Minimal surfaces associated with nonpolynomial contact symmetry flows
Arthemy V. Kiselev
利用对称性分析构造了空间中两个无穷序列的极小曲面。特别地，得到了填充三叶结的自相交极小曲面的显式公式。
Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.
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On the symmetry structure of the minimal surface equation
构造了二维极小曲面方程的可交换非多项式接触对称的无穷序列。利用Noether恒等式得到了$n$维极小面积曲面方程的局部和非局部守恒律.
A Congruence Theorem for Minimal Surfaces in $S^{5}$ with Constant Contact Angle
利用Gauss-Codazzi-Ricci方程，给出了具有常接触角的$S^5$中极小曲面的一个同余定理.更准确地说，我们证明了高斯Codazzi-Ricci方程的极小曲面在$S^5$与恒定的接触角满足一个方程的拉普拉斯全纯角。此外，我们还将给出具有常接触角的$S^5$中平坦极小曲面的特征。
Minimal annuli with constant contact angle along the planar boundaries
我们发现，一个浸没的最小环，有两个平面的边界曲线，沿该表面满足这些平面与恒定的接触角，是悬链线的一部分。
Minimal Surfaces in $S^3$ with Constant Contact Angle
我们通过活动标架和切触结构方程给出了S ~ 3中Clifford环面的一个刻画。更精确地说，我们证明了S3中具有常接触角的极小曲面一定是Clifford环面。最后给出了这个结果的一些应用，并讨论了一些例子。
Second order Contact of Minimal Surfaces
给出了$R^3$的二阶切触丛中的极小曲面方程$Q$的复结构和$Q\0$上的标准向量值全纯微分形式$Omega$. $R^3$中的极小曲面$M$对应于$Q$中的复解析曲线$C$，其中高斯映射的导数将$M$发送到$C$，并且$M$等于$Ω$对$C$的积分的实部。有限拓扑型且无穷远点平坦的完备极小曲面对应于$Q$中的代数曲线。
Equivariant Gluing Constructions of Contact Stationary Legendrian Submanifolds of the (2n+1)-Sphere
$S^{2n+1}$的接触平稳勒让德子流形是在接触变形下体积平稳的勒让德子流形。最简单的接触平稳勒让德子流形（实际上是极小的和勒让德的）是实的赤道$n$ -球面$S_0$。本文给出了一种构造$S^{2n+1}$的接触稳定（但不是极小）Legendrian子流形的方法，该方法是通过将$S_0$的足够多个$U(n+1)$ -旋转副本在适当的横截相交的孤立点处粘在一起.由此产生的子流形是非常对称的;在几何上类似于$S_0$的副本通过窄颈相互连接的“项链”，并在$S^{2n + 1}$周围缠绕了大量的时间，然后在自身上闭合;并且在拓扑上等价于$S^1 \times S^{n-1}$。此外，它们代表了$S^{2n + 1}$的接触定常嵌入子流形的全新例子，从而产生了$C^{n+1}$中嵌入Hamilton定常锥的全新例子。
New minimal surfaces in the sphere from capillary minimal cones
该研究通过双倍自由边界极小锥构造了球面上的新极小曲面，并展示了毛细功能在单相自由边界与极小曲面之间的联系。
We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $ϕ$. For the normal case, we prove that a $ϕ$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $ϕ$-invariant submanifold $N$ everywhere transverse to both the Reeb vector fields but not orthogonal to them, we prove that it is minimal if and only if the angle between the tangential component $ξ$ (with respect to $N$) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves…
Contact Angle for Immersed Surfaces in $S^{2n+1}$
本文介绍了接触角的概念。本文推导了$S^{2n+1}$中极小曲面的Laplacian曲率和高斯曲率的计算公式，并给出了广义Clifford环面是$S^5$中唯一具有常接触角和Kaehler角的非高斯极小曲面的一个特征.
