We investigate the gradient flow of the $L^2$ norm of the Riemannian curvature on surfaces. We show long time existence with arbitrary initial data, and exponential convergence of the volume normalized flow to a constant scalar curvature metric when the initial energy is below a constant determined by the Euler characteristic of the underlying surface.
The gradient flow of the $L^2$ curvature energy near the round sphere
Long time existence of the (n-1)-plurisubharmonic flow
Long time existence of Minimizing Movement solutions of Calabi flow
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