Area minimizing surfaces in flat tori
Antonio Ros Universidad de Granada
A surface in a complete -manifold is area-minimizing mod if it has least area among all surfaces, orientable or nonorientable in the same homology class. These surfaces present a rich and interesting geometry, even in flat or positively curved -manifolds. For instance, if is flat, Fischer-Colbrie and Schoen, Do Carmo and Peng, and Pogorelov proved that complete two-sided stable minimal surfaces are flat, but Ross proved that some nonorientable quotient of the classic Schwarz P and D surfaces are estable, and we proved that an area minimizing surface in is either planar or a quotient of the Helicoid. We will review some results about this problem and we will prove that area minimizing surfaces in flat -tori are planar.