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.
| Original language | English |
|---|---|
| Pages (from-to) | 577-597 |
| Number of pages | 21 |
| Journal | Geometric and Functional Analysis |
| Volume | 15 |
| Issue number | 3 |
| DOIs | |
| State | Published - Jun 2005 |
Bibliographical note
Funding Information:V.B. partially supported by DFG-Forschergruppe ‘Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis’. C.C. supported by NSF grant DMS 02-02536 and the MSRI. S.V.I. supported by grants CRDF RM1-2381-ST-02, RFBR 02-01-00090, and NS-1914.2003.1. M.G.K. supported by the Israel Science Foundation (grants nos. 620/00-10.0 and 84/03).
Funding
V.B. partially supported by DFG-Forschergruppe ‘Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis’. C.C. supported by NSF grant DMS 02-02536 and the MSRI. S.V.I. supported by grants CRDF RM1-2381-ST-02, RFBR 02-01-00090, and NS-1914.2003.1. M.G.K. supported by the Israel Science Foundation (grants nos. 620/00-10.0 and 84/03).
| Funders | Funder number |
|---|---|
| DFG-Forschergruppe | |
| MSRI | CRDF RM1-2381-ST-02 |
| National Science Foundation | DMS 02-02536 |
| Russian Foundation for Basic Research | NS-1914.2003.1, 02-01-00090 |
| Israel Science Foundation | 620/00-10.0, 84/03 |