point. In other words, d g (x, {q 1 , · · · , q l }) > σ; b) ifx = q j for some 1 ≤ j ≤ l, choose σ = σ j where σ j is as defined in Definition 1.1. Then the neighborhood ofx is a quotient ball B σ (x)/Γ. Define Ωx ,σ = π * j (B σ (x)/Γ) to be the lifting-up space. Denote the lifting-up functions and metric still by u, f , K and g.
First, let us recall a lemma from [LZ99].
then we have p > 3 − ε and for some local maximum point of u in Ωx ,σ \ S, denoted as x 0 , where d g (x, S) denotes the distance of y to S, and d g (x, S) = 1 if S = ∅.
Proof. The case that K is a positive constant and Ωx then there exists some integer N = N (u) ≥ 1 and N local maximum points of u denoted as {x 1 , · · · , x N } ⊂ Ωx ,σ , such that: Proof. The case that K is a positive constant and Ωx ,σ is replaced by a compact 3-dimensional manifold M is proved in [LZ99, Proposition 5.1]. Briefly, that proof was completed by an induction process as following: first apply [LZ99, Lemma 5.1] with S = ∅ to get the first local maximum point of u, denoted by x 1 ; assuming we have already got local maximum points {x 1 , · · · , x l }, apply [LZ99, Lemma 5.1] with S = ∪ l j=1 B r j (x j ) to get the next local maximum point. If the condition (5.2) is not satisfied in any step, the process stops. This process must stop after a finite number of times because each time Br j (x j ) |∇u| 2 is greater than some positive universal constant and their sum j Br j (x j ) |∇u| 2 ≤ M |∇u| 2 which is finite. It is not hard to verify the set {x 1 , · · · , x N } constructed by the process satisfies all the above properties. In our case, we can continue the same process by inductively applying our Lemma 5.1 with S as mentioned above. Everything follows the same way and our proposition is proved.
Next, we can rule out bubble accumulation.
Proposition 5.3. Let (M, g), u, K, f, Ωx ,σ be as defined in the beginning of this section. Let ε, R ′ , C 0 , C 1 and {x 1 , · · · , x N } be as defined in Proposition 5.2. If ε is sufficiently small and R ′ is sufficiently large, then there exists a positive constantC, which only depends on M , g, f C 1 (M ) , inf M K, K C 2 (M ) , ε and R ′ , such that if max Ωx,σ u ≥ C 0 , then d g (x j , x l ) ≥C, for all j = l.
Proof. The proof is similar to [KMS09,Proposition 8.3] and [LZ99, Proposition 5.2]. We will prove it by contradiction. Suppose that such a constantC does not exist, then there exist sequences p k → p ∈ (3 − ε, 3] and {u k } with max Ωx,σ u k ≥ C 0 and lim k→∞ min j =l d g (x j (u k ), x l (u k )) = 0. (5.6) Without loss of generality, assume that For each k, take normal coordinates {x i } centered at point x 1 (u k ) and consider change of variables Then v k satisfies where (h k ) ij (y) = g ij (δ k y),K k (y) = K(δ k y) andf k (y) = f (δ k y). If x j (u k ) ∈ B √ δ k (x 1 ), denote by y j (u k ) = δ −1 k x j (u k ) the y-coordinate of point x j (u k ). By the proof of [KMS09, Proposition 8.3], we have y 1 (u k ) = 0, y 2 (u k ) →ȳ 2 with |ȳ 2 | = 1, {0,ȳ 2 } are isolated simple blow-up points for {v k }, and where S ′ denotes the set of blow-up points for {v k } and a 1 , b 1 > 0 are some positive constants. On the other hand,K HenceK k converges to the constant K(0) in the C 2 loc norm, where K(0) by definition is the K value at the limit point of x 1 (u k ) as k → ∞, possibly by passing to a subsequence. Applying Proposition 4.10 to the blow-up sequence {v k }, we get b 1 ≤ 0, which contradicts b 1 > 0. Therefore our proposition is proved.
Corollary 5.4. Let (M, g) be a compact Riemannian 4-dimensional orbifold with positive scalar curvature. Suppose {f k } is a sequence of positive C 1 functions converging in the C 1 loc topology to a positive function f . Also suppose {K k } is a sequence of positive C 2 functions converging in the C 2 loc topology to a positive function K ∞ . Let {u k } be a sequence of positive solutions of equation (3.5) on M with g k = g and max M u k → ∞. Then p k → 3 and the set of blow-up points is finite and consists only of isolated blow-up points. Moreover, if blow-up occurs at a singular point, i.e.
x k →x and u k (x k ) → ∞ wherex is a singular point, then there exists an integer N ∈ N * such that for any k > N , x k =x.
