A geometric inequality for length and volume in complex projective plane

 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 an area-minimizing 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.