Nicolaos Kapouleas, Seong-Deog Yang
We construct embedded closed minimal surfaces in the round three-sphere, resembling two parallel copies of the Clifford torus, joined by m^2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus.
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Bifurcations of the Clifford Torus as Willmore Surfaces in Berger Spheres
通过估计Morse指标并应用分岔理论，从Clifford环面分岔出新的对称Willmore环面。
Building on work of Kapouleas and Yang, we construct sequences of minimal surfaces embedded in the round 3-sphere which converge to the Clifford torus counted with multiplicity two and have second fundamental form blowing up at every point of the torus and genus tending to infinity. Each surface in a given sequence resembles a pair of tori close to the limit torus and joined by many catenoidal bridges arranged over a rectangular lattice on the limit. The collection of sequences is indexed by the ratio of the lengths of the lattice edges, which may be any prescribed positive rational. Unlike th…
New minimal surface doublings of the Clifford torus and contributions to questions of Yau
该研究利用前期一般性定理构造了三维球面中不受透镜空间提升限制的新型Clifford环面极小倍化曲面，得到指定亏格嵌入极小曲面数量的二次下界，并在两类倍化曲面情形下验证了丘成桐的第一特征值猜想。
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.
We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.
In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus $\mathbb{S}^1(1)\times\mathbb{S}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in $\mathbb{C}^2$ with $|A|^2\leq 2$, which gives an affirmative answer to Castro-Lerma's conjecture. We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in $\mathbb{C}^2$.
On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimi…
The Clifford tori in the 3-sphere are a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) surfaces. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each sub-lattice point; and then one can show that a perturbation of this approximate submanifold exists which …
The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tori connected to each other by small catenoidal bridges at a sufficiently symmetric configuration of points can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each point; and then o…
We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions there exist two spin structures which give non-vanishing Floer cohomology of the Clifford torus, and in…
The Neumann problem on the Clifford torus in $\mathbb{S}^3$
本文研究CR流形$\mathbb{S}^3$中Clifford环面界定的子集上CR Yamabe算子的Neumann问题，并探讨寻找零Tanaka--Webster标量曲率且边界具常$p$-平均曲率的切触形式的Yamabe型问题。
Quantitative Stability of the Clifford Torus as a Willmore Minimizer
研究证明在特定条件下，Clifford torus是Willmore极小值的定量稳定形式。
We construct the spectral curve and the Baker--Akhiezer function for the Dirac operator which corresponds to the Clifford torus via the Weierstrass representation. By constructing this Baker--Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker--Akhiezer functions corresponding to singular spectral curves
In this paper we introduce the notion of contact angle for an immersed surface in three dimensional sphere. We deduce formulas for the Laplacian and for the Gaussian curvature, and we classify minimal surfaces in $S^3$ with constant contact angle. Also, we give an example of a minimal surface in $S^3$ with non constant contact angle.
The Marques-Neves theorem asserts that among all the torodial (i.e. genus 1) closed surfaces, the Clifford torus has the minimal Willmore energy $\int H^2 \, dA$. % It is a natural conjecture that if one prescribes the isoperimetric Since the Willmore energy is invariant M{ö}bius transformations, it can be shown that there is a one-parameter family, up to homotheties, of genus 1 Willmore minimizers. It is then a natural conjecture that such a minimizer is unique if one prescribes its isoperimetric ratio. In this article, we show that this conjecture can be reduced to the positivity question of…
Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $\mathbb{S}^{n+1}$. Denote by $S$ the squared norm of the second fundamental form of $M$. We prove that there exists a positive constant $γ(n)$ depending only on $n$ such that if $|H|\leqγ(n)$ and $β(n,H)\leq S\leqβ(n,H)+\frac{n}{23}$, then $S\equivβ(n,H)$ and $M$ is one of the following cases: (i) $\mathbb{S}^{k}(\sqrt{\frac{k}{n}})\times \mathbb{S}^{n-k}(\sqrt{\frac{n-k}{n}})$, $\,1\le k\le n-1$; (ii) $\mathbb{S}^{1}(\frac{1}{\sqrt{1+μ^2}})\times \mathbb{S}^{n-1}(\fracμ{\sqrt{1+μ^2}})$. Here $β(n,H)=n+\frac{n^3}{2(n-1)}H…
Bifurcations of Clifford tori in ellipsoids
证明了在2-环面作用下不变的3维椭球包含无穷多个不同的浸入极小环面，这些环面在偏心率的稠密集上从体积最大的2-环面轨道分岔，并在圆作用下保持不变。
A peculiarity of the geometry of the euclidean 3-sphere $\S3$ is that it allows for the existence of compact without boundary minimally immersed surfaces. Despite a wealthy of examples of such surfaces, the only known tori minimally embedded in $\S3$ are the ones congruent to the Clifford torus. In 1970 Lawson conjectured that the Clifford torus is, up to congruences, the only torus minimally embedded in $\S3$. We prove here Lawson conjecture to be true. Two results are instrumental to this work, namely, a characterization of the Clifford torus in terms of its first eingenfunctions (\cite{MR})…
This note is motivated by Y.G. Oh's conjecture that the Clifford torus $L_n$ in $\mathbb{C}P^n$ minimizes volume in its Hamiltonian deformation class. We show that there exist explicit positive constants $a_n$ depending on the dimension with $a_2=3/π$ such that for any Lagrangian torus $L$ in the Hamiltonian class of $L_n$ we have $vol(L) \geq a_n vol (L_n)$. The proof uses the recent work of C.H. Cho on Floer homology of the Clifford tori. A formula from integral geometry enables us to derive the estimate. We wish to point out that a general lower bound on the volume of $L$ exists from the wo…
We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.
