Willmore Function on Curvatures of The Curve-Surface Pair Under Mobius Transformation

abstract: We find a geometric invariant of the curve-surface pairs on Willmore functions with the mean and Gauss curvatures. Similar to the work in [5,19], in this work, we define Willmore functions on curve– surface pair and give new characterizations about Willmore functions with necessary and sufficient condition with strip theory in Euclidean 3-space for the first time. In this paper Willmore function on curvatures of the curve-surface pair under Möbiüs transformation is provided invariant.


Introduction
Möbiüs differential geometry is a classical subject that was extensively developed in the nineteenth and early twntieth centuries, culminating with the publication of Blascke's Vorlesungenüber Differentialgeometrie III:Differentialgeometrie der Kreise und Kugeln [3] in 1929.
In 3-dimensional Euclidean Space, a regular curve is described by its curvatures k 1 and k 2 and also a curve-surface pair is described by its curvatures k n , k g and t r . The relations between the curvatures of a curve-surface pair and the curvatures of the curve can be seen in many differential books and papers. Möbius transformations are the automorphisms of the extended complex plane C ∞ : C ∪ {∞} , that is the metamorphic bijections [24]. M : C ∞ → C ∞ . A möbius transformation M has the form The set of all Möbius transformations is a group under composition. The Möbius transformation with c = 0 form the subgroup of similarities. such transformations have the form The transformation J(Z) = 1 Z is called an inversion. Every Möbius transformation M of the form (2) is a composition of finitely many similarities and inversions [5,9].
In this paper we provide that Willmore function on curve-surface of the curve-surface pair under Möbiüs transformation is invariant.

The Curve-Surface pairs
In this section, we give some basic definitions from differential geometry and curve-surface pairs Definition 2.1. Let M and α be a surface in E 3 and a curve in M ⊂ E 3 . We define a surface element of M is the part of a tangent plane at the neighbour of the point. The locus of these surface element along the curve α is called a curve-surface pair and is shown as (α, M ).
be the curve and curve-surface pair's vector fields . The curve-surface pair's tangent vector field, normal vector field and binormal vector field is given by We know that a curve α has two curvatures κ and τ . A curve has a strip and a strip has three curvatures k n , k g and t r . Let k n , k g and t r be the -b, c, a [4,6]. From (3)

Willmore Function on Curvatures of the Curve-Surface Pair Under Möbiüs
The most outstanding problem in Möbiüs differential geometry is the Willmore Conjecture [5,19]. This conjecture is most naturally formulated in terms of surfaces in R 3 rather than S 3 . Let f : M 2 → R 3 be a compact surface immersed in R 3 [5,19]. Let κ and τ denote principal curvatures of f, H = (κ + τ )/2 and K = κτ denote the mean and Gauss curvatures of f, respectively [5,19]. In 1965 Willmore [5,19] proposed the study of the functional. So it can be written τ (f, M 2 ) on the curve surface pair where dA is the area form on (f, M 2 ) induced by the immersion f. Several authors includinf Fubini [21], Thomsen[22] and White [23] have proven that the two form H 2 − KdA is Möbiüs invariant. It so-called Willmore functional. Now it is: is Möbiüs invariant on curve-surface pair. Thus the Gauss-Bonnet Theorem states that , where χ(f, M 2 ) is the Euler characteristic of (f, M 2 ),we have and then τ (f, M 2 ) = W (f, M 2 ) + 2πχ(f, M 2 ) is also Möbiüs invariant. Note that √ b 2 + c 2 + a + b´c−bcb 2 +c 2 2 2 − b 2 + c 2 a + b´c − bcb 2 + c 2 = 1 4 b 2 + c 2 − a + b´c − bcb 2 + c 2 2 so the Willmore funtional on curve-surface pair has the property that its integrand is non-negative, it vanishes at umbilic point where √ b 2 + c 2 = a + b´c−bcb 2 +c 2 .

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.