Embedded minimal and constant mean curvature annulus touching spheres
Sung-Ho Park
本文证明了一个紧的嵌入的最小或常中曲率环是旋转的，它具有非零的高斯曲率，它与两个相同半径的球相切或与一个球相切，并以常接触角与一个平面相交。
We show that a compact embedded minimal or constant mean curvature annulus with non-vanishing Gaussian curvature which is tangent to two spheres of same radius or tangent to a sphere and meeting a plane in constant contact angle is rotational.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Quantitative Spectral Stability for an Embedded Annulus under Coupled Curve Shortening and $2$D Ricci Flows
该研究通过变分公式和调和容量方法，分析了嵌入环形区域在耦合曲线缩短流和二维里奇流下的谱稳定性，并建立了谱隙估计与几何亏损的关系。
In this paper we use the Alexandrov Reflection Method to obtain a characterization to embedded CMC capillary annulus $Σ^2 \subset \mathbb{B}^3$. In especial, but using a new strategy, we present a new characterization to the critical catenoid. Precisely, we show that $Σ\subset \mathbb{B}^3$ being an embedded minimal free boundary annulus in $\mathbb{B}^3$ such that $\partial Σ$ is invariant under reflection through a coordinates planes, then $Σ$ is the critical catenoid. This work is part of the second author thesis which was written in 2019.
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。
On uniqueness of free boundary minimal annuli in geodesic balls of $\mathbb{S}^3_+$ and $\mathbb{H}^3$
本文研究了球面S³₊和双曲空间H³中测地球内自由边界极小环的唯一性，证明其与临界旋转环全等。
We show that an embedded minimal annulus $Σ^2 \subset B^3$ which intersects $\partial B^3$ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique free boundary minimal annulus in $B^n$ with lowest Steklov eigenvalue equal to 1. We also give more general criteria which imply that a free boundary minimal surface in $B^3$ invariant under a group of reflections has lowest Steklov eigenvalue 1.
We first consider a uniqueness problem for embedded free boundary minimal annuli in the three-dimensional Euclidean unit half-ball. Then, we obtain symmetry properties for compact embedded free boundary minimal surfaces in the unit ball. Finally, we obtain several uniqueness results for the critical catenoid under symmetry conditions on the boundary. For example, we show that if an embedded free boundary minimal annulus whose boundary consists of two congruent components and has a reflection symmetry by a plane, then it is congruent to the critical catenoid.
A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere
研究球冠中自由边界极小旋转环面族，证明相关半径函数\(R\)的性质及环面退化情况，指出纳夫 - 朱唯一性假设对某些\(R>\pi/2\)不成立，通过罗宾缺陷等式检测退化，给出环面维度等式。
Free boundary minimal surfaces and the reflection principle
本文利用极小曲面关于球面的90度角反射原理，证明了球中嵌入自由边界极小环面的唯一性，即必为临界悬链面。
In this paper we study the moduli space of properly Alexandrov-embedded, minimal annuli in $\mathbb{H}^2 \times \mathbb{R}$ with horizontal ends. We say that the ends are horizontal when they are graphs of $\mathcal{C}^{2, α}$ functions over $\partial_\infty \mathbb{H}^2$. Contrary to expectation, we show that one can not fully prescribe the two boundary curves at infinity, but rather, one can prescribe the bottom curve, but the top curve only up to a translation and a tilt, along with the position of the neck and the vertical flux of the annulus. We also prove general existence theorems for m…
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。
Axial minimal surfaces in S^2 x R are helicoidal
我们证明了，如果一个完整的，适当嵌入，有限拓扑极小曲面在S^2 x R包含一条线，那么它的端点是渐近螺旋面，如果表面是一个环，它必须是螺旋面。
