Leonor Ferrer, Francisco Martin
In this paper we describe a new deformation that connects minimal disks with planar ends with minimal disks with helicoidal ends. In this way, we are able to construct a family of complete minimal surfaces with helicoidal ends that contains the singly periodic genus one helicoid of Hoffman, Karcher and Wei.
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
We prove that if a complete, properly embedded, finite-topology minimal surface in S^2 x R contains a line, then its ends are asymptotic to helicoids, and that if the surface is an annulus, it must be a helicoid.
We prove that for each positive integer g, there exists a complete minimal surface of genus g that is properly embedded in three-dimensional euclidean space and that is asymptotic to the helicoid.
For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in $R^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in $R^3$ of every prescribed genus occur as such limits of examples in $S^2\times R$.
A Note on Helicoidal Singular Minimal Surfaces
本文研究在单参数螺旋运动下不变的奇异极小曲面，证明其螺旋轴必正交于给定单位向量、参数α=-1，且曲面必为正圆柱面。
Helicoidal minimal surfaces of prescribed genus, I
本文证明 S^2×R 中存在任意亏格、两端渐近于指定螺距螺旋面的完备真嵌入极小曲面，并说明其在半径趋于无穷时收敛为 R^3 中无穷远呈螺旋状的极小曲面。
In this article, we construct two one-parameter families of properly embedded minimal surfaces in a three-dimensional Lie group $\widetilde{E(2)}$, which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrass-type representation introduced by Meeks, Mira, Pérez and Ros for minimal surfaces in Lie groups of dimension three. In the …
Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves
该研究证明了三维球面中由轴距连续函数给定平均/高斯曲率的球面螺旋面的存在唯一性，推广了欧氏空间belicoid与catenoid的共轭关系，以球面曲线的球面角动量为关键工具确定Lawson意义下的伴随曲面。
We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at their limiting behaviour as the pitch of the helicoidal motion goes to 0 and compare it with the limiting behaviour of the classical helicoidal minimal surfaces. Finally, we give a classification of the immersed cylinders in the family of constant mean curvature helicoidal surfa…
In this paper we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space R^3 to the case of helicoidal surfaces in the Bianchi-Cartan-Vranceanu (BCV) spaces, i.e. in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension $4$ or $6$, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mea…
The Dirichlet problem for the minimal surface equation on unbounded helicoidal domains of $\mathbb{R}^{m}$
该研究针对$\u211d^{n+1}$中螺旋投影为外域的无界$G$-不变$C^{2,\u03b1}$-域，证明了零边界极小曲面Dirichlet问题存在一族梯度边界上确界为$|s|$的$G$-不变解，并补充了解的无穷远行为信息。
We consider families of embedded, screw motion invariant minimal surfaces in $\R^3$ which limit to parking garage structures. We derive balance equations for the nodal limit and regenerate to obtain surfaces corresponding to solutions. We thus prove the existence of many new examples with helicoidal or planar ends.
The Clifford torus is a product surface in $\mathbb S^3$ and it is helicoidal. It will be shown that more minimal submanifolds of $\mathbb S^n$ have these properties.
Compactness and Willmore Energy of Helicoidal Minimal Surfaces in the 3-Sphere
本文解决了三维球面中双参数螺旋极小曲面族$\text{Hel}_c^h$的紧性判定问题，给出紧成员的刻画条件并计算了对应紧浸入曲面的Willmore能量。
In a previous paper the author introduced the notion of TreadmillSled of a curve, which is an operator that takes regular curves in R^2 to curves in R^2. This operator turned out to be very useful to describe helicoidal surfaces, for example, it provides an interpretation for the profile curve of helicoidal surfaces with constant mean curvature similar to the well known interpretation of the profile curve of Delaunay's surfaces using conics. Recentely, Palmer and Kuhns used the TreadmillSled to classify all helicoidal surfaces with constant anisotropic mean curvature coming from axially symmet…
For any m > 0, we construct properly embedded minimal surfaces in H^2 x R with genus zero, infinitely many vertical planar ends and m limit ends. We also provide examples with an infinite countable number of limit ends. All these examples are vertical bi-graphs.