Proof. By the assumption of f k and K k , there exists a constant C 2 such that for large k, (5.14) Then v k satisfies where a 2 = b 2 = 1, where Γ = {e} ifx is a smooth point, but Γ is the quotient group ifx is a singular point.
On the other hand, because K k converges to K ∞ in the C 2 loc norm, we know (5.17) HenceK k converges to the constant K ∞ (x) in the C 2 loc norm. Applying Proposition 4.10 to the blow-up sequence {v k }, we obtain b 2 ≤ 0, which contradicts b 2 = 1. Thereforex is an isolated simple blow-up points for {u k }.
Corollary 5.6. Let (M, g), f k , K k , g k be as defined in Corollary 5.4. Assume u k is a sequence of positive functions satisfying equation (3.5), then Condition 3.7 is a necessary condition for any blow-up point.
Proof. This follows immediately from combining Corollary 5.4 and Proposition 5.5.
Proof of Theorem 1.11. To prove the upper bound u ≤ C under assumption (1.10) in Theorem 1.11, suppose the contrary. Then there exist p k → 3 and {u k } satisfying where {K k } is a sequence of positive C 2 functions converging in the C 2 loc topology to a positive function K ∞ . Let x k denote the point where u k obtains a maximum, after possibly passing to a subsequence, we may assume x k →x is a blow-up point. By Corollary 5.6, the sequence (u k , x k ) satisfies Condition 3.7, where g k in equation (3.5) is the metric conformal to g with conformal normal coordinates centered at x k . By Proposition 4.11, we know which contradicts against assumption (1.10) in Theorem 1.11. Therefore we know that u ≤ C for u and C as stated in Theorem 1.11. Next, assume u obtains sup M u at a point P , then ∆u(P ) ≤ 0. By (1.3), u(P ) ≥ R g (P )/(6K(P )) ≥ c 0 . By the Harnack inequality, inf M u ≥ (1/c 1 )u(P ) ≥ c 0 /c 1 ≥ 1/C, (5.20) for sufficiently large C. By standard elliptic estimates, we conclude that u C 2,α (M ) ≤ C. Furthermore, because for p < 3, there always exist subcritical solutions for any K (the proof on manifolds remains valid in the orbifold setting; see for example [ES86,LP87]). Take a sequence of subcritical solutions and let p → 3, due to (1.9), they limit to a critical solution for p = 3. Thus the second part of Theorem 1.11 is proved. The first part of Theorem 1.11 can be proved similarly, by fixing p k = 3 in the above proof.

Variational methods
Let (M, g) be a compact Riemannian n-orbifold with singularities Σ Γ = {(q 1 , Γ 1 ), · · · , (q l , Γ l )} and positive scalar curvature R g . Let K be a positive smooth function on M . In this section, we will study equation (1.3) using a variational method. Consider the energy functional , (6.1) for 1 < p ≤ n+2 n−2 and u ∈ W 1,2 (M ). Define the minimal energy to be E(p, K) = inf Let Q(S n ) denote the Sobolev quotient of S n , which is also the minimal energy on S n for K ≡ 1, that is, 3) The following theorem generalizes [ES86, Proposition 1.1] to the orbifold case, and also generalizes [Aku12, Theorem 3.1] from the case K = constant to the case of variable K.
Theorem 6.1. Let (M, g) be a compact Riemannian n-orbifold with positive scalar curvature and singularities Σ Γ = {(q 1 , Γ 1 ), · · · , (q l , Γ l )}. Let K be a positive smooth function on M . Define the modified maximum value of K Then the following inequality always holds then there exists a positive smooth solution u of (1.3) with p = n+2 n−2 , such that Proof. We first prove the inequality (6.5). The quantity B K is attained at either a singular point or a smooth point. Consider any point q with |Γ| 2 n−2 K(q) = B K , where Γ is the orbifold group at q if q is a singular point, or Γ = {e} if q is a smooth point. Take a conformal mapping g q = u 4 n−2 q g such that g q is the conformal normal metric centered at the point q. Consider the Green function with the power series expansion G q = 1 4n(n − 1)V ol(S n−1 ) (r 2−n q + H q ), (6.8) where r q is the geodesic distance from q based on the metric g q and H q is the higher order term. Take a family of test functions Chapter 1]. If q is an orbifold point, we estimate integrals in J n+2 n−2 by lifting everything up to the universal cover near the orbifold point q. Therefore, compared to [MM21, Proposition 5.1], our estimation (6.10) has an extra factor of c(n)/|Γ| 2 n . By a direct computation, we know that c(n)ĉ 0 = Q(S n ). (6.12) Clearly, we have (6.13) By our assumption, we have |Γ| 2 n K(q) n−2 n = B n−2 n K , hence (6.5) is proved. We next show that the strict inequality (6.6) implies the existence result. The proof is a modification of [Aku12, Theorem 3.1]. If we have proved the case that M has only one singularity, the more general cases can be proved by an induction on the number of singularity points. Hence we may assume M has only one singularity (q, Γ). Let X = M − {q}. Note that n−2 (u, K, X). (6.14) Let B ρ denote the open geodesic ball centered at q of radius ρ. Define Note that on the manifold with boundary (N, ∂N ) = (X − B 1/k , ∂B 1/k ), when we apply integration by parts to any function in C ∞ c (X \ B 1/k ), the boundary integral term always vanishes. As a result, the variational method used in [ES86, Proposition 1.1] remains valid here. Thus, (6.18) implies that for each k ≥ k 0 , there exists a non-negative J n+2 Denote the zero extension of each u k to M by also the same symbol u k . Suppose the sequence {u k } has a uniform C 0 -bound, i.e., there exists a constant C > 0 such that u k C 0 (M ) ≤ C for k ≥ k 0 , then there exists a non-negative J n+2 n−2 (·, K, M )-minimizer u ∈ W 1,2 (M ) with ||u|| C 0 (M ) ≤ C, such that u k → u weakly in W 1,2 (M ), u k → u strongly in L 2 (M ).
