Higher order Seiberg-Witten functionals and their associated gradient flows

TL;DR

This paper introduces higher order Seiberg-Witten functionals on closed spin^c manifolds, studies their gradient flows, and establishes short and long time existence results, highlighting curvature concentration phenomena in critical dimensions.

Contribution

It generalizes Seiberg-Witten functionals to include higher derivatives and analyzes the associated gradient flows, providing new existence and energy estimates.

Findings

Proves short time existence of the gradient flows.

Establishes energy and derivative estimates along the flows.

Shows long time existence in sub-critical dimensions and identifies curvature concentration as an obstruction in critical dimensions.

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.