Construction of embedded area-minimizing surfaces via a topological type induction scheme

In this paper we present a new scheme of constructing embedded area-minimizing surfaces in 3-manifolds based on the classic parametric approach, and apply it to resolve the problems of existence of embedded area-minimizing surfaces with free boundary in a given boundary homology class or in a given boundary homotopy class. As a consequence, we establish a general existence result on embedded minimal surfaces in a 3-dimensional manifold with boundary. We also develop existence and immersion theories of least area surfaces with a xed topological type, which serve as a basis for our new scheme. There are two main approaches to construction (existence) of surfaces in R3 or a general 3-dimensional Riemannian manifold which minimize area in a suitable lass of surfaces. One is the parametric approach, i.e. the approach of parametrized surfaces or mappings; the other is the geometric measure theory approach, i.e. the approach of rectiable currents. The classical Plateau problem of nding least area surfaces of a given topological type (e.g. least area discs) with a given Jordan curve boundary is the best-known example of the parametric approach. (We adopt the convention that least area" means area-minimizing under a topological type restriction.) However, the least area surfaces surfaces of a given topological type (e.g. least area discs) with a given Jordan curve boundary is the best-known example of the parametric approach. (We adopt the convention that least area" means area-minimizing under a topological type restriction.) However, the least area surfaces obtained via this approach are generally not embedded. Furthermore, it leaves until now the question of absolutely minimizing area, i.e. minimizing area among surfaces of all topological types unanwsered. In contrary, a main success of the geometric measure theory approach is to produce embedded absolutely area-minimizing surfaces in several important problems. There are also some restrictions of this approach, e.g. in general it is less adpated than the parametric approach to problems in the mapping set-up, such as problems involving homotopy groups. We note that there are variants of the above general picture. In [MY1, MY2], the authors established the embedded character of least area discs with a given Jordan curve boundary which lies in a mean convex boundary (of a 3-manifold), and the embedded character of least area discs with free boundary in a mean convex boundary. These results do not address the issu