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

The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks By Tobias H. Colding and William P. Minicozzi II* 0. Introduction This paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 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 . We show here that if the curvature of such a disk. | Annals of Mathematics The space of embedded minimal surfaces of fixed genus in a 3-manifold II Estimates off the axis for disks By Tobias H. Colding and William P. Minicozzi II Annals of Mathematics 160 2004 69 92 The space of embedded minimal surfaces of fixed genus in a 3-manifold II Multi-valued graphs in disks By Tobias H. Colding and William P. Minicozzi II 0. Introduction This paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed but arbitrary closed 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. We show here that if the curvature of such a disk becomes large at some point then it contains an almost flat multi-valued graph nearby that continues almost all the way to the boundary. This will be proved by showing the existence of small multi-valued graphs near points of large curvature and then using the extension result for multi-valued graphs proved in the first paper in this series. 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 like a helicoid. Recall that a double spiral staircase consists of two spiral staircases that spiral together around a common axis one inside the other. 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 it is a double spiral staircase we will first prove that it is built out of N-valued graphs where N is a fixed number. These N-valued graphs are like a single spiral staircase .