Embedding a Riemann surface in the sphere - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:56:37Z http://mathoverflow.net/feeds/question/96789 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96789/embedding-a-riemann-surface-in-the-sphere Embedding a Riemann surface in the sphere uncooltoby 2012-05-12T18:23:15Z 2012-05-12T19:24:48Z <p>Assume we have a Riemann surface, the underlying topological surface of which is a sphere with (possibly uncountably many) points removed. Can we always conformally embed this Riemann surface in the Riemann sphere? If not, can someone suggest a counter example?</p> http://mathoverflow.net/questions/96789/embedding-a-riemann-surface-in-the-sphere/96792#96792 Answer by Igor Rivin for Embedding a Riemann surface in the sphere Igor Rivin 2012-05-12T19:07:09Z 2012-05-12T19:07:09Z <p>Unless I am confused, why not add the points back, map the resulting surface to the Riemann sphere conformally by unifomization, then remove the images of the offending points?</p> http://mathoverflow.net/questions/96789/embedding-a-riemann-surface-in-the-sphere/96795#96795 Answer by Misha for Embedding a Riemann surface in the sphere Misha 2012-05-12T19:24:48Z 2012-05-12T19:24:48Z <p>See e.g. <a href="http://www.math.tifr.res.in/~pablo/download/book/chp3.ps" rel="nofollow">here</a>: </p> <p>Theorem 3.2.7. Any planar connected Riemann surface is biholomorphic to an open subset of $S^2$.</p> <p>The proof is very straightforward: Exhaust a genus $0$ surface $S$ by relatively compact domains $D_n$ each of which necessarily has genus $0$. For each $D_n$ find a conformal embedding $f_n$ to $S^2$. Now, normalize the family of mappings $f_n$ to to send a point $x\in D_1$ to a fixed point $z\in {\mathbb C}$ and to have unit derivative (in a chart) at $x$. Then use normality of the family of maps $f_n$ to get the limit (for a subsequence). Lastly, check that the limit is injective. This is the same argument Caratheodory used in his proof of uniformization theorem. </p>