Bubbling and regularity issues in geometric non-linear analysis

Details Bubbling and regularity issues in geometric non-linear analysis Bubbling and regularity issues in geometric non-linear analysis

2003-04-24

arxiv.org/abs/math/0304396v1

Proceedings of the ICM, Beijing 2002, vol. 3, 197--208

Mathematics

Analysis of PDEs

Scientific paper

Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which an invariance group (similitudes, conformal groups) acts. This common feature generates, in all these different situations, the same non-linear effect. One observes a strict splitting in space between an almost linear regime and a dominantly non-linear regime which has two major characteristics : it requires a quantized amount of energy and arises along rectifiable objects of special geometric interest (geodesics, minimal surfaces, J-holomorphic curves, special Lagrangian manifolds, mean-curvature flows...).

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