Date of Degree

9-2015

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor(s)

Zheng Huang

Subject Categories

Mathematics

Keywords

Bernstein type results; graphical self-shrinker; high codimension

Abstract

A self-shrinker characterizes the type I singularity of the mean curvature flow. In this thesis we concern about some Bernstein type results of graphical self-shrinkers with high codimension in Euclidean space.

There are two main tools in our work. The first one is structure equations of graphical self-shriners in terms of parallel forms (Theorem 2.3.6). This is motivated by M.T.Wang's work ([Wan02]) on graphical mean curvature flows with arbitrary codimension in product manifolds. The second one is an integration technique (Lemma 2.4.5) based on the fact that every graphical graphical self-shrinker has the polynomial volume growth (Corollary 2.4.4). Because of it the derivations of all results are independent of the maximal principle of elliptic equations.

A general process we attack the rigidity of graphical self-shrinkers mainly consists of the following two steps:

a) derive the structure equation of graphical self-shrinkers under certain geometric conditions;

b) apply the integration technique to establish the minimality of the graphical self-shrinkers.

The rigidity follows from the well-known fact that every minimal, complete, smooth self-shrinker is a plane through 0 (Theorem 2.2.4). An example is given in section 2.5 to illustrate the above process.

This thesis is organized as follows. Chapter 1 is devoted to our main results and some geometric background. We also discuss some Bernstein type results on minimal submanifolds in Euclidean space and the long time existence of graphical mean curvature flows in product manifolds.

In Chapter 2, we construct main tools to explore graphical self-shrinkers. They includes structure equations of self-shrinkers in term of parallel forms, the integration technique and other technique results. In Chapter 3 we discuss the rigidity of graphical self-shrinker surfaces in $R^4$ with codimension two. In Chapter 4 we investigate the rigidity of graphical self-shrinker's with arbitrary codimension under certain geometric conditions. In Chapter 5, we give a new proof of the fact that a Lagrangian self-shrinker of zero Maslov class is a plane through 0 if its Lagrangian angle has a upper bound or lower bound. Here we use the structure equation of Lagrangian angles (Lemma 5.1.3.)

Included in

Mathematics Commons

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.