Refined asymptotics for constant scalar curvature metrics with isolated singularities

Abstract

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, ∆u + n(n−2) 4 u n+2 n−2 = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohoẑaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.

Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities

“…As in [32], we also obtain higher order expansions for solutions to (8) in the case 2k ≤ n. Theorem 2. Let w(t, θ) be a solution to (8) on {t > t 0 } × S n−1 in the Γ + k class, where n ≥ 3, 2 ≤ k ≤ n/2, and the constant c is normalized to be 2 −k n k , and let w * (t) = ξ h (t + τ ) be the radial solution to (8) on R × S n−1 in the Γ + k class for which (10) holds.…”

“…We will first summarize some needed preliminary properties for solutions to (8) and (8 ′ ) in section 2, then provide a proof for Theorem 1 in section 3, using Theorem 3 and several other ingredients, the proof for which we supply in this section. In section 4, we provide the analysis for the linearized operator for (8) at entire radial solutions, and use them to provide an alternative proof for Theorem 1 along the approach of [32]. We will also provide a proof for Theorem 2 here.…”

“…Several preliminary properties for solutions to (7) To adapt either of the approaches in [4] or [32] to our situation, we will need several key properties of solutions derived in [38]. The following is a special case of Theorem 1.2 in [38].…”

“…Similar to properties of the linearized operator to the scalar curvature operator used in [32], we have the following properties for the L j 's.…”

The k-Yamabe problem

“…He also developed a series of local estimates and fine local analysis for possible blow-ups of solutions to equation (1.6) [52,53,54,57]. Some of his estimates can be found in [35]. His estimates were used by many researchers in the study of semilinear elliptic equations, including the compactness problem, and eventually led to the resolution of the compactness of the solution set for dimensions n = 3 in [47], n = 4, 5 in [17], n = 6, 7 in [45,49], and also the compactness when n ≤ 11 under the assumption of positive mass theorem in [46].…”