Relation To Pu's Inequality
The case of simply-connected fillings is equivalent to Pu's inequality for the real projective plane RP2. Recently the case of genus-1 fillings was settled affirmatively, as well (see Bangert et al). Namely, it turns out that one can exploit a half-century old formula by J. Hersch from integral geometry. Namely, consider the family of figure-8 loops on a football, with the self-intersection point at the equator (see figure at the beginning of the article). Hersch's formula expresses the area of a metric in the conformal class of the football, as an average of the energies of the figure-8 loops from the family. An application of Hersch's formula to the hyperelliptic quotient of the Riemann surface proves the filling area conjecture in this case.
Read more about this topic: Filling Area Conjecture
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