In Riemannian geometry, we know that every 2d manifold is locally conformally flat thanks to local existence of isothermal coordinates; what could we say about surfaces for which these coordinates exist globally ?
Are they "Riemann surfaces of parabolic type" (-> uniformisation thm) ? The reason would be that they are conformally equivalent to the complex plane C.
And can they have negative curvature ?
Thank you very much.
I'm interested in PDE methods for isometric immersions of 2d negative curvature surfaces, that is, solving the system of 2 Gauss-Codazzi eqns. coupled to the Gauss curvature expression. Negative curvature makes such a system hyperbolic so that it reformulates as a Cauchy problem and one may find (usually weak) solns.
As Gauss-Codazzi simplifies notably when the metric is conformal, i was thinking about taking advantage of this for finding solns more easily. Also, curvature usually should decay rather quickly to zero.
I think that every oriented Riemannian 2-manifold M admits a global conformal map into the Riemann sphere. Just pass to the universal covering and patch the local isothermic systems together. However, this is one-to-one only on the universal covering.
A global conformal coordinate system means that M is an open subset of the complex plane. Whenever the image not the full plane, it is biholomorphic to the unit disk, i.e. it is hyperbolic, not parabolic.
Best regards
Jost
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