Seminar in Real and Complex Geometry

Thursday, January 2, 2020, 16:00-18:00, Kaplun building, room 324

Mikhail Katz (Bar Ilan)

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.