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the firs…
Minimal surfaces in contact Sub-Riemannian manifolds
在本文中，我们考虑一般的次黎曼结构的余秩1非完整向量分布，并介绍了相关的正则体积和“水平”面积形式。与经典情形一样，亚黎曼极小曲面可以定义为“水平”面积泛函的临界点。我们推导出一个内在方程的极小曲面相关的一般Sub-Riemann结构的co-rank 1的正则体积形式和“水平”正常。该构造允许描述一般的次黎曼流形中的次黎曼极小曲面，并且可以容易地推广到更大余秩的非完整向量分布的情形.接触矢量分布的情况下，特别是$(2,3)$的情况下，进行了更详细的研究。在后一种情况下，次黎曼极小曲面的几何形状由其特征点的结构确定（即，超曲面接触水平分布的点）和特征曲线。结果表明，海森堡群$H^1$中亚黎曼极小曲面的特征点的已知分类对于与一般接触$(2,3)$分布相关联的极小曲面是成立的。证明了在$(2,3)$情形下，次黎曼极小曲面是扩展状态空间中某常微分方程组的积分曲面.在某些特殊情况下，该系统的柯西问题可以显式求解。考虑海森堡群和旋转平移群中的次黎曼极小曲面，我们给出了我们的结果。
K-contact manifolds with minimal closed Reeb orbits
该研究利用Boothby-Wang纤维化构造单连通K-切触流形，给出其同胚于奇维球面的充要条件，构造了具最少闭Reeb轨道但不同胚于奇维球面的流形实例，丰富了K-切触流形的分类研究。
We prove the existence of minimal surfaces in a bounded convex subset of $\mathbb R^3$, $\mathcal M$, intersecting the boundary of $\mathcal M$ with a fixed contact angle. The proof is based on a min-max construction in the spirit of Almgren-Pitts for the capillarity functional.
A New Characterization of the Clifford Torus
本文引入了三维球面中浸没面接触角的概念。我们推导出的拉普拉斯和高斯曲率的公式，我们分类的极小曲面在$S^3$与常接触角。并给出了一个接触角为非常数的$S^3$中极小曲面的例子。
A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${\mathcal O}_{{\mathbb P}^1}(1)^{\oplus p} \oplus {\mathcal O}_{{\mathbb P}^1}^{\oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C,$ which is the germ of submanifolds ${\mathcal C}^C_x \subset {\mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small…
Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) case
本文研究了一般次黎曼流形中具有余秩1的次黎曼结构的极小曲面。这些表面可以被定义为与规范{ \it horizontal}面积形式相关联的所谓{\it horizontal}面积泛函的临界点。我们推导出的内在方程在一般情况下，然后考虑在更详细的二维表面接触流形的3维。我们表明，在这种情况下，极小曲面是一类特殊的二维曲面的水平球面丛的基础流形上的投影。极小曲面的通有奇性导致了这种投影的奇性，并给出了它们的一个完整的局部分类。我们用海森堡群和旋转平移群中的例子来说明我们的结果
A Remark on Compact Minimal Surfaces in $S^5$ with Non-Negative Gaussian Curvature
在本文中，我们使用接触角的概念对$S^5$中具有非负高斯曲率的紧致极小曲面进行分类。
A Characterization of Minimal Surfaces in $S^5$ with Parallel Normal Vector Field
本文证明了具有常接触角和平行法向量场的球面$S^5$中紧致极小曲面的全纯角必为常值。
A Wide Class of Contact-Complex Riemannian Submersions
本文研究全空间为局部共形近拟Sasakian流形的接触复黎曼淹没，确定纤维的Chinea-Gonzalez类与基空间的Gray-Hervella类，并利用O'Neill不变量讨论水平分布可积性与纤维极小性。
Parallel mean curvature surfaces with constant contact angle along free boundaries
本文研究了具有常接触角的平行平均曲率曲面，通过分类和构造例子验证了相关定理的准确性。
In \cite{Luo}, the present author proved that if $L$ is a contact stationary Legendrian surface in $\mathbb{S}^5$ with the canonical Sasakian structure and the square length of its second fundamental form belongs to $[0,2]$. Then we have that $L$ is either totally umbilical or is a flat minimal Legendrian torus. In this addendum we further prove that if $L$ is a totally umbilical contact stationary Legendrian surface in $\mathbb{S}^5$ , then $L$ is totally geodesic.
