Mass transport generated by a flow of Gauss maps

Let A⊂Rd, d?2, be a compact convex set and let be a probability measure on A equivalent to the restriction of Lebesgue measure. Let be a probability measure on equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν=μ○T⁻¹ and T=φ⋅n, where is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth φ the level sets of φ are governed by the Gauss curvature flow , where K is the Gauss curvature. As a by-product one can reprove the existence of weak solutions to the classical Gauss curvature flow starting from a convex hypersurface.     

On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S⁴ and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to P ³.Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.     

Surfaces with harmonic inverse mean curvature in space forms

We define surfaces with harmonic inverse mean curvature in space forms and generalize a theorem due to Lawson by which surfaces of constant mean curvature in one space form isometrically correspond to those in another. We also obtain an immersion formula, which gives a deformation family for these surfaces.

     

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.     

Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map

For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative on the Grassmann bundle ^ := G_m(T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rⁿ this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ G_m(T_pN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.     

Harmonic Gauss maps of submanifolds of arbitrary codimension of the Euclidean space and sphere and some applications

It is proven results about existence and nonexistence of unit normal sections of submanifolds of the Euclidean space and sphere, which associated Gauss maps, are harmonic. Some applications to constant mean curvature hypersurfaces of the sphere and to isoparametric submanifolds are obtained too.     

The nature of singularities in mean curvature flow of mean-convex sets

This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact mean-convex region of or of an -dimensional riemannian manifold. If , the moving boundary is shown to be very nearly convex in a spacetime neighborhood of any singularity. In particular, the tangent flows at singular points are all shrinking spheres or shrinking cylinders. If , the same results are shown up to the first time that singularities occur.

     

Singularities of symplectic and Lagrangian mean curvature flows

In this paper, we study the singularities of the mean curvature ?ow from a symplectic surface or from a Lagrangian surface in a K?hler-Einstein surface. We prove that the blow-up ?ow ∑s∞ at a singular point(X₀, T₀) of a symplectic mean curvature ?ow Σ_t or of a Lagrangian mean curvature ?ow Σ_t is a nontrivial minimal surface in ?4, if ∑-∞∞ is connected.     

On an extension of the H k mean curvature flow

In this note,we generalize an extension theorem in [Le-Sesum] and [Xu-Ye-Zhao] of the mean curvature flow to the Hk mean curvature flow under some extra conditions.The main difficulty in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the Hk mean curvature flow,and to do a suitable Moser iteration process.These two problems are overcome by imposing some extra conditions which may be weakened or removed in our forthcoming paper.On the other hand,we derive some estimates for the generalized mean curvature flow,which have their own interesting.     

The value distribution of the hyperbolic Gauss map

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in , and prove that it cannot omit more than four points unless the surface is a horosphere.

     

The size of the singular set in mean curvature flow of mean-convex sets

We prove that when a compact mean-convex subset of (or of an -dimensional riemannian manifold) moves by mean-curvature, the spacetime singular set has parabolic hausdorff dimension at most . Examples show that this is optimal. We also show that, as , the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most . If , the convergence is everywhere smooth and hence after some time , the moving surface has no singularities

     

Self-similar solutions to the hyperbolic mean curvature flow

This article concerns the self-similar solutions to the hyperbolic mean curvature flow (HMCF) for plane curves, which is proposed by Kong, Liu, and Wang and relates to an earlier proposal for general flows by LeFloch and Smoczyk. We prove that all curves immersed in the plane which move in a self-similar manner under the HMCF are straight lines and circles. Moreover, it is found that a circle can either expand to a larger one and then converge to a point, or shrink directly and converge to a point, where the curvature approaches to infinity.     

Anisotropic inverse harmonic mean curvature flow

We study the evolution of convex hypersurfaces H（., t） with initial H（., 0） = 0H0 at a rate equal to H - f along its outer normal, where H is the inverse of harmonic mean curvature of H（., t）, H0 is a smooth, closed, and uniformly convex hypersurface. We find a θ^＊ 〉 0 and a sufficient condition about the anisotropic function f, such that if θ 〉 θ^＊, then H（.,t） remains uniformly convex and expands to infinity as t →∞ and its scaling, H（-, t）e^-nt, converges to a sphere. In addition, the convergence result is generalized to the fully nonlinear case in which the evolution rate is log H - log f instead of H - f.     

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.     

The volume preserving mean curvature flow near spheres

By means of a center manifold analysis we investigate the averaged mean curvature flow near spheres. In particular, we show that there exist global solutions to this flow starting from non-convex initial hypersurfaces.