Volume 15, issue 3 (2011)

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Isosystolic genus three surfaces critical for slow metric variations

Stéphane Sabourau

Geometry & Topology 15 (2011) 1477–1508
Abstract

We show that the two piecewise flat surfaces with conical singularities conjectured by E Calabi as extremal surfaces for the isosystolic problem in genus $3$ are critical with respect to some metric variations. The proof relies on a new approach to study isosystolic extremal surfaces.

Keywords
systole, systolic inequality, extremal surface, flat surface with conical singularities, Calabi surface, calibration, Busemann function
Mathematical Subject Classification 2000
Primary: 53C23
Secondary: 53C20, 53C22, 53C38