Isosystolic genus three surfaces critical for slow metric variations
Let \(M\) be a non-simply-connected closed surface. The systolic area of a Riemannian metric \(g\) on \(M\) is \(\sigma(g)=\mathrm{area}\,(g) / \mathrm{sys}\,(g)^2\), where \(\mathrm{sys}\,(g)\) is the infimum of the lengths of noncontractible loops on \(M\). The metric \(g\) is extremal if it realizes the minimum of the systolic area for a given topological surface. Very few such metrics are known at present, and none on the surfaces of genus higher than \(1\). E. Calabi described two highly symmetric piecewise flat metrics on surfaces of genus 3 and conjectured that one of them is extremal [\textit{E. Calabi}, in: Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger, Luminy, France, 12--18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 167--204 (1996; Zbl 0884.58029)]. The author provides some evidence in support of this conjecture by showing that both metrics are critical with respect to variations of certain type, called slow metric variations. The slowness condition is defined in terms of the differential of the Busemann function. It remains unknown whether this condition is satisfied for all metric variations.
scientific article; zbMATH DE number 5940697
scientific article