Concentration-compactness phenomena in conformal geometry
Luca Martinazzi · Repository for Publications and Research Data (ETH Zurich) · 2009
Consider a smooth Riemannian manifold (M, g) of arbitrary even dimension 2m, and a sequence of conformal metrics g k = e 2u k g on M , u k ∈ C ∞ (M ).In this work we study the concentration-compactness behaviour of this sequence of metrics, under the assumption that their volumes are equibounded and their Qcurvatures Q 2m g k converge uniformly or even in C 0 to a given continuous function Q 0 .We start by taking (M, g) to be R 2m with the Euclidean metric.Then, in analogy with a 4-dimensional result of Adimurthy, F. Robert and M. Struwe, we show that, in case of non-compactness and up to subsequences, the metrics vanish in the limit uniformly locally outside a rectifiable set of dimension at most 2m -1.