Refined asymptotics for constant scalar curvature metrics with isolated singularities

Details Refined asymptotics for constant scalar curvature metrics with isolated singularities Refined asymptotics for constant scalar curvature metrics with isolated singularities

1998-07-08

arxiv.org/abs/math/9807038v1

Mathematics

Differential Geometry

To appear, Inventiones Mathematicae

Scientific paper

10.1007/s002220050285

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, \Delta 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, by Caffarelli, Gidas and Spruck, 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 Pohozaev 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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