Homologically versus homotopically area minimizing surfaces in 3-manifolds

Abstract.

We construct a Riemannian metric on the 3-torus $T^3$ such that no closed surface minimizing area in its homology class is incompressible, i.e., each such surface is of genus greater than one. In particular, for such a Riemannian metric, the homotopically area minimizing 2-tori constructed in [5] do not minimize area in their homology classes. The example is easily generalized to arbitrary 3-manifolds. The constructed Riemannian metric can be chosen to be conformally equivalent to any arbitrary given one.