Higher order Seiberg–Witten functionals and their associated gradient flows

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 L²-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.