Geodesics for a Cone Metric - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:12:00Z http://mathoverflow.net/feeds/question/90847 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90847/geodesics-for-a-cone-metric Geodesics for a Cone Metric Doogies 2012-03-10T21:42:48Z 2012-03-11T01:51:52Z <p>Here is a question that I hope/suspect is elementary but cannot find a reference for. Suppose we are given a surface, S, with a conformally Euclidean metric, |f(z)||dz|, where f(z) is meromorphic. Puncture the surface S at all of the zeroes and poles of f(z) (at the "cone points"), and denote the resulting surface S*.</p> <p>Is it true that there exists a unique geodesic in every homotopy class of curves in S*?</p> <p>Thanks!</p> http://mathoverflow.net/questions/90847/geodesics-for-a-cone-metric/90854#90854 Answer by Robert Kucharczyk for Geodesics for a Cone Metric Robert Kucharczyk 2012-03-10T22:28:04Z 2012-03-10T22:36:15Z <p>The answer to the question <em>as stated</em> is "no": take for $S$ the plane $\mathbb{C}$ with the metric induced by the differential $z\mathrm{d}z$; then $S^{\ast }=\mathbb{C}\smallsetminus {0}$ and there is simply no geodesic from $1$ and $\mathrm{i}$. Or, if you want closed loops and free homotopy, there is no geodesic in the free homotopy class of the loop around $0$. Intuitively, it is clear what happens: if you draw a curve and try to pull it straight, you are forced to go through the cone point at the origin.</p> <p>If, however, you consider the surface $S$ itself and extend the definition of a geodesic so as to work on Euclidean metrics with cone points as well, the answer is "yes" <em>if $S$ is complete as a metric space</em>. The usual definition of geodesics in this context is one which works for all metric spaces: locally isometric maps from an interval. The statement you asked for is then a consequence of the Arzelà-Ascoli theorem.</p> <p>These things are treated in detail in the textbook <em>Quadratic differentials</em> by Kurt Strebel.</p> http://mathoverflow.net/questions/90847/geodesics-for-a-cone-metric/90858#90858 Answer by Misha for Geodesics for a Cone Metric Misha 2012-03-10T23:39:06Z 2012-03-11T01:51:52Z <p>Maybe you are asking for uniqueness, not existence? Then the answer is yes because your metric admits a locally CAT(0) completion in the answer below. This uniqueness result, originally, I think, due to Teichmuller, should be also in Strebel's book and is based on a Gauss-Bonnet calculation. </p>