A Survey on Classical Minimal Surface Theory (draft)

Preface.
We present in this monograph a survey of recent spectacular successes in classical minimal surface theory. Many of these successes were covered in
our survey article \The classical theory of minimal surfaces" that appeared in the Bulletin of the AMS [125]. The focus of our BAMS article was to describe the work that led up to the theorem that the plane, the helicoid, the catenoid and the one-parameter family of Riemann minimal examples
are the only properly embedded, minimal planar domains in R3. The proof of this result depends primarily on work of Colding and Minicozzi [37, 34],
Collin [38], Lopez and Ros [107], Meeks, Perez and Ros [136] and Meeks and Rosenberg [148]. Partly due to limitations of length of our BAMS article,
we omitted the discussion of many other important recent advances in the theory. The central topics missing in [125] and which appear here include
the following ones:
1. The topological classication of minimal surfaces in R3 (Frohman and Meeks [63]).
2. The uniqueness of Scherk's singly-periodic minimal surfaces (Meeks and Wolf [155]).
3. The Calabi-Yau problem for minimal surfaces based on work by Nadirashvili [166] and Ferrer, Martn and Meeks [56].
4. Colding-Minicozzi theory for minimal surfaces of nite genus [24].
5. The asymptotic behavior of minimal annular ends with innite total cur-
vature (Meeks and Perez [126]).
6. The local removable singularity theorem for minimal laminations and its applications: quadratic decay of curvature theorem, dynamics theorem
and the local picture theorem on the scale of topology (Meeks, Perez and Ros [134]).

Besides the above items, every topic that is in [125] appears here as well, with small modications and additions. Another purpose of this monograph
is to provide a more complete reference for the general reader of our BAMS article where he/she can nd further discussion on related topics covered
in [125], as well as the proofs of some of the results stated there.

- William H. Meeks, III