Timelike minimal surface in $\mathbb{E}^3_1$ with arbitrary ends
本文证明了具有任意多个弱完整端点的类时极小曲面的存在性，并讨论了简单端点的渐近行为及所构造曲面的奇点集拓扑。
In this paper, we consider complete non-catenoidal minimal surfaces of finite total curvature with two ends. A family of such minimal surfaces with least total absolute curvature is given. Moreover, we obtain a uniqueness theorem for this family from its symmetries.
We construct constant mean curvature surfaces in euclidean space by gluing n half Delaunay surfaces to a non-degenerate minimal n-noid, using the DPW method.
Genus three embedded doubly periodic minimal surfaces with parallel ends
本文构造了一族亏格为三的嵌入双周期极小曲面，具有四个平行端口，并解决了二维周期问题。
In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus yet maintain the symmetry of their genus-zero counterparts constructed by Jorge-Meeks and Xu.
In this article, we study the modified defect relations of the Gauss map of complete minimal surfaces in $\mathbb R^3$ and $ \mathbb R^4$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto~[J. Differential Geometry \textbf{29} (1989), 245-262] for (the whole) complete minimal surfaces. We thus give some improvements of the previous results for the Gauss maps of complete minimal surfaces restricted on annular ends.
The ends of a complete embedded minimal surface of {\em finite total curvature} are well understood (every such end is asymptotic to a catenoid or to a plane). We give a similar characterization for a large class of ends of {\em infinite total curvature}, showing that each such end is asymptotic to a helicoid. The result applies, in particular, to the genus one helicoid and implies that it is embedded outside of a compact set in ${\mathbb R}^3$.
We obtain isometric minimal helicoidal and rotational surfaces using generalized Bour's theorem in three dimensional Minkowski space. In addition, we show that the surfaces preserve minimality when their Gauss maps identically equal, choosing any diffentiable functions on the profile curve.
In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar ends like the Lagrangian catenoid. Then we show the existence of embedded minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we construct genus $g$ examples in $\mathbb{R}^4$ with $d$ embedded planar ends such that $g\geq 1$ and $g+2\leq d\leq 2g+1$. These examples include a family of embed…
In this paper we construct an example of a complete immersed minimal surface in $\mathbb{R}^3$ of genus one with two embedded catenoid-type ends, one Enneper-type end and total Gauss curvature $-16π.$ The proof of the existence of this example, was obtained using the Weierstrass representation, the theory of elliptic functions and explicitly solving the period problem.
A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type is necessarily the double of the spectral curve of an elliptic KP soliton. The simplest possible examples of minimal tori with planar ends are related to 1--gap Lamé potentials, the simplest non--trivial algebro geometric KdV potentials. If one allows for translational periods, …
Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}. The degenerate limits of surfaces in $\mathcal{K}$ are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examp…
Singly periodic minimal Wente torus with ends
研究在\(\mathbb{R}^3\)中构造由菱形晶格参数化且由球面曲率线叶状化的单周期极小曲面，以此找到新的自由边界和毛细极小环面例子。
Spectral splitting theorem and ends of minimal hypersurfaces
本文通过构造加权最小测地线，证明了非负双曲曲率流形中有限指数极小超曲面具有有限端，推广了Li-Wang关于非负截面曲率流形的结果。
We describe a 3-parametric family $\mathcal{K}$ of properly embedded minimal tori with four parallel ends in quotients of $\mathbb{R}^3$ by two independent translations, which we will call the \textit{Standard Examples.} These surfaces generalize the examples given by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}. $\mathcal{K}$ can be endowed with a natural structure of a self-conjugated 3-dimensional real analytic manifold diffeomorphic to $\mathbb{R}\times(\mathbb{R}^2-\{(\pm 1,0)\})$ whose degenerate limits are the catenoid, the helicoid, the simply and doubly periodic Scherk minimal s…
We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.
