Steady States for One Dimensional Conformal Metric Flows
  We define two conformal structures on $S^1$ which give rise to a different view of the affine curvature flow and a new curvature flow, the ``$Q$-curvature flow". The steady state of these flows are studied. More specifically, we prove four sharp inequalities, which state the existences of the corresponding extremal metrics. 
Yilong Ni, Meijun Zhu