Dissertations, Theses, and Capstone Projects

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Zheng Huang

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Bernstein type results; graphical self-shrinker; high codimension


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.)

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