TAILIEUCHUNG - Đề tài " The space of embedded minimal surfaces of fixed genus in a 3-manifold III; Planar domains "

Annals of Mathematics This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any fixed small scale. Although the focus of this paper, general planar domains, is more in line with [CM6], we will prove a result here (namely, Corollary III. | Annals of Mathematics The space of embedded minimal surfaces of fixed genus in a 3-manifold III Planar domains By Tobias H. Colding and William P. Minicozzi II Annals of Mathematics 160 2004 523 572 The space of embedded minimal surfaces of fixed genus in a 3-manifold III Planar domains By Tobias H. Colding and William P. Minicozzi II 0. Introduction This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 3-manifold. In CM3 - CM5 we describe the case where the surfaces are topologically disks on any fixed small scale. Although the focus of this paper general planar domains is more in line with CM6 we will prove a result here namely Corollary below which is needed in CM5 even for the case of disks. Roughly speaking there are two main themes in this paper. The first is that stability leads to improved curvature estimates. This allows us to find large graphical regions. These graphical regions lead to two possibilities Either they close up to form a graph Or a multi-valued graph forms. The second theme is that in certain important cases we can rule out the formation of multi-valued graphs . we can show that only the first possibility can arise. The techniques that we develop here apply both to general planar domains and to certain topological annuli in an embedded minimal disk the latter is used in CM5 . The current paper is third in the series since the techniques here are needed for our main results on disks. The above hopefully gives a rough idea of the present paper. To describe these results more precisely and explain in more detail why and how they are needed for our results on disks we will need to briefly outline those arguments. There are two local models for embedded minimal disks by an embedded disk we mean a smooth injective map from the closed unit ball in R2 The first author was partially supported by NSF Grant DMS 9803253 and an Alfred P. Sloan Research .

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