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.

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We prove that many complete, noncompact, constant mean curvature (CMC) surfaces f : Σ → R 3 are nondegenerate; that is, the Jacobi operator ∆ f + |A f | 2 has no L 2 kernel. In fact, if Σ has genus zero with k ends, and if f (Σ) is embedded (or Alexandrov immersed) in a halfspace, then we find an explicit upper bound for the dimension of the L 2 kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres.

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