To complete the proof, it is sufficient to show a uniform C 0 -bound for the sequence {u k }. For each k, take the absolute maximum point x k ∈ X of u k and denote M k ≡ u k (x k ). Taking a subsequence if necessary, there exists a pointx ∈ M such that lim k→∞ x k =x. (6.24) Suppose that there is not a uniform C 0 -bound for {u k }, that is lim k→∞ M k = ∞. There will be two cases. Case 1:x = q (blow-up occurs at the singular point). In this case, we consider the universal cover of a small neighborhood around q. Let {x k } be a sequence of lifting-up points of {x k } in the same branch of the lifting-up space. Letũ k ,K denote the lifting-up functions of u k , K, respectively. In the lifting-up space, for each k, let {x i } be a normal coordinate system in a small ball B σ (x k ) centered at eachx k . Consider the change of variablesỹ = M Multiplyingṽ with (6.26) and integrating by parts, we obtain Together with (6.29), we have (6.31) Hence we may choose normal coordinates {x i } centered atq such that s = |x| + O(|x| 3 ). Also note that g LEB(n) = (n +r 2 ) 2 ·ǧ LEB(n) = (s −2 + n) 2 ·ǧ LEB(n) is scalar-flat, hence g LEB(n) is the conformal blow-up ofǧ LEB(n) at the pointq, as in Definition 1.8. We will next study In the U (2)-invariant case, Theorem 1.11 specializes to the following.
Consequently, in this case there exists a U (2)-invariant solution u of (7.3) with p = 3.
Proof. Assume {u k } ⊂ X n is a family of U(2)-invariant solutions of (7.3) with corresponding exponents p k → 3 as k → ∞. Assume u k blows up at a pointx when k → ∞. Supposex is in the set {s = s 0 } for some s 0 > 0. Because u k is U(2)-invariant, u k blows up on the entire set {s = s 0 }, which is either a hypersurface if s 0 < ∞ or the CP 1 component if s 0 = ∞. However, either case is impossible, because Proposition 5.3 implies that blow-up points are isolated. Therefore, u k must blow up at the orbifold pointq. It is clear that if K ∈ X n,+ implies (1.10), and K ∈ X n,− implies (1.8). By the proof and statement of Theorem 1.11, this proposition is proved.
(7.16) By (7.5), it is clear that v(s) = O(s 2 ) near s = 0. Also because the curvature and volume term forǧ are bounded near s = 0, the first integral term on the right hand side goes to zero as ε → 0. For the second integral term on the right hand side, since v(s) is a radial function, the only nonvanishing component of (∇v) i is (∇v) 0 = v ′ (s) along the ∂/∂s direction. On the other hand, by the Bianchi identity, Since K(s) is a radial function, the only nonvanishing component of (∇K) i is (∇K) 0 = K ′ (s) along the ds direction. Hence which implies Eǧ = 0. By (7.18), we know K ′ (s) ≡ 0, so that K = constant. By [Via10, Theorem 1.3], such a solution v does not exist.
For the last part, assume K(s) ∈ X n . Near s = 0, K has a power series expansion Proof of Theorem 1.18. By Proposition 7.1, and homotopy invariance of the Leray-Schauder degree, deg(F p,K,n , Ω Λ,n , 0) is equal to a constant in either case (1) or (2) in Theorem 1.18. For any n ∈ N * , Theorem 7.3 implies that deg(F 3,K,n , Ω Λ,n , 0) = 0, (7.38) for K ∈ X n,− satisfying the assumptions of Theorem 7.3. Homotopy invariance implies that the degree is zero for any K ∈ X n,− , which proves (2) of Theorem 1.18. Part (1) of Theorem 1.18 is proved by a standard subcritical degree counting argument; see [Sch91].