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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.

Given a nondegenerate minimal hypersurface Σ in a Riemannian manifold, we prove that, for all ε small enough there exists uε, a critical point of the Allen-Cahn energy Eε(u) = ε 2 |∇u| 2 + (1 − u 2 ) 2 , whose nodal set converges to Σ as ε tends to 0. Moreover, if Σ is a volume nondegenerate constant mean curvature hypersurface, then the same conclusion holds with the function uε being a critical point of Eε under some volume constraint.