Complete minimal surfaces with Cantor ends in minimally convex domains
本文综述了共形Calabi-Yau问题的近期历史，并证明了在$\mathbb{R}^3$的任意极小凸区域中，任意紧Riemann曲面去掉一个Cantor集后可作为完备真极小曲面的复结构。
Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than $π$. In the case of general angles, we prove that there exist at most four such minimal disks, we give a sufficient condition of existence in terms of a system of three equations of degree 2, and we give explicit formulas for the Weierstrass data in terms of hypergeometric functions. Finally, we construct constant-mean-curvature-one trinoids in hyperbolic space by the method of the conjugate cousin immersion.
We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We give the precise asymptotic behavior of an end of such a surface. This asymptotic information is very much dependent on the way the ambient metric decays to the Euclidean one. Our results apply in particular to minimal surfaces.
Catenoid limits of singly periodic minimal surfaces with Scherk-type ends
本文构造了任意亏格和偶数个Scherk端的单周期极小曲面族，其极限为带节点的多层垂直平面，节点打开条件由Stieltjes多项式根的平衡方程决定。
We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has at least two ends implies that $M$ has finite stability index which is bounded by a constant that only depends on its genus.
Morse index, topology, and ends of minimal surfaces with noncompact free boundary
本文为具有非紧自由边界的完备双侧极小曲面建立 Morse 指标下界，结合局部能量恒等式与 Riemann--Roch 计数，并处理内部端与边界端的截断问题，推广了紧边界情形的结果。
An end-to-end construction of doubly periodic minimal surfaces
本文基于Traizet再生方法，构造了亏格为2n-1且具有4个平行端的双周期极小曲面族，证明其光滑收敛于2n份Scherk曲面，填补了亏格大于3的该类曲面研究空白。
Minimal surfaces in $\mathbb{H}^2\times\mathbb{R}$ with finite total curvature and embedded ends
本文研究 $\mathbb{H}^2\times\mathbb{R}$ 中具有嵌入端和有限全曲率的完备极小曲面，分类了全曲率为 $-4\pi$ 和 $-6\pi$ 的渐近边界，并通过渐近 Plateau 问题构造了新的极小曲面族。
We construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known surface for each genus. We illustrate multiple new examples for each genus g>2. In the parallel ends case, the known examples limit as a foliation of parallel planes with nodes. We construct a new example for each genus g>2 that limit as g-1 singly periodic Scherk surfaces glued between two doubly periodic Scherk surfaces and also as a singly periodic surface wit…
We prove the existence of nonperiodic, properly embedded minimal surfaces in $\mathbb{R}^2\times\mathbb{S}^1$ with genus zero, infinitely many ends and one limit end (in particular, they have infinite total curvature).
Complete Minimal Surfaces in $\mathbb{R}^4$ with Three Embedded Planar Ends
本文利用广义 Weierstrass 表示，研究 $\mathbb{R}^4$ 中具有三个嵌入平面端的完备极小曲面，证明了亏格为 1 时曲面的 $J$-全纯性与唯一性，并给出了高亏格下的不存在性结果。
In this paper, we study Alexandrov-embedded r-noids with genus 1 and horizontal ends. Such minimal surfaces are of two types and we build several examples of the first one. We prove that if a polygon bounds an immersed polygonal disk, it is the flux polygon of an r-noid with genus 1 of the first type. We also study the case of polygons which are invariant under a rotation. The construction of these surfaces is based on the resolution of the Dirichlet problem for the minimal surface equation on an unbounded domain.
In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^m on annular ends. This work is a continuation of previous work of Dethloff-Ha. We thus give an improvement of the results on annular ends of complete minimal surfaces of Jin-Ru.
We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are well-understood surfaces of revolution -- the catenoid cousins. In contrast to this, we show that, unlike the case of minimal surfaces in Euclidean 3-space, there do exist complete connected immersed constant mean curvature 1 surfaces with two ends in hyperbolic space that are not …
We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in R^N. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codimensionn meeting any n-plane passing through the origin in at most k points has no more c(n,N)k ends.
Bryant \cite{Bryant84} classified all Willmore spheres in $3$-space to be given by minimal surfaces in $\mathbb R^3$ with embedded planar ends. This note provides new explicit formulas for genus 0 minimal surfaces in $\mathbb R^3$ with $2k+1$ embedded planar ends for all $k\geq4.$ Peng and Xiao claimed these examples to exist in \cite{PengXiao2000}, but in the same paper they also claimed the existence of a minimal surface with 7 embedded planar ends, which was falsified by Bryant \cite{Bryant88}.
