Coordinate on the boundary of Riemann surface

Question

Let $\Sigma$ be a Riemann surface with boundary.

Question: Is there canonical way to parameterise the boundary components up to shift?

By shift I mean change of coordinate $\phi$ to $\phi + c$.

Comment

Such a coordinate would definitely exist if there was canonical metric of the boundary. Then such parameterisation is just geodesic.

There is a canonical metric on a Riemann surface (coming from universal covering). Unfortunately, this metric tends to infinity on the boundary.

Answer 1

There is indeed a canonical metric on $\Sigma$, namely, a complete hyperbolic metric such that the boundary is totally geodesic and such that the length of each compact boundary component equals $1$. This is easily proved by doubling $\Sigma$ across its boundary, and applying the empty boundary case. So yes, boundary components are canonically parameterized up to shift.