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In this paper, we obtain the uniqueness of general codimension Dirichlet problem for minimal surface system in restricted classes. The condition is in terms of singular values and in particular covers the classical hypersurface case and earlier results in higher codimension. To prove the uniqueness result, a natural way is to consider the geodesic homotopy of two solutions. However, the singular values for linear combination of maps are not clear. We apply majorization techniques from convex optimisation to overcome the difficulties.

We generalize Colding and Minicozzi’s work (2012) on the stability of hypersurface self-shrinkers to higher co-dimension. The first and second variation formulae of the F F -functional are derived and an equivalent condition to the stability in general co-dimension is found. We also prove that R n \mathbb R^n is the only stable product self-shrinker and show that the closed embedded Lagrangian self-shrinkers constructed by Anciaux are unstable.

We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)] (2008), 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, $p \ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow., 14 pages, to appear in Journal of Geometry and Physics

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $\Sigma$ is the graph of a (strictly) distance-decreasing map, then $\Sigma$ is (strictly) stable. It is known that a minimal graph of codimension one is stable without assuming the distance-decreasing condition. We give another criterion for the stability in terms of the two-Jacobians of the map which in particular covers the codimension one case. All theorems are proved in the more general setting for minimal maps between Riemannian manifolds., Comment: 13 pages

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 \'{O} Murchadha in their 2003 paper., Comment: 28 pages, 3 figures. This paper is submitted and accepted in Annales Henri Poincar\'e

We show a bridge principle for harmonic diffeomorphisms between closed surfaces with higher genus.

The compactness theorem of the closed embedded minimal surfaces of fixed genus in a 3-dimensional closed Riemannian manifoldN is proved, providedN is simply connected and the nonpositive value set of Ricci curvature is sufficiently concentrated within finite balls and the minimal surfaces are uniformly away from these balls.

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.

Hamiltonian stationary Lagrangians are Lagrangian submanifolds that are critical points of the volume functional under Hamiltonian deformations. They can be considered as a generalization of special Lagrangians or Lagrangian and minimal submanifolds. Joyce, Schoen and the author show that given any compact rigid Hamiltonian stationary Lagrangian in $\C^n$, one can always find a family of Hamiltonian stationary Lagrangians of the same type in any compact symplectic manifolds with a compatible metric. The advantage of this result is that it holds in very general classes. But the disadvantage is that we do not know where these examples locate and examples in this family might be far apart. In this paper, we derive a local condition on Kahler manifolds which ensures the existence of one family of Hamiltonian stationary Lagrangian tori near a point with given frame satisfying the criterion. Butscher and Corvino ever proposed a condition in n=2. But our condition appears to be different from theirs. The condition derived in this paper not only works for any dimension, but also for the Clifford torus case which is not covered by their condition., 21 pages

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regularity of Lagrangian mean curvature flow. Given two transverse Lagrangian planes R^n in C^n with sum of characteristic angles less than pi, we show there exists a Lagrangian self-expander asymptotic to this pair of planes. The Maslov class of these self-expanders is zero. Thus they can serve as local models for surgeries on Lagrangian mean curvature flow. Families of self-shrinkers and self-expanders with different topologies are also constructed. This paper generalizes the work of Anciaux, Joyce, Lawlor, and Lee and Wang., Comment: 33 pages. (v2) minor corrections. to appear in Journal of Differential Geometry

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth oriented Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a weak formulation of the mean curvature flow. For any coprime pair $(p,q)$ other than $(2,1)$, we construct such a solution that resolves any single Schoen-Wolfson cone $C_{p,q}$. This thus provides an evidence to Schoen-Wolfson's conjecture that the $(2,1)$ cone is the only area-minimizing cone. Higher dimensional generalizations are also obtained., 19 pages

Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian deformations. It is called Hamiltonian stable if in addition the second variation of volume under Hamiltonian deformations is nonnegative. Our main result is that if L is a compact, Hamiltonian stationary Lagrangian in C^n satisfying the extra condition of being Hamiltonian rigid, then for any M,w,g as above there exist compact Hamiltonian stationary Lagrangians L' in M contained in a small ball about some p in M and locally modelled on tL for small t>0, identifying M near p with C^n near 0. If L is Hamiltonian stable, we can take L' to be Hamiltonian stable. Applying this to known examples L in C^n shows that there exist families of Hamiltonian stable, Hamiltonian stationary Lagrangians diffeomorphic to T^n, and to (S^1 x S^{n-1})/{1,-1}, and with other topologies, in every compact symplectic 2n-manifold (M,w) with compatible metric g., 25 pages. (v2) minor changes, final version to appear in American Journal of Mathematics

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]., 18 pages

An $n$ dimensional minimal submanifold $\Sigma$ of $\R^{n+m}$ is called non-parametric if $\Sigma$ can be represented as the graph of a vector-valued function $f:D\subset \R^n \mapsto \R^m$. This note provides a sufficient condition for the stability of such $\Sigma$ in terms of the norm of the differential $df$., Comment: submitted

Let $(N,g_{0})$ be a Kahler-Einstein surface with the first Chern class negative and assume that there exists a branched Lagrangian minimal surfaces with respect to the metric $g_{0}$. We show that when the Kahler-Einstein metric is changed in the same component (i.e. the complex structure is changed), the Lagrangian minimal surface can be deformed accordingly. To get the result, we first obtain a theorem on the deformation of the branched minimal surfaces in a complete Riemannian $n$-manifold and also generalize a result of J. Chen and G. Tian on the limit of adjunction numbers., LaTeX, 29 pages, to appear in JDG

This thesis consists of four chapters and an appendix. The first chapter is dedicated to the fundamental ideas of the theory of Ricci flow, which shows how our works are connected to the whole story. In the second chapter, we construct a solution of Ricci flow on a rotationally symmetric manifold such that it remains a complete manifold at the maximal time. We also derive a noncollapsing property for certain ancient solutions near their maximal times. Both of these two results are related to the regularity of limits of solutions. In the third chapter, we show that a first order Shi-type estimate holds for Ricci tensor on manifolds which satisfy the weak Bianchi inequality. The last chapter is concerned with expanding gradient Ricci solitons. There we discuss the classification problem and show that every tangent cone at infinity of an expanding soliton with fast-than-quadratic-decay curvature must be ℝⁿ., Cette these se compose de quatre chapîtres et une annexe. Le premier chapître est consacre à des idées fondamentales de la theorie du flot de Ricci, qui montre comment nos travaux sont reliés a l'histoire entière. Dans le deuxième chapître, nous construisons une solution du flot de Ricci sur une variete a symétrie de rotation de telle sorte qu'il reste un collecteur complet a l'heure maximale. Nous dérivons également le non-effondrement pour certaines solutions anciennes à proximité de leur temps maximal. Chacun de ces deux resultats sont liés à la régularité des limites des solutions. Dans le troisième chapître, nous montrons qu'une estimation de type Shi d'ordre un est valable pour tenseur de Ricci sur des variétés qui satisfont l'inégalité Bianchi faibles. Le dernier chapître s'interesse aux gradient solitons de Ricci qui sont en expansion. Nous discutons du problème de classification et montrons que chaque cône tangent à l'infini d'un soliton expansion à "fast-than-quadratic-decay" courbure doit être ℝⁿ.