A Note on Curvatures and Rank 2 Seiberg–Witten Invariants
In this paper, we investigate rank 2 Seiberg–Witten equations which were introduced and studied in Massamba and Thompson (J Geom Phys 56:643–665, 2006). We derive some lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with non-trivial rank 2 Seiberg–Witten invariants. Existence of Einstein and anti-self-dual metrics on some compact oriented 4-manifolds is also discussed.
KeywordsSeiberg–Witten equation Weyl curvature Kähler metric Einstein metric Anti-self-dual metric
2010 Mathematics Subject Classification:81T13; 70S15; 58D27
The author would like to express his sincere gratitude to Professor Norbert M. Hounkonnou for his continuous support and invaluable friendship over years. He is also grateful to G. Thompson for invaluable discussions and support. Finally, the author thanks the referee for his/her valuable comments and suggestions. This work is based on the research supported wholly/in part by the National Research Foundation of South Africa (Grant no: 95931).
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