Exterior free-boundary minimal surfaces of every topological type with at least two ends
该研究构造出具有至少两个端点、任意拓扑类型的连通可定向真嵌入外部自由边界极小曲面，通过两种粘合方法实现，解决了此类曲面的拓扑类型覆盖问题。
The space of embedded minimal surfaces of fixed genus in a 3-manifold III; Planar domains
本文是第三次在一个系列中，我们描述了空间的所有嵌入的极小曲面的固定属在一个固定的（但任意的）封闭的3-流形。在[CM 3]-[CM 5]中，我们描述了曲面在任何固定的小尺度上都是拓扑圆盘的情况。为了描述一般的平面区域（在[CM 6]中），除了[CM 3]-[CM 5]的结果之外，我们还需要嵌入稳定环的关键估计，这是本文的主要结果。这个估计断言，这样的环是一个图形远离其边界，如果它只有一个内部边界组件，如果这个组件位于一个小（外在）球。
We construct a family of compact free boundary minimal annuli immersed in the unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in $\mathbb{B}^3$ foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embe…
We develop new methods to compare the span $\mathcal{C}(Σ)$ of the coordinate functions on a free boundary minimal submanifold $Σ$ embedded in the unit $n$-ball $\mathbb{B}^n$ with its first Steklov eigenspace $\mathcal{E}_{σ_1}(Σ)$. Using these methods, we show that $\mathcal{C}(A)=\mathcal{E}_{σ_1}(A)$ for any embedded free boundary minimal annulus $A$ in $\mathbb{B}^3$ invariant under the antipodal map, and thus prove that $A$ is congruent to the critical catenoid. We also confirm that $\mathcal{C}=\mathcal{E}_{σ_1}$ for any free boundary minimal surface embedded in $\mathbb{B}^3$ with the …
Non-collapsing in fully nonlinear curvature flows
我们考虑由完全非线性流演化的嵌入超曲面，其中运动的法向速度是齐次的，是主曲率的凹函数或凸函数，并证明了一个非崩溃估计：精确地说，如果速度是凹的，那么给出接触超曲面的每一点的最大内部球面的曲率的函数是线性化流方程的子解.如果速度是凸的，那么对于外部球体也有类似的陈述。特别地，如果超曲面以正速度运动，并且速度在主曲率中是凹的，则只要解存在，最大接触内球面的曲率就以速度的倍数为界。证明使用了一个最大值原理应用到一个函数的两个点上的演变超曲面。我们说明了处理这些功能所需的技术，在一个证明的超曲面流的包容性原则。
In this paper, we provide a sufficient condition for a curve on a surface in $\mathbb{R}^3$ to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball $\mathbb{B}^3$. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville ty…
A solution to a problem of Ghomi and Raffaelli on closed asymptotic curves
本文构造了具有严格负高斯曲率和单射高斯映射的嵌入开环带，其中包含法向自链数为零的闭渐近曲线，从而肯定解决了 Ghomi 和 Raffaelli 的问题 1.1。
In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $Σ$ is a complete, properly embedded CMC surface in the Euclidean space satisfying this inequality, then $Σ$ is either a sphere or a right circular cylinder. Next, we show that if $Σ$ is a free boundary CMC surface in the Euclidean 3-ball satisfying the same inequality, then either $Σ$ is a totally umbilical disk or an annulus of revolution. These results complete the picture about gap theorems f…
On the total curvature of minimal annuli in $R^3$ and Nitsche's conjecture
给出了Meeks III和H. Rosenberg：对于$t\ge 0$，让$P_t$表示高度$t$在$x_1,x_2$平面上的水平面。假设$M \subset R^3$是一个极小环，其边界包含在$P_0$中，且$M$与每个$P_t$相交于一条简单闭曲线.则$M$具有有限的全曲率。因此，我们证明了$R^3$中每个适当嵌入的具有多个端点的有限拓扑的极小曲面具有有限的全曲率。
Alexandrov's Soap Bubble theorem dates back to $1958$ and states that a compact embedded hypersurface in $\mathbb{R}^N$ with constant mean curvature must be a sphere. For its proof, A.D. Alexandrov invented his reflection priciple. In $1982$, R. Reilly gave an alternative proof, based on integral identities and inequalities, connected with the torsional rigidity of a bar. In this article we study the stability of the spherical symmetry: the question is how much a hypersurface is near to a sphere, when its mean curvature is near to a constant in some norm. We present a stability estimate that s…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。