Every closed surface of genus at least 18 is Loewner

Abstract

In this paper, we obtain an improved upper bound involving the systole and area for the volume entropy of a Riemannian surface. As a result, we show that every orientable and closed Riemannian surface of genus $g\geq 18$ satisfies Loewner's systolic ratio inequality. We also show that every closed orientable and nonpositively curved Riemannnian surface of genus $g\geq 11$ satisfies Loewner's systolic ratio inequality.