Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow

This paper has been withdrawn by the author for further modification.

In this work, we consider the area non-increasing map between manifolds with positive curvature. By exploring the strong maximum principle along the graphical mean curvature flow, we show that an area non-increasing map between positively curved manifolds is either homotopy trivial, Riemannian submersion, local isometry or isometric immersion. This confirm the speculation of Tsai-Tsui-Wang. We also use Brendle's sphere Theorem and mean curvature flow coupled with Ricci flow t...

Mean curvature flows of hypersurfaces have been extensively studied and there are various different approaches and many beautiful results. However, relatively little is known about mean curvature flows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then introduces mean curvature flows in general dimensions and co-dimensions. The related techniques in the so called "blow-up" analysis are also discussed. At the e...

We consider the graphical mean curvature flow of maps ${\bf f}:\mathbb{R}^m\to\mathbb{R}^n$, $m\ge 2$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken derived in their seminal paper [10]. In the case of uniformly area decreasing maps ${\bf f}:\mathbb{R}^m\to\mathbb{R}^2$, $m\ge 2$, we use this maximum principle t...

We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author.

We consider a gradient flow associated to the mean field equation on $(M,g)$ a compact riemanniann surface without boundary. We prove that this flow exists for all time. Moreover, letting $G$ be a group of isometry acting on $(M,g)$, we obtain the convergence of the flow to a solution of the mean field equation under suitable hypothesis on the orbits of points of $M$ under the action of $G$.

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.

Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a {\it quasi-conformal} map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichm\"uller theory, there is an 1-1 correspondence between the set of Beltrami differentials ...

We proved a Bernstein theorem of ancient solutions to mean curvature flow.

In this paper we study a neighborhood of generic singularities formed by mean curvature flow (MCF). We limit our consideration to the singularities modelled on $\mathbb{S}^3\times\mathbb{R}$ because, compared to the cases $\mathbb{S}^k\times \mathbb{R}^{l}$ with $l\geq 2$, the present case has the fewest possibilities to be considered. For various possibilities, we provide a detailed description for a small, but fixed, neighborhood of singularity, and prove that a small neigh...