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Mu-Tao Wang

Publications and source records attributed to Mu-Tao Wang.

86 records · Page 5Linked to original sources
arXiv2004

Stability and curvature estimates for minimal graphs with flat normal bundles

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minimal graphs of dimension no greater than six and any codimension is flat if the the normal bundle is flat and the density at infinity is finite. Such a Bernstein type theorem holds in any dimension if we assume additionally growth conditions on the volume element.

math.DG↗
arXiv2003

The Mean Curvature Flow Smoothes Lipschitz Submanifolds

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regularity theorem of Ecker and Huisken for Lipschitz hypersurfaces. In particular, any submanifold of the Euclidean space with a continuous induced metric can be smoothed out by the mean curvature flow. The smallness assumption is necessary in the higher codimension case in view of an example of Lawson and Osserman. The stationary phase of the mean curvature flow corresponds to minimal submanifolds. Our result thus generalizes Morrey's classical theorem on the smoothness of $C^1$ minimal submanifolds.

math.DG↗
arXiv2003

Gauss Maps of the Mean Curvature Flow

Let $F:Σ^n \times [0,T)\to \R^{n+m}$ be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps $γ:(Σ^n, g_t)\to G(n, m)$ form a harmonic heat flow with respect to the time-dependent induced metric $g_t$. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on $G(n,m)$ produces a subsolution of the nonlinear heat equation on $(Σ, g_t)$. We also show the condition that the image of the Gauss map lies in a totally geodesic submanifold of $G(n, m)$ is preserved by the mean curvature flow. Since the space of Lagrangian subspaces is totally geodesic in G(n,n), this gives an alternative proof that any Lagrangian submanifold remains Lagrangian along the mean curvature flow.

math.DG↗
arXiv2003

Remarks on a Class of Solutions to the Minimal Surface System

We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical results for the minimal surface equation can be generalized. We also conjecture the solvability of Dirichlet problems within the category of area-decreasing maps.

math.DG↗
arXiv2002

A Stability Criterion for Nonparametric Minimal Submanifolds

An $n$ dimensional minimal submanifold $Σ$ of $\R^{n+m}$ is called non-parametric if $Σ$ 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 $Σ$ in terms of the norm of the differential $df$.

math.DG↗
arXiv2002

Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.

math.DG↗
arXiv2002

Subsets of Grassmannians Preserved by Mean Curvature Flows

Let $M=Σ_1\times Σ_2$ be the product of two compact Riemannian manifolds of dimension $n\geq 2 $ and two, respectively. Let $Σ$ be the graph of a smooth map $f:Σ_1\mapsto Σ_2$, then $Σ$ is an $n$-dimensional submanifold of $M$. Let ${\frak G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of all $n$-dimensional subspaces of the tangent space of $M$. The Gauss map $γ:Σ\mapsto \frak{G} $ assigns to each point $x\in Σ$ the tangent space of $Σ$ at $x$. This article considers the mean curvature flow of $Σ$ in $M$. When $Σ_1$ and $Σ_2$ are of the same non-negative curvature, we show a sub-bundle $\frak{S}$ of the Grassmannian bundle is preserved along the flow, i.e. if the Gauss map of the initial submanifold $Σ$ lies in $\frak{S}$, then the Gauss map of $Σ_t$ at any later time $t$ remains in $\frak{S}$. We also show that under this initial condition, the mean curvature flow remains a graph, exists for all time and converges to the graph of a constant map at infinity . As an application, we show that if $f$ is any map from $S^n$ to $S^2$ and if at each point, the restriction of $df$ to any two dimensional subspace is area decreasing, then $f$ is homotopic to a constant map.

math.DG↗
arXiv2002

Mean curvature flow in higher codimension

The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity, global existence and convergence of the flow in various ambient spaces and codimensions were proved. We shall explain the results obtained as well as the techniques involved. The potential applications in symplectic topology and mirror symmetry will also be discussed.

math.DG↗
arXiv2002

On graphic Bernstein type results in higher codimension

Let Σbe a minimal submanifold of \R^{n+m} that can be represented as the graph of a smooth map f:\R^n-->\R^m. We apply a formula we derived in the study of mean curvature flow to obtain conditions under which Σmust be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.

math.DG↗
arXiv2001

Long-time Existence and Convergence of Graphic Mean Curvature Flow in Arbitrary Codimension

Let f:Σ_1 --> Σ_2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in the product of Σ_1 and Σ_2 by the mean curvature flow. Under suitable conditions on the curvature of Σ_1 and Σ_2 and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map f_t and f_t converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschtz initial data.

math.DG↗
arXiv2001

Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

Let Σbe a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold M. We consider the evolution of Σin the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms ω' and ω'' that determine different orientations andΣis symplectic with respect to both ω' and ω'', we prove the mean curvature flow of Σexists smoothly for all time. In the positive curvature case, the flow indeed converges at infinity.

math.DG↗
arXiv2001

Deforming Area Preserving Diffeomorphism of Surfaces by Mean Curvature Flow

Let f:Σ_1 --> Σ_2 be an area preserving diffeomorphism between compact Riemann surfaces of constant curvature. The graph of f can be viewed as a Lagrangian submanifold in Σ_1\times Σ_2. This article discusses a canonical way to deform f along area preserving diffeomorphisms. This deformation process is realized through the mean curvature flow of the graph of f in Σ_1\times Σ_2. It is proved that the flow exists for all time and the map converges to a canonical map. In particular, this gives a new proof of the classical topological results that O(3) is a deformation retract of the diffeomorphism group of S^2 and the mapping class group of a Riemman surface of positive genus is a deformation retract of the diffeomorphism group .

math.DG↗