Nonexistence of Blow-up Flows for Symplectic and Lagrangian Mean Curvature Flows

In this paper we mainly study the relation between $|A|^2, |H|^2$ and cosα (α is the Kähler angle) of the blow up flow around the type II singularities of a symplectic mean curvature flow. We also study similar property of an almost calibrated Lagrangian mean curvature flow. We show the nonexistence of type II blow-up flows for a symplectic mean curvature flow satisfying $|A|^2≤λ|H|^2$ and $cosα≥δ>1-\frac{1}{2λ}(½≤α≤ 2)$, or for an almost calibrated Lagrangian mean curvature flow satisfying $|A|^2≤λ|H|^2$ and $cosθ≥δ>max\ {0,1-\frac{1}{λ}}(\frac34≤λ≤ 2)$, where θ is the Lagrangian angle.     

The Second Type Singularities of Symplectic and Lagrangian Mean Curvature Flows

This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a Kähler surface. The relation between the maximum of the Kähler angle and the maximum of |H|² on the limit flow is studied. The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.     

The Generalized Weierstrass System for Nonconstant Mean Curvature Surfaces and the Nonlinear Sigma Model

A study of the generalized Weierstrass system which can be used to induce mean curvature surfaces in three-dimensional Euclidean space is presented. A specific transformation is obtained which reduces the initial system to a two-dimensional Euclidean nonlinear sigma model. Some aspects of integrability are discussed, in particular, a connection with a version of the sinh-Gordon equation is established. Finally, some specific solutions are given and a systematic way of calculating multisoliton solutions is presented.     

On Einstein Manifolds of Positive Sectional Curvature

Let (M, g) be a compact oriented four-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M, g) is ₂, with its standard Fubini–Study metric.     

Hamiltonian Systems Derived from Constant Mean Curvature Surfaces in Hyperbolic Three-Space

We give a representation formula for surfaces of constant mean curvature in Euclidean or hyperbolic space, which is a natural generalization of Weierstrass-Enneper representation formula. The data (two functions) used in our formula should satisfy a certain system of differential equations. The system can be interpreted as an infinite dimensional Hamiltonian system. We investigate two finite-dimensional reductions in detail.     

一类推广的中曲率水平集的运动

研究了根据一个光滑曲面的主曲率函数的经典运动短时存在及唯一性,推广了Evans和Spruck的结果。     

Generalized Mean Curvature Flow in Carnot Groups

In this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4 Chen , Y. , Giga , Y. , Goto , S. ( 1991 ). Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations . J. Diff. Geom. 33 : 749 – 786 .[Crossref], [Web of Science ®] , [Google Scholar]] and [12 Evans , L. C. , Spruck , J. ( 1991 ). Motion of level sets by mean curvature. I . J. Diff. Geom. 33 ( 3 ): 635 – 681 . [Google Scholar]]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow.     

Curvature Estimates for Strongly Stable Domains on Surfaces with Constant Mean Curvature in Space Forms

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三维空间形式中常平均曲率曲面的稳定性

本文利用共形度量高斯曲率的估计研究了三维空间形式N^3(C）中具常平均曲率曲面的区域稳定性。     

Euclidean Complete Affine Surfaces with Constant Affine Mean Curvature

The purpose of this paper is to prove that alocally strongly convex, Euclidean complete surface with constantaffine mean curvature is also affine complete. Consequently weobtain a classification of locally strongly convex, Euclideancomplete surfaces with constant affine mean curvature.     

A Relation between Mean Curvature Flow Solitons and Minimal Submanifolds

We derive a one to one correspondence between conformal solitons of the mean curvature flow in an ambient space N and minimal submanifolds in a different ambient space where equals ℝ × N equipped with a warped product metric and show that a submanifold inN converges to a conformal soliton under the mean curvature flow in N if and only if its associatedsubmanifold in converges to a minimal submanifold under a rescaled mean curvature flow in . We then define a notion of stability for conformal solitons and obtain L^p estimates as well as pointwise estimates for the curvature of stable solitons.     

Surfaces of Constant Mean Curvature Bounded by Convex Curves

This paper proves that an embedded compact surface in the Euclidean space with constant mean curvature H bounded by a circle of radius 1 and included in a slab of width is a spherical cap. Also, we give partial answers to the problem when a surface with constant mean curvature and planar boundary lies in one of the halfspaces determined by the plane containing the boundary, exactly, when the surface is included in a slab.     

Surfaces of Constant Mean Curvature Bounded by Two Planar Curves

In this paper we study constant mean curvature compact surfaces with two Jordan curves in parallel planes as boundary and we investigate the point at which the surface inherits the symmetries of its boundary.     

On Einstein Hermitian Surfaces

We show that every compact Einstein Hermitian surface with constant conformal scalar curvature is a Kahler surface and that, in contrast to the compact case, there exits a noncompact Einstein Hermitian and non-Kahler surface with constant conformal scalar curvature.     

Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

The main result of this paper states that the traceless second fundamental tensor A⁰ of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, _M |A⁰|ⁿ dv_M < , in a simply-connected space form (c), with non-positive curvature c, goes to zero uniformly at infinity. Several corollaries of this result are considered: any such hypersurface has finite index and, in dimension 2, if H ² + c > 0, any such surface must be compact.     

A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R ⁿ . We establish such a maximum principle for parabolic surfaces in R³ with nonzero constant mean curvature and bounded Gaussian curvature.     

Mean Curvature Flow Closes Open Ends of Noncompact Surfaces of Rotation

We discuss the motion of noncompact axisymmetric hypersurfaces Γ _t evolved by mean curvature flow. Our study provides a class of hypersurfaces that share the same quenching time with the shrinking cylinder evolved by the flow and prove that they tend to a smooth hypersurface having no pinching neck and having closed ends at infinity of the axis of rotation as the quenching time is approached. Moreover, they are completely characterized by a condition on initial hypersurface.     

New Homogeneous Einstein Metrics of Negative Ricci Curvature

We construct new homogeneous Einstein spaces with negativeRicci curvature in two ways: First, we give a method for classifying andconstructing a class of rank one Einstein solvmanifolds whose derivedalgebras are two-step nilpotent. As an application, we describe anexplicit continuous family of ten-dimensional Einstein manifolds with atwo-dimensional parameter space, including a continuous subfamily ofmanifolds with negative sectional curvature. Secondly, we obtain newexamples of non-symmetric Einstein solvmanifolds by modifying thealgebraic structure of non-compact irreducible symmetric spaces of rankgreater than one, preserving the (constant) Ricci curvature.     

The Inverse Mean Curvature Flow in Rotationally Symmetric Spaces

In this paper, the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied. It is proved that the flow converges to a unique geodesic sphere, i.e., every principle curvature of the hypersurfaces converges to a same constant under the flow.     

单位球面中具平行单位平均曲率向量的子流形

设M是n-维闭黎曼流形,等距浸入(n+p)-维单位球空间S^n+p,具有平行的单位平均曲率向量。若S≤min{2n/3,2(n-1)^1/2},其中S是M的第二基本形式长度的平方,则M是S^n+p的一个(n+1)-维全测地子流形Sⁿ⁺¹中的超曲面。