Extremal Length of Topologically Essential Paths in Projective Plane
In the above examples, the extremal which maximized the ratio and gave the extremal length corresponded to a flat metric. In other words, when the Euclidean Riemannian metric of the corresponding planar domain is scaled by, the resulting metric is flat. In the case of the rectangle, this was just the original metric, but for the annulus, the extremal metric identified is the metric of a cylinder. We now discuss an example where an extremal metric is not flat. The projective plane with the spherical metric is obtained by identifying antipodal points on the unit sphere in with its Riemannian spherical metric. In other words, this is the quotient of the sphere by the map . Let denote the set of closed curves in this projective plane that are not null-homotopic. (Each curve in is obtained by projecting a curve on the sphere from a point to its antipode.) Then the spherical metric is extremal for this curve family. (The definition of extremal length readily extends to Riemannian surfaces.) Thus, the extremal length is .
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