Filling area conjecture and ovalless real hyperelliptic surfaces

Details Filling area conjecture and ovalless real hyperelliptic surfaces Filling area conjecture and ovalless real hyperelliptic surfaces

2004-05-30

arxiv.org/abs/math/0405583v2

Mathematics

Differential Geometry

21 pages, 3 figures, to appear in Geometric and Functional Analysis (GAFA)

Scientific paper

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real surfaces of even positive genus. Here the singular points are Weierstrass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics.

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