On Deciding Whether a Surface is Parabolic or Hyperbolic [PDF]
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http://math.dartmouth.edu/~doyle/docs/milnor/milnor.pdf  


Peter G. Doyle  
An application of Rayleigh's method to the classical type problem for Riemann surfaces, which is really just the same random walk problem, only instead of walking around on an infinite graph the walker is diffusing around on a surface. The aim will be to convince you that if you want to figure out whether or not a random walker gets lost, you should use Rayleigh's method, which gives us a way of understanding and extending Milnor's investigations of the relation between growth of the Gaussian curvature and the conformal type of a surface. This is just one example of how Rayleigh's method can be used to determine the conformal type of a surface. A PostScript file is available from Doyle's site.  


Levels:  College, Research 
Languages:  English 
Resource Types:  Articles 
Math Topics:  Differential Geometry 
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