关于一类极小曲面系统解的注记

关于欧氏四维空间中完备极小曲面的高斯像的注记

Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space

系统研究欧氏四维空间中完备极小曲面的高斯映射像，给出例外值最大数量的几何解释，并为拉格朗日曲面及不可定向曲面提供最优结果。

关于收缩梯度Ricci孤子中极小超曲面的注记

Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

本文证明在标量曲率满足R≥(n-1)λ的收缩梯度Ricci孤子中，任何紧致双侧光滑稳定极小超曲面的第二基本形式和法向Ricci曲率必须为零，且面积极小超曲面的存在会导致流形分裂。

A remark on the Heat Equation and minimal Morse Functions on Tori and Spheres

Let (M,g) be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of points p,q in M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each "generic" initial condition f0, the solution to the Heat Equation is such that for sufficiently large t, f(.,t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.

Bour's spacelike maximal and timelike minimal surfaces in the 3-dimensional Minkowski space

Bour's minimal surface has remarkable properties in three dimensional Minkowski space. We reveal the definite and indefinite cases of the Bour's surface using Weierstrass representations, and give some differential geometric properties of the astonishing maximal and minimal surfaces.

Long-time behavior of 3 dimensional Ricci flow -- B: Evolution of the minimal area of simplicial complexes under Ricci flow

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of the paper. We remark that in this paper we will mostly be dealing with non-singular Ricci flows. The existence of surgeries will not play an important role.

Minimal surfaces - variational theory and applications

Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more remarkably, minimal surfaces (or minimal submanifolds) have encountered striking applications in other fields, like three-dimensional topology, mathematical physics, conformal geometry, among others. Even though it has been the subject of intense activity, many basic open problems still remain. In this lecture we will survey recent advances in this area and …

On the height function of the affine minimal translation surfaces

In this paper, using the Weierstrass-Enneper formula and the hodographic coordinate system, we find the relationships between the Ramanujan identity and the generalized class of Scherk surfaces known as affine Scherk surfaces. We find the Dirichlet series expansion of the affine Scherk surface. We also obtain some of the probability measures of affine Scherk surface with respect to its logarithmic distribution. Next, we classify the affine minimal translation surfaces in $\mathbb{L}^3$ and remarked the analogous forms in $\mathbb{L}^3.$

关于高余维下广义Calabi-Yau问题的注记

Remarks on the generalised Calabi-Yau problem in higher codimension

该研究引入更灵活的凸性概念，得到适用于高余维极小子流形全测地投影无界性研究的新型Omori-Yau最大值原理，并将其应用于多类映射与相关定理的探究，推进广义Calabi-Yau问题相关研究。

关于临界悬链面与$\Bbb S^2$中超定特征值问题的注记

Remarks on the critical catenoid and an overdetermined eigenvalue problem in $\Bbb S^2$

本文证明临界悬链面在更弱假设（浸入、可能带分支、嵌入边界）下仍唯一，并应用于$\Bbb S^2$上超定特征值问题，得出解区域为赤道对称的旋转环面。

Remarks on the stability operator for MOTS

Small deformations of marginally outer trapped surfaces (MOTS) are studied by using the stability operator introduced by Andersson-Mars-Simon. Novel formulae for the principal eigenvalue are presented. A characterization of the many marginally outer trapped tubes (MOTT) passing through a given MOTS is given, and the possibility of selecting a privileged MOTT is discussed. This is related to the concept of `core' of a black hole: a minimal region that one should remove from the spacetime in order to get rid of all possible closed trapped surfaces. In spherical symmetry one can prove that the sp…

具有悬链端点和指定通量的亏格为零的极小曲面的普遍存在性

General existence of minimal surfaces of genus zero with catenoidal ends and prescribed flux

对于欧氏空间中完备极小曲面的每一端，定义了通量向量。众所周知，所有端点上的通量矢量之和为零。考虑下面的逆问题：对于每个平衡的n-向量，找到一个n-端悬链线，它将这些向量实现为通量。在这里，一个n端悬链面是亏格为零的完全极小曲面，其端点渐近于悬链面。本文证明了对于任意大于4的n个平衡向量，几乎所有的n个平衡向量的反问题都可解。假设“几乎所有”是必要的，因为不存在已知的特殊平衡向量。我们注意到，在n=4的情况下，作者（dg-ga/9709006）得到了相同的结果。Lopez和Berbanel处理了n=3的情况。

Remark on the Novikov conjecture and the homotopy invariance of the Hirzebruch L-class

We use K-area homology to summarize some results about the Novikov conjecture and the Hirzebruch L-class. In fact, we provide necessary and sufficient conditions for closed manifolds to have a homotopy invariant L-class. In order to obtain additional properties we also introduce the cohomology of infinite K--area which preserves the cup product of singular cohomology.

On a remarkable class of left-symmetric algebras and its relationship with the class of Novikov algebras

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G,G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x,y].z=0. We derive some basic characterizations of such left-symmetric algebras and we highlight their relationships with the so-called Novikov algebras and derivation algebras.