We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467
On characteristic foliations of metric contact-symplectic structures
本文研究了接触辛结构的特征流形，证明了其积分曲线为测地线，并构造了非完全测地的特征流形例子。
Let $(M^5,α,g_α,J)$ be a 5-dimensional Sasakian Einstein manifold with contact 1-form $α$, associated metric $g_α$ and almost complex structure $J$ and $L$ a contact stationary Legendrian surface in $M^5$. We will prove that $L$ satisfies the following equation \begin{eqnarray}\label{equ} -Δ^νH+(K-1)H=0, \end{eqnarray} where $Δ^ν$ is the normal Laplacian w.r.t the metric $g$ on $L$ induced from $g_α$ and $K$ is the Gauss curvature of $(L,g)$. Using equation \eqref{equ} and a new Simons' type inequality for Legendrian surfaces in the standard unit sphere $\mathbb{S}^5$, we prove an integral ine…
Regularity of stable capillary minimal hypersurfaces
本文研究了半空间中具有接触角θ∈(0,π)和维度n≥2的稳定毛细极小超曲面的正则性和紧性理论，并由此得到嵌入的完整稳定毛细极小超曲面的广义伯恩斯坦定理。核心创新是通过精心选择适当的倾斜过剩函数，消除了稳定性不等式中出现的边界项，进而建立边界分层定理。
Stable Capillary Minimal Hypersurfaces in $\R^4_+$
本文证明在 $\R^4_+$ 中，当接触角满足 $|\cot\theta|<0.45$ 时，完备双侧稳定的毛细极小超曲面必为平坦。
On surfaces in three dimensional contact manifolds
本文在切触流形的曲面上引入了两个概念。第一个是横截度（DOT），它度量曲面的切空间与接触平面之间的横截性。第二个量，称为曲率的横向（COT），被设计为给出一个比较原则的DOT沿特征曲线的界限下的COT。特别地，这给出了假定COT由正常数限制的特征曲线的长度的估计。我们表明，表面与常数COT存在，我们分类的海森堡群中的所有图形消失COT。这是通过证明具有零COT的图的方程可以分解为两个一阶偏微分方程来实现的，其中一个是后向无粘Burgers方程。最后证明了Heisenberg群中的p-极小图方程也有这样的分解。此外，我们还可以利用这种分解写出一个解在正则点附近的显式公式。
We establish two main inequalities; one for the norm of the second fundamental form and the other for the matrix of the shape operator. The results obtained are for cosymplectic manifolds and, for these, we show that the contact warped product submanifolds naturally possess a geometric property; namely $\mathcal{D}_1$-minimality which, by means of the Gauss equation, allows us to obtain an optimal general inequality. For sake of generalization, we state our hypotheses for nearly cosymplectic manifolds, then we obtain them as particular cases for cosymplectic manifolds. For the other part of th…
Properly embedded and immersed minimal surfaces in the Heisenberg group
本文研究了三维Heisenberg群中适当嵌入和浸入的p（pseudohermitian）-极小曲面.从最近的工作程，黄，Malchiodi，和杨，我们知道，这样的曲面必须直纹面。根据它们的拓扑结构，有两种类型的这样的表面：带型和环型。我们给出了这些曲面的显式表达式。在带型中，有一类所谓螺旋面型的适当嵌入的p-极小曲面。对螺旋面型p-极小曲面进行了分类。这类p-极小曲面包含了任意平面上的所有p-极小图（接触平面除外）。此外，我们给出了这样的p-极小曲面没有奇点的一个充要条件。对于一般的完全浸入p-极小曲面，我们证明了一个半空间定理，并给出了一个适当的判据。
A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. In this paper, we provide a characterization of Besse contact forms for convex contact spheres and Riemannian unit tangent bundles in terms of $S^1$-equivariant spectral invariants. Furthermore, for restricted contact type hypersurfaces of symplectic Euclidean spaces, we give a sufficient condition for the Besse property via the Ekeland-Hofer capacities.
Straight ruled surfaces in the Heisenberg group
我们推广了Garofalo和Pauls的一个结果：嵌入Heisenberg群的水平极小光滑曲面是局部直纹曲面，即它由沿平滑曲线与水平向量场相切的直线组成.我们还证明了任何水平极小曲面都局部接触同构于复平面。
The systolic ratio of a contact form $α$ on the three-sphere is the quantity \[ ρ_{\mathrm{sys}}(α) = \frac{T_{\min}(α)^2}{\mathrm{vol}(S^3,α\wedge dα)}, \] where $T_{\min}(α)$ is the minimal period of closed Reeb orbits on $(S^3,α)$. A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that $ρ_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a…
The Calabi-Yau problem for minimal surfaces with Cantor ends
本文证明任意连通紧致或带边黎曼面都含有一个康托集，其补集可完备共形极小浸入到有界区域中的 $\mathbb R^3$，并将类似结论推广到多种全纯、勒让德及超极小浸入情形。
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