Math 286: Topics in Differential Geometry

Course Description

This class will attempt to survey some of the recent work of Colding and Minicozzi regarding compactness properties of embedded minimal surfaces in three-manifolds. We will begin with a review of some of some basic (smooth) minimal surface theory, with particular emphasis on the role of stability.

Using the tools developed in the first part of the course we will illustrate some standard compactness results. In particular we will prove the celebrated theorem of Choi and Schoen showing smooth compactness for minimal surfaces of fixed genus in a three-manifold with positive Ricci curvature.

In the last part of the course we will turn to the theory developed by Colding and Minicozzi in order to characterize the failure of smooth compactness for embedded minimal disks in $R^3$. We will sketch their results and then focus on the proof of one of key steps in their program, namely the one-sided curvature estimate. This estimate establishes an interior curvature bound for an embedded minimal disk under very weak assumptions and is of independent interest.

A copy of the syllabus may be found here

Class Notes

Here is a version of the notes. Comments and corrections will be greatly appreciated.

References

Below is a list of references

• H. I. Choi and R. Schoen. “The Space Of Minimal Embeddings Of a Surface Into a Three-dimensional Manifold Of Positive Ricci Curvature.” Invent. Math. 81, no. 3 (1985) : 387-394.
   
• T. H. Colding and W. P. Minicozzi II. Minimal Surfaces. New York: Courant Institute of Mathematical Sciences New York University, 1999.
   
• T. H. Colding and W. P. Minicozzi II. “The Space of Embedded Minimal Surfaces of Fixed Genus in a 3-manifold I; Estimates Off the Axis For Disks.” Ann. Of Math. (2) 160, no. 1 (2004) : 27-68.
   
• ---. “The Space Of Embedded Minimal Surfaces Of Fixed Genus In a 3-manifold II; Multi-valued Graphs In Disks.” Ann. Of Math. (2) 160, no. 1 (2004) : 69-92.
   
• ---. “The Space of Embedded Minimal Surfaces of Fixed Genus in a 3-manifold III; Planar Domains.” Ann. Of Math. (2) 160, no. 2 (2004) : 523-572.
   
• ---. “The Space of Embedded Minimal Surfaces of Fixed Genus in a 3-manifold IV; Locally Simply Connected.” Ann. Of Math. (2) 160, no. 2 (2004) : 573-615.
   
• ---. “The Space of Embedded Minimal Surfaces of Fixed Genus in a 3-manifold V; Fixed Genus.” Preprint. http://arxiv.org/abs/math/0509647
   
• ---. “Multivalued Minimal Graphs and Properness of Disks.” Int. Math. Res. Not. 21 (2002) : 1111-1127.