tailieunhanh - Đề tài " The space of embedded minimal surfaces of fixed genus in a 3-manifold I; Estimates off the axis for disks "

This paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 (with the flat metric). This study is undertaken here and completed in [CM6]. These local results are then applied in [CM7] where we describe the general structure of fixed genus surfaces in 3-manifolds. There are two local models for. | Annals of Mathematics The space of embedded minimal surfaces of fixed genus in a 3-manifold I Estimates off the axis for disks By Tobias H. Colding and William P. Minicozzi II Annals of Mathematics 160 2004 27 68 The space of embedded minimal surfaces of fixed genus in a 3-manifold I Estimates off the axis for disks By Tobias H. Colding and William P. Minicozzi II 0. Introduction This paper is the first in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 with the flat metric . This study is undertaken here and completed in CM6 . These local results are then applied in CM7 where we describe the general structure of fixed genus surfaces in 3-manifolds. 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 into R3 . One model is the plane or more generally a minimal graph the other is a piece of a helicoid. In the first four papers of this series we will show that every embedded minimal disk is either a graph of a function or is a double spiral staircase where each staircase is a multi-valued graph. This will be done by showing that if the curvature is large at some point and hence the surface is not a graph then it is a double spiral staircase. To prove that such a disk is a double spiral staircase we will first prove that it is built out of N-valued graphs where N is a fixed number. This is initiated here and will be completed in the second paper. The third and fourth papers of this series will deal with how the multi-valued graphs fit together and in particular prove regularity of the set of points of large curvature - the axis of the double spiral staircase. The reader may find it useful to also look at the survey CM8 and the .

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