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Filling area conjecture and ovalless real hyperelliptic surfaces. (English) Zbl 1082.53033
Authors’ abstract: 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.

MSC:
53C20 Global Riemannian geometry, including pinching
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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