# Sebastian

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## Registered User

 Name Sebastian Member for 3 years Seen 4 hours ago Website Location Age
 Mar12 awarded ● Yearling Mar5 comment Branched Regular Cover over 4-times punctured sphereSorry, I was to vague about this: there exists holomorphic coordinates $y$ and $z$ on $\Sigma$ and $P^1$ centered at a branch point resp. branch image such that $f$ is given by $z=y^{g+1}.$ Then, the 4 sheets are exchanged by going around $y=0,$ and such a deck transformations $\varphi_{g+1}$ extends to all of $\Sigma$ holomorphicallysuch that $f\circ\varphi_{g+1}=f.$ This topic is covered in most Riemann surface books, for example the recent one by Donaldson. Mar5 comment Branched Regular Cover over 4-times punctured sphereThis is just a consequence of Riemann-Hurwitz and the construction of the Lawson surface $\xi_{g,1}:$ the $(g+1)-$symmetry is isometric and orientation preserving (both on the surface and in 3 space), hence it gives a holomorphic symmetry of the Riemann surface. By construction, it has 4 fix points, and all are of order (g+1). The quotient (as a Riemann surface) $\xi_{g,1}/\mathbb Z_{g+1}$ is the projective line. Mar5 comment Branched Regular Cover over 4-times punctured sphereThere is also a visual proof of this fact: Topological (the Riemann surface structure of $\Sigma_g$ is determined by the cross-ratio of the four branch images, in the case of the Lawson surface it is 2), the map is the same as the quotient by the $g+1$ symmetry of the Lawson minimal surface $\xi_{g,1}.$ But this symmetry is given by rotation by $2\pi/(g+1)$ around a great circle, and one sees that $f$ is regular. Dec19 asked Grauert’s theorem for infinite dimensional Frechet Lie groups Dec17 awarded ● Nice Answer Dec6 comment Higgs bundle and stable bundleA detailed answer to your question in the case of Riemann surfaces is given in Proposition 3.3 of Hitchin's "Self-duality equations on a Riemann surface". Dec6 answered Real analytic submanifolds of $\mathbb{R}^{n}$ Dec5 awarded ● Necromancer Dec5 answered Noteworthy achievements in and around 2010?