Deformations of surfaces preserving conformal or similarity invariants
  In this paper we study Moebius applicable surfaces, i.e., conformally immersed surfaces in Moebius 3-space which admit deformations preserving the Moebius metric.   We show new characterizations of Willmore surfaces, Bonnet surfaces and Harmonic inverse mean curvature surfaces in terms of Moebius or similarity invariants. 
Atsushi Fujioka and Jun-ichi Inoguchi