Singular Yamabe problem for scalar flat metrics on the sphere

Let $\Omega\subset \mathbb S^n$ be a domain of unit $n$-sphere and $\mathring{g}$ the standard metric of $\mathbb S^n$, $n\ge 3$. We show that there exists conformal metric $g$ with vanishing scalar curvature such that $(\Omega, g)$ is complete if and only if the Bessel capacity $\mathcal C_{\alpha, q}(\mathbb S^n\setminus \Omega)=0$, where $\alpha=1+\frac2n$ and $q=\frac n2$. Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as ideas used to characterize the existence of negative scalar metric developed by Labutin.


Introduction
Let Ω be an open subset of unit sphere S n , n ≥ 3 and • g the standard metric of S n .We want to characterize the open sets Ω with the following property: there exists a metric g, conformal to • g such that (Ω, g) is complete and g has vanishing scalar curvature.This question was studied by Schoen and Yau [Sch87], [SchY88].If we are given a compact Riemannian manifold (M, g) then the question of existence of conformal deformation of metric into complete metric ḡ with constant scalar curvature is known as the Yamabe problem [Yam60].Yamabe's original approach was to formulate this as a variational problem.Later contributions of Trudinger [T67], Aubin [A76] and Schoen [Sch84] helped to complete Yamabe's original approach.
If we impose structural assumptions on K := S n \ Ω, then it is known that there is a g with above properties.In particular, if K is a finite union of Lipschitz submanifolds of dimension k ≤ (n − 2)/2 then this is indeed the case, see [De92], [KaN93], .Some discussion on this and related open problems are contained in [McO98].For a survey of related results see [G16], [La05] and references therein.
The aim of this work is to give a complete characterization of open set Ω without any assumption on the structure of K = S n ⊂ Ω.Such characterization for negative scalar case was given by Labutin [La05].In what follows cap(•) := C 1+ 2 n , n 2 (•) stands for Bessel's capacity (see section 2 for precise definition).Our main result is Theorem 1.1.Let Ω ⊂ S n , n ≥ 3, be an open set and K = S n \ Ω.Then the following properties are equivalent: (i) In Ω there exists a scalar flat complete metric conformal to The proof is based on a characterization of Bessel's capacities in terms of the Wolff potential [HWo83].One of the main ingredients of the proof is the representation of positive harmonic functions in terms of Martin kernels [ArG], [He].

Background
In this section we recall some well known facts from conformal geometry which can be found in [SchY].Let (M, g) be a Riemannian manifold of dimension n ≥ 3. The operator (2.1) is called conformal Laplacian.Here R g is the scalar curvature of the metric g and ∆ g is the Laplace-Beltrami operator.L g has remarkable properties: under conformal change of metric g = φ More generally, let M be another manifold with the metric g, and let f : M → M be a diffeomorphism changing the metrics conformally.If From (2.2) and (2.3) it follows that Next we define the stereographic projection to be σ : Since we consider the scalar flat case, i.e.R g = 0, then (2.5) yields (2.10) Introduce the function Then from (2.3) we obtain (2.12) Since R gE = 0 then we get 2.1.Characterization of capacity.Let α > 0, 1 < q ≤ n α and C α,q (E) be the Bessel capacity of E ⊂ R n [HWo83].For given Radon measure µ we can consider the Wolff potentials defined as Then we say that a set E is (α, q)-thin at x 0 ∈ E if and only if there is a Radon measure µ such that In what follows we take α = 1 + 2 n , q = n 2 and denote cap( Then for this choice of parameters the Wolff potential takes the form In view of theorem 4 [HWo83] we have Proposition 2.1.Let K ⊂ R n be a compact set such that cap(K) = 0. Then there exists a Radon measure µ, µ = 1, such that suppµ ⊂ K and We can use the stereographic projection and without loss of generality assume that the north pole N ∈ Ω such that σ(K) ⊂ B(0, 1/2) and there is u : R n \ σ(K) such that g = u g is complete where ∆u = 0, in Ω = {u > 0}.
We claim that there exists a Radon measure µ with suppµ ⊂ K, where k(x, y) is the Martin kernel (see [ArG] Theorem 8.4.1 or Chapter 12 [He] p 251) and k is locally integrable in R n × R n .Moreover, there two universal constants c 1 , c 2 such that Suppose cap(K) > 0. By Proposition 2.1 the Wolff potential of µ must be finite at some point x 0 ∈ K. Without loss of generality we assume that x 0 = 0 and µ is a probability measure such that We first establish a technical Lemma 3.1.Let u > 0 be as above and suppµ ⊂ B(0, 1/2).Then there is a constant C > 0 such that As for I 1 in (3.6) we have Combining For n = 3 we take a sequence of smooth functions f i weakly converging to µ in D m := B (0, ρ m−2 ) \ B(0, ρ m+2 ) (see Lemma 0.2 [Lan]) then applying lemma 7.12 from [GT] to After letting i → ∞ this yields Moreover, denoting m(t) = µ(B(0, t)) and using integration by parts together with Cauchy-Schwarz inequality we get implying that Hence and then from Fubini's theorem we get as above The estimate for I 3 follows from integration by parts.
Finally, let us consider the case n ≥ 5.
As for I 3 one can easily see that and consequently after integration by parts we get and the proof of lemma is complete.

Proof of (i)⇒(ii) in Theorem 1.1.
We claim that there exists a smooth curve c Observe that (3.8) is impossible if u 4/(n−2) g E is complete, Hence to finish the proof we have to establish (3.8).But from (3.5) the existence of such curve can be deduced as in [La05] p 23 and therefore the result follows.
4. Proof of (ii) ⇒ (i): cap(K) = 0 implies existence of metric Proof of (ii)⇒(i) in Theorem 1.1.In what follows we assume, without loss of generality, that the north pole N ∈ Ω.Since σ(K) ⊂ R n is the image of K under stereographic projection then it is compact such that cap(σ(K)) = 0.
From Proposition 2.1 it follows that there is a probability measure µ, such that suppµ ⊂ σ(K) and To see this we use a version of Hopf-Rinow theorem formulated in terms of divergent paths [PV16].A continuous path c : [0, ∞) → Ω is said to be a divergent path if, for every compact set E ⊂ Ω, there exists t 0 ≥ 0 such that c(t) ∈ E for every t > t 0 .(Ω, g) is called "divergent paths complete" (or complete with respect to divergent paths) if every locally Lipschitz divergent path has infinite length.Using this version of Hopf-Rinow theorem one can see that (Ω, g) is complete if and only if every smooth (or even Lipschitz) divergent path has infinite length.
Let us take a divergent path c in Ω, and denote c : By assumption c is a divergent path, therefore there exists x 0 ∈ K such that (4.4) dist g (x 0 , c(t k )) → 0 for a sequence {t k }, k → +∞.
For m ∈ N, we let γ m = D m ∩ c where If m ≥ m 0 for sufficiently large m 0 it follows from the smoothness of c that γ m is at most a countable union of open smooth curves.Moreover, from (4.4) we see that γ m = ∅, and For y ∈ B( x 0 , 2 −(k+2) ) and x ∈ D k we have |x − y| ≤ 1 2 k + 1 2 k+2 = 5 2 k+2 .Therefore Comparing the inequalities for L g (c) and W and recalling (4.1) W(µ, x 0 ) = +∞.