Suppose that M is a complete minimal surface with finite total curvature. If M is embedded in $\mathbf{R}^3$, then we observe that M viewed from infinity looks like a plane passing through the origin. Using this observation, we can prove that M is a flat plane or M has at least two topological ends. In particular, if M is not a flat plane, then M is not simply connected. Of course, there are examples, such as Enneper’s surface, which are simply connected. Therefore, the embedding property is an essential part of this topological result. Is this idea clear ?
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$\begingroup$ It's not clear what your question is. This sketch of an argument that the plane is the only simply-connected, complete minimal surface of finite total curvature in $\mathrm{R}^3$ is the standard one. Are you asking whether your thoughts on the subject are clear? Your observations are well-known statements about minimal surfaces in $\mathbb{R}^3$, but they all require proofs, starting with 'viewed from infinity$\ldots$'. $\endgroup$– Robert BryantCommented Aug 30, 2014 at 10:53
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