The Conformal Arclength Functional

Abstract A generic smooth curve of the three dimensional Möbius space admits a natural parameter (or conformal arclength). Integrating the conformal arclength we get a conformally invariant variational problem. In the present paper we study the extremal curves of this variational problem. We derive the associated Euler‐Lagrange equations and we get the natural equations of the extremal curves. The natural equations are integrated and the explicit solutions are given.

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