In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal complete end E with finite total curvature is properly immersed and that the Gaussian curvature of E is locally bounded in terms of the geodesic distance to its boundary.
We prove that the ends of a properly immersed simply or one connected minimal surface in H(2)xR contained in a slab of height less than πof H(2)xR, are multi-graphs. When such a surface is embedded then the ends are graphs. When embedded and simply connected, it is an entire graph.
In this article, we study the ramification of the Gauss map of complete minimal surfaces in R^3 and R^4 on annular ends. We obtain results which are similar to the ones obtained by Fujimoto and Ru for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement of the results on annular ends of complete minimal surfaces of Kao.
Given a complete isometric immersion $ϕ: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^n_w$, we determine a set of conditions on the extrinsic curvatures of $P$ that guarantees that the immersion is proper and that $P$ has finite topology, in the line of the paper "On Submanifolds With Tamed Second Fundamental Form", (Glasgow Mathematical Journal, 51, 2009), authored by G. Pacelli Bessa and M. Silvana Costa. When the ambient ma…
Minimal surfaces with closed curvature lines
研究有限总曲率的非定向极小曲面，其端面由闭曲率线覆盖，证明此类曲面的刚性，排除单端面的存在。
In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb H^2$ without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in $\mathbb H^2$ or a hyperbolic translation along a geodesic in $\mathbb H^2.$ In fact, we prove that if a properly immersed minimal surface in $\mathbb H^2\times\mathbb R/G$ has finite total curvature then its total curvature is a multiple of $2π,$ and…
For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there are no complete two-sided immersed minimal surfaces in $\mathbb{R}^3$ of index two, complete embedded minimal surface with index three, or comple…
The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmonic surfaces, which in general don't have the same topological limitations of minimal surfaces and can have complicated embedded ends. In this paper we demonstrate the flexibility of using the Weierstrass representation for harmonic surfaces to combine embedded harmonic ends togeth…
In 1988, Karcher generalized the family of singly periodic Scherk minimal surfaces by constructing, for each natural $n\geq 2$, a $(2n-3)$-parameter family of singly periodic minimal surfaces with genus zero and $2n$ Scherk-type ends in the quotient, called {\it saddle towers}. They have been recently classified by Pérez and Traizet \cite{PeTra1} as the only properly embedded singly periodic minimal surfaces in $\R^3$ with genus zero and finitely many Scherk-type ends in the quotient. In this paper we obtain as a limit of saddle towers: the catenoid; the doubly periodic Scherk minimal surface …
We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry group the natural $\bfZ_2$ extension of the dihedral group $D_n$. The surfaces are constructed by proving existence of the conjugate surfaces. We extend this method to cases where the conjugate surface of the fundamental piece is noncompact and is not a graph over a convex plane domain.
We construct a complete, embedded minimal surface in euclidean 3-space which has unbounded Gaussian curvature. It has infinite genus, infinitely many catenoidal type ends and one limit end.
A Ricci surface is defined as a Riemannian surface $(M,g_M)$ whose Gauss curvature satisfies the differential equation $KΔK + g_M(dK,dK) + 4K^3=0$. Andrei Moroianu and Sergiu Moroianu proved that a Ricci surface with non-positive Gauss curvature admits locally a minimal immersion into $\mathbb{R}^3$. In this paper, we are interested in studying non-compact orientable Ricci surfaces with catenoidal ends. We use an analogue of the Weierstrass data to obtain some classification results for such Ricci surfaces. We also give an existence result for positive genus Ricci surfaces with catenoidal ends…
We compute the flux of Killing fields through ends of constant mean curvature 1 in hyperbolic space, and we prove a result conjectured by Rossman, Umehara and Yamada : the flux matrix they have defined is equivalent to the flux of Killing fields. We next give a geometric description of embedded ends of finite total curvature. In particular, we show that we can define an axis for these ends that are asymptotic to a catenoid cousin. We also compute the flux of Killing fields through these ends, and we deduce some geometric properties and some analogies with minimal surfaces in Euclidean space.
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。