Calibration for semicircular hyperbolic geodesics ($\Phi (x,y)=\mathrm{arsinh}(x/y),|\nabla \Phi |=1/y$)

In the upper half-plane $y>0$, the hyperbolic length element is $ds/y$. The function $\Phi =\mathrm{arsinh}(x/y)$ has gradient $(1,-x/y)/\sqrt{x^2+y^2}$ and norm $1/y$. The Cauchy-Schwarz inequality gives $\int ds/y\ge \Phi (B)-\Phi (A)$ for any oriented path from $A$ to $B$. Equality holds along a semicircle $x^2+y^2=R^2$ traversed with increasing $x$, because its tangent is parallel to that gradient. This is a calibration proof of global minimality, beyond the stationary-path conclusion of the Beltrami identity. Translating $x$ gives the analogous proof for semicircles with other boundary centers.