Conformally invariant skew curves for the total skew curvature on surfaces of $$\mathbb {R}^3$$

Abstract In differential geometry, curvature-based functionals, such as the total Gaussian curvature, the Willmore energy, and the total geodesic torsion, play a central role in both theoretical investigations and practical applications. In this paper, we study geometric properties of the extremal curves for the next functional $$\begin{aligned} \mathfrak {F}:=\int {\sqrt{H^2-K}}ds=\int {\frac{(k_1-k_2)}{2}}ds, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>∫</mml:mo> <mml:msqrt> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:msqrt> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mo>∫</mml:mo> <mml:mfrac> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where ds is the arc element on S and $$k_1,k_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>k</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> are the principal curvatures. First, we establish that a necessary and sufficient condition for a surface to be a Dupin cyclide is that its lines of curvature and the extremal curves of functional $$\mathfrak {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> intersect at a constant angle. Secondly, we demonstrate that the extremal curves of the functional $$\mathfrak {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> are invariant under inversion. Finally, we show that the determination of functional extremal curves of $$\mathfrak {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> for any cone, general cylinder, and surfaces of revolution can be reduced to quadratures.