This paper is devoted to investigating the Möbius differential geometry of a new class of surfaces, named the surfaces with closed Möbius form. The main theorem shows that a surface with closed Möbius form can be determined by a smooth function satisfying a 5th order partial differential equation presented in this paper. As an application of the main theorem, the isothermic surfaces with closed Möbius form are classified.

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The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entropy functional we give another proof of the monotonicity by establishing a pointwise formula. Moreover, we show that the increase are strict unless the metrics are Einstein.