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- W1928590248 abstract "We introduce the notion of a special monopole class on a four-manifold. This is used to prove restrictions on the smooth structures of Einstein manifolds. As an application we prove that there are Einstein four-manifolds which are simply connected, spin, and satisfy the strict Hitchin-Thorpe inequality, and which are homeomorphic to manifolds without Einstein metrics. Seiberg-Witten theory has been developed as a theory of moduli spaces of solutions to the monopole equations by imitating the development for the anti-self-dual Yang-Mills equations. Therefore, in th e applications the use of the whole moduli space has been emphasized, following the ideas of Donaldson in Yang-Mills theory. Nevertheless, unlike Donaldson theory, Seiberg-Witten theory has incorporated geometric arguments exploiting a single solution to the monopole equations for explicit constructions, without necessarily studing all its deformations. Rather than imit ating Donaldson theory, these arguments are more reminiscent of Yau's work o n the Cal- abi conjectures, where a single Einstein metric is construc ted and used as input in geometric arguments, or of the use of minimal surfaces in three- dimensional topology or in the theory of manifolds of positive scalar curva- ture. Going back even further, there is a clear parallel with the use of har- monic forms and harmonic spinors in geometry. Of course, those are solu- tions of linear partial differential equations, rather tha n non-linear ones, and one might speculate that the mildness of the non-linearity of the Seiberg- Witten equations manifests itself in the way the solutions are used 1 . For arguments using the existence of a solution to the monopole equa- tions rather than the non-triviality of a topological invar iant, Kronheimer (25)" @default.
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- W1928590248 date "2004-01-01" @default.
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- W1928590248 doi "https://doi.org/10.1155/s1073792804131802" @default.
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