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A framework for Morse theory for the Yang-Mills functional. (English) Zbl 0665.58006
It is well known that if a functional defined on a non-compact Banach manifold satisfies the Palais-Smale condition then the hessian data from the set of critical points of the functional can be used to compute topological properties of the Banach manifold in question. This condition is not satisfied by the Yang-Mills functional. Even in the simplest cases this functional does not satisfy the usual hypothesis of Morse theory. However, in this paper, the author partially recovers the Morse theory for the Yang-Mills functional by studying its restriction to a countable set of finite dimensional, non-compact varieties.
The author proves that below to fixed energy E only a finite number of these varieties need be considered. In the paper it is also shown that these varieties are naturally parametrized with topological data from the 4-dimensional manifold and with the critical points of the Yang-Mills functional and that the connections on these varieties obey a priori estimates given by solutions of the Yang-Mills equations.
Reviewer: M.Tucsnak

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: DOI EuDML
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