An inequality for length and volume in the complex projective plane

Abstract

In the 1950s, Carl Loewner proved an inequality relating the shortest closed geodesics on a 2-torus to its area. Many generalisations have been developed since, by Gromov and others. We show that the shortest closed geodesic on a minimal surface S for a generic metric on CP 2 is controlled by the total volume, even though the area of S is not. We exploit the Croke-Rotman inequality, Gromov's systolic inequalities, the Kronheimer-Mrowka proof of the Thom conjecture, and White's regularity results for area minimizers.