一组标定曲面在同构下面积极小的充分条件

$L^2$ 曲率界对于模2面积最小化流形

$L^2$ Curvature Bounds for Area-Minimizing Currents Mod(2)

本文证明了模2面积最小化流形的第二基本形式满足仅依赖于维度、余维数及质量上界的内部 $L^2$ 界，并由此推出有界面积的此类曲面属于有限多个微分同胚类，回答了相关文献中的问题。

Elastic distance between curves under the metamorphosis viewpoint

We provide a new angle and obtain new results on a class of metrics on length-normalized curves in $d$ dimensions, represented by their unit tangents expressed as a function of arc-length, which are functions from the unit interval to the $(d-1)$-dimensional unit sphere. These metrics are derived from the combined action of diffeomorphisms (change of parameters) and arc-length-dependent rotation acting on the tangent. Minimizing a Riemannian metric balancing a right-invariant metric on diffeomorphisms and an $L^2$ norm on the motion of tangents leads to a special case of "metamorphosis", which…

Minimizing immersions of a hyperbolic surface in a hyperbolic $3$-manifold

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps …

genus-zero 表面的度量方法

A Metric for genus-zero surfaces

本文提出一种比较 genus-zero 表面形状的新方法，引入对称扭曲能量度量，并证明任何两个 genus-zero 表面之间存在最小化该能量的共形微分同胚。证明该能量给出 genus-zero 里emannian 表面空间的度量。

Hasse原理在几何变分问题中的应用：通过面积最小子流形的示例

The Hasse Principle for Geometric Variational Problems: An Illustration via Area-minimizing Submanifolds

本文探讨了几何变分问题中Hasse原理的应用，通过面积最小子流形展示了如何从实同调和模n同调中恢复积分同调中的信息，并回答了Almgren等人提出的问题。

$\mathbf{R}^n$ 中密度为二的完全面积极小曲面是二次的或平面的

Entire area-minimizing surfaces in $\mathbf{R}^n$ of density two are quadratic or planar

本文分类了无穷远处密度为2的完全面积极小曲面，证明其要么是平面，要么是二次全纯多项式的零点集（在刚体运动下），且包含于$\mathbf{R}^4$的仿射拷贝中。

面积几乎最小化流的平坦内部奇点

Flat interior singularities for area almost-minimizing currents

本文通过构造两族面积几乎最小化整系数流的例子，证明 Bombieri 关于内部正则集稠密的结果是尖锐的，其平坦奇点集可包含任意闭且内部为空的平面子集。

拉格朗日中曲率流的奇性：零Maslov类情形

Singularities of Lagrangian mean curvature flow: zero-Maslov class case

研究了当初始条件为零Maslov类拉格朗日量时，$\C^n$中拉格朗日中曲率流的奇异性.我们首先要说明，在这种情况下，奇点是不可避免的。更确切地说，我们构造拉格朗日与任意小的拉格朗日角和拉格朗日是哈密顿同位素的平面，然而，开发有限的时间奇点下中曲率流。然后，我们证明了两个定理关于切流在奇点时的初始条件是零马斯洛夫类拉格朗日。第一个（定理A）指出，在奇点处的重标度流弱收敛到面积最小化拉格朗日锥的有限并。第二个定理（定理B）指出，在初始条件是几乎校准和有理拉格朗日的附加假设下，重新标度流的连通分量收敛到单个面积最小化拉格朗日锥，而不是面积最小化拉格朗日锥的可能非面积最小化并集。后一个条件是稠密的拉格朗日与生成的$H_1(L,\Z)$。

关于1-调和函数

On 1-Harmonic Functions

常数1 $-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over $ \mathbb{R} $; and every 7-dimensional $ SO（2）\times SO（6）$-invariant absolutely area-minimizing integral current in $ \mathbb{R}^8 $ is real analytic. The assumption on the $ SO（2）\times SO（6）$-invariance cannot be removed, due to the first counter-example in $ \mathbb{R}^8 $的1-调和方程的整个子解的特征，由Bjereri，De Girogi和Giusti证明.

Degeneration of 7-dimensional minimal hypersurfaces which are stable or have bounded index

A 7-dimensional area-minimizing embedded hypersurface $M$ will in general have a discrete singular set. The same is true if $M$ is stable, or has bounded index, provided $H^6(sing M) = 0$. We show that if $M_i$ are a sequence of such minimal hypersurfaces which are minimizing, stable, or have bounded index, then $M_i$ can limit to a singular $M$ with only very controlled geometry, topology, and singular set. We show one can always "parameterize" a subsequence $i'$ with controlled bi-Lipschitz maps $ϕ_{i'}$ taking $ϕ_{i'}(M_{1'}) = M_{i'}$. As a consequence, we prove the space of smooth, closed…