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Higher order Seiberg–Witten functionals and their associated gradient flows

  • Hemanth Saratchandran
Article
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Abstract

We define functionals generalising the Seiberg–Witten functional on closed \(spin^c\) manifolds, involving higher order derivatives of the curvature form and spinor field. We then consider their associated gradient flows and, using a gauge fixing technique, are able to prove short time existence for the flows. We then prove energy estimates along the flow, and establish local \(L^2\)-derivative estimates. These are then used to show long time existence of the flow in sub-critical dimensions. In the critical dimension, we are able to show that long time existence is obstructed by an \(L^{k+2}\) curvature concentration phenomenon.

Mathematics Subject Classification

53C07 53C21 53C23 

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Notes

Acknowledgements

The author thanks the anonymous referee for their comments and suggestions on the initial submission of the paper. The author wishes to acknowledge support from the Bejing International Centre for Mathematical Research, and the Jin Guang Mathematical Foundation.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Beijing International Center for Mathematical ResearchBeijingPeople’s Republic of China

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