Yng-Ing Lee

Hamiltonian stationary cones and self-similar solutions in higher dimension

In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimension. We construct new higher dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfson cones. Hamiltonian stationary self-shrinkers and self-expanders that are asymptotic to these Hamiltonian stationary cones are also constructed. They can also be glued together to produce eternal solutions of the Brakke flow without mass loss. Finally, we show the same conclusion holds for those Lagrangian self-similar examples recently found by Joyce, Tsui and the first author in [JLT].

Mean curvature flow of the graphs of maps between compact manifolds

We make several improvements on the results of M.-T. Wang (2002) and his joint paper with M.-P. Tsui (2004) concerning the long time existence and convergence for solutions of mean curvature flow in higher codimension. Both the curvature condition and lower bound of ∗Ω are weakened. New applications are also obtained.

The stability of self-shrinkers of mean curvature flow in higher co-dimension

In this paper, we generalize Colding and Minicozzis work \cite{CM} on the stability of self-shrinkers in the hypersurface case to higher co-dimensional cases. The first and second variation formulae of the

On a generalized 1-harmonic equation and the inverse mean curvature flow

F

Spacelike Spherically Symmetric CMC Foliation in the Extended Schwarzschild Spacetime

-functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in \cite{An} are unstable.

Stability of the minimal surface system and convexity of area functional

We introduce and study generalized 1-harmonic equations (1.1). Using some ideas and techniques in studying 1-harmonic functions from Wei (2007) [1], and in studying nonhomogeneous 1-harmonic functions on a cocompact set from Wei (2008) [2, (9.1)], we find an analytic quantity w in the generalized 1-harmonic equations (1.1) on a domain in a Riemannian n-manifold that affects the behavior of weak solutions of (1.1), and establish its link with the geometry of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for p-subharmonic functions of constant p-tension field, p≥n, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.

A Bridge Principle for Harmonic Maps

We first summarize the characterization of smooth spacelike spherically symmetric constant mean curvature (SS-CMC) hypersurfaces in the Schwarzschild spacetime and Kruskal extension. Then use the characterization to prove special SS-CMC foliation property, and verify part of the conjecture by Malec and Ó Murchadha (Phys Rev D (3) 68:124019, 2003).

Self-similar solutions and translating solutions

Abstract. We study the convexity of the area functional for the graphs of maps with respect to the singular values of their differentials. Suppose that f is a solution to the Dirichlet problem for the minimal surface system and the area functional is convex at f . Then the graph of f is stable. New criteria for the stability of minimal graphs in any co-dimension are derived in the paper by this method. Our results in particular generalize the co-dimension one case, and improve the condition in the 2003 paper of the first author and M.-T. Wang from | ∧2 df | ≤ 1 p−1 to | ∧2 df | ≤ 1 √ p−1 , where p is an upper bound of the rank of df , and the condition in the 2008 paper of the first author and M.-T. Wang from √ det(I + (df)T df) ≤ 43 40 to √ det(I + (df)T df) ≤ 2.

A Bridge Principle for Harmonic Diffeomorphisms between Surfaces

We prove a bridge principle for harmonic maps between generalmanifolds.

The closed embedded minimal surfaces in an almost positively curved three manifold

In this note, I provide some detailed computation of constructing translating solutions from self-similar solutions for Lagrangian mean curvature flow discussed in [6] and explore the related geometric meanings. This method works for all mean curvature flows and has great potential to find other new translating solutions.