Supersymmetric Topological Sigma Models and Doubling Spaces
本文在加倍几何框架中提出分析环面上拓扑 B 模型的新方法，通过纳入秩一 D 膜并利用其相交理论计算 BRST 上同调，为理解环面导出范畴提供了新视角。
For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with maximal symmetry along these two circles. Near each of the circles, after rescaling, the sequences converge smoothly on compact subsets to a Karcher-Scherk tower of order $k$. Sequences of the second type desingularize a collection of the same $k$ Clifford tori supplemented by an…
Four constructions of constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere are given, which should be considered analogues of `classical' constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symm…
The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori which is contained in a hypersphere in the complex Euclidean plane are constructed explicitly. Then it is shown that they are homogeneous tori. For the construction, flat quaternionic connections of Hamiltonian stationary Lagrangian tori are considered and a spectral curve of an asso…
In 1965 Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $R^3$ is at least $2π^2$ and attains this minimal value if and only if the torus is a Möbius transform of the Clifford torus. This was recently proved by Marques and Neves. In this paper, we show for tori there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.
Our goal is to generalize the Choe-Hoppe helicoid and Clifford cones in Euclidean space. By sweeping out $L$ indpendent Clifford cones in ${\mathbb{R}}^{2N+2}$ via the multi-screw motion, we construct minimal submanifolds in ${\mathbb{R}}^{L(2N+2)+1}$. Also, we sweep out the $L$-rays Clifford cone (introduced in Section 2.3) in ${\mathbb{R}}^{L(2N+2)}$ to construct minimal submanifolds in ${\mathbb{R}}^{L(2N+2)+1}$. Our minimal submanifolds unify various interesting examples: Choe-Hoppe's helicoid of codimension one, cone over Lawson's ruled minimal surfaces in ${\mathbb{S}}^{3}$, Barbosa-Dajc…
We show that any embedded minimal torus in S^3 is congruent to the Clifford torus. This answers a question posed by H.B. Lawson, Jr., in 1970.
On the second variation of the biharmonic Clifford torus in S^4
本文研究 S^4 中真双调和 Clifford 环面的第二变分，计算其双调和指数与零化度，分析广义 Jacobi 算子核，并推广到极小浸入与双调和小超球面的复合。
We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.
A proof of the Willmore-type conjecture in $\mathbb{C}P^2$
本文证明了 $\mathbb{C}P^2$ 中 Lagrangian 环面的 Willmore 型猜想，并构造非 Lagrangian 形变否定一般情形。
Isoperimetric Ratios of Toroidal Dupin Cyclides
本文研究矩形Clifford环面的等周比，发现其闭式公式且函数严格递增，但等周比不能唯一确定其欧几里得形状，不同于方形情况。
Yau's conjecture for the stacked Clifford tori of Wiygul
本文证明Wiygul堆叠Clifford环面的Yau猜想，通过反射引理和谱分析，证明其第一Laplace特征值为2，推广了粘合构造的验证方法。
In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Σ$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $Σ$, a minimal surface $\breve{M}$ resembling two copies of $Σ$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $φ$ on $Σ$ is a singular solution of the Jacobi equation with logarithmic singularities which in the constru…
Uniqueness of embedded minimal Lagrangian tori in $\mathbb{CP}^2$
本文证明复射影平面中任意嵌入极小拉格朗日环面必全等于克利福德环面，方法为利用五维球面归一化水平提升构造两点函数。
Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell m^2(N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by $k\ell m^2 (N-1)$ small catenoidal tunnels, with $k \ell m^2$ tunnels join…
Minimal hypersurfaces in $\mathbb{S}^{4}(1)$ by doubling the equatorial $\mathbb{S}^{3}$
本文利用PDE粘合与线性化加倍方法，在$\mathbb{S}^4(1)$中构造加倍赤道$\mathbb{S}^3$的极小超曲面，并在$\mathbb{R}^4$中构造加倍球面自收缩子的平均曲率流自收缩子，回答了Yau与Hsiang的长期问题。
We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .
Ejiri's torus in $S^5$ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in $S^{n}$ by reducing them into elastic curves in $S^3$, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in $S^5$ attains the minimum $2π^2\sqrt{3}$. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also sho…
In this note, we generalize a characterization of the Clifford torus due to Ros. Let $f:M\rightarrow S^{n+1}$ be an embedded closed minimal hypersurface. Assume there are $(n+2)$ great hyperspheres of $S^{n+1}$ perpendicular to each other, such that $M$ is symmetric with respect to them. Let $S$ denote the square of the length of the second fundamental form of $f$ and let $\bar S=\frac{1}{Vol(M)}\int_{M} Sd M$ be the average of $S$. Then $\bar S\geq n$ with equality holding if and only if $f$ is the Clifford torus $C_{m,n-m}$. It can be rewritten as a Simons' type theorem: If $0\leq \int_M (n-…
On mean curvature flow solitons in the sphere
本文研究了球面上均曲率流孤立子，构造了非极小完备例子并证明了相关收缩定理。
We show that the number of genus $g$ embedded minimal surfaces in $\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as $g\rightarrow\infty$, and arise from a two-parameter min-max scheme in lens spaces. More generally, by stabilizing and flipping Heegaard foliations we produce index at most $2$ minimal surfaces with controlled topological type in arbitrary Riemannian three-manifolds.
We study conformally flat hypersurfaces $f\colon M^{3} \to \Q^{4}(c)$ with three distinct principal curvatures and constant mean curvature $H$ in a space form with constant sectional curvature $c$. First we extend a theorem due to Defever when $c=0$ and show that there is no such hypersurface if $H\neq 0$. Our main results are for the minimal case $H=0$. If $c\neq 0$, we prove that if $f\colon M^{3} \to \Q^{4}(c)$ is a minimal conformally flat hypersurface with three distinct principal curvatures then $f(M^3)$ is an open subset of a generalized cone over a Clifford torus in an umbilical hypers…
In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minimal surface or an inverted Chen's minimal graph must be close to these surfaces in the $W^{2,2}$-norm. Moreover, we apply these results to prove a corresponding rigidity result for complete, connected and non-compact surfaces.
On the index of minimal 2-tori in the 4-sphere
本文研究球面中极小2维环面的Morse指数问题，证明了4维球面中极小2维环面指数下界为6及等号成立条件，还给出Hopf微分消失的n维球面中极小2维环面的指数下界与紧例。
In this paper we introduce the notion of contact angle. We deduce formulas for Laplacian and Gaussian curvature of a minimal surface in $S^{2n+1}$ and give a characterization of the generalized Clifford Torus as the only non-legendrian minimal surface in $S^5$ with constant Contact and Kaehler angles.
Using Takahashi theorem we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spectral properties of the metrics on these surfaces are investigated. These metrics include i) both metrics extremal for the first non-trivial eigenvalue on the torus, i.e. the metric on the Clifford torus and the metric on the equilateral torus and ii) the metric maximal for the fir…
We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.
In this paper, we study $n$-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an $n$-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with $f_3$ constant is isometric to the totally geodesic sphere or the Clifford torus if $S\leq 1.8252 n-0.712898$, where $S$ denotes the squared norm of the second fundamental form of this hypersurface.
On the Rigidity of Closed CMC Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar Curvature
该研究证明$\boldsymbol{\text{S}^5(1)}$中具有常数量曲率的闭CMC超曲面若某点仅两个主曲率则为等参Clifford环面，加Willmore条件后可完整分类为几类等参超曲面，采用无迹张量恒等式等方法完成证明。
Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $τ_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ of tori and Klein bottles minimally immersed in spheres. It was remarked that this family includes surfaces carrying all extremal metrics for the first non-trivial eigenvalue of the Laplace-Beltrami operator on the torus and on the Klein bottle: the Clifford torus, the equilateral torus and surprisingly the bipolar Lawson Klein bottle $\tildeτ_{3,1}$. In the present paper …
On the Willmore energy of flat $n$-tori in $\mathbb{R}^N$ and Chen's conjecture for $n$-tori
本文证明了平坦 $n$-环面 Willmore 能量的尖锐下界，并确认 Clifford $n$-环面为唯一极小化子，从而证实 Chen 猜想；但 $n\geq3$ 时该猜想对一般浸入环面不成立。
Using a new estimate for the Peng-Terng invariant and the multiple-parameter method, we verify a rigidity theorem on the stronger version of Chern Conjecture for minimal hypersurfaces in spheres. More precisely, we prove that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfies $0\leq S-n\leq\frac{n}{18}$, then $S\equiv n$ and $M$ is a Clifford torus.
The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a 4-sphere by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for the Lawson's {3,1}-torus. We conjecture that an extremal metric …
We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global behavior may be periodic or the curve may be dense in a Clifford torus embedded in the three-sphere. This behavior is very different from that of helices in three-dimensional Euclidean space, which also have constant curvature and torsion.
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mean curvature surfaces in S^3(a), 1/3 <= a < 1 proving that they are spheres or the minimal Clifford torus in S^3(1/3). This allows to solve the isoperimetric problem in these Berger spheres.
Using the Lawson's existence theorem of minimal surfaces and the symmetries of the Hopf fibration, we will construct symmetric embedded closed minimal surfaces in the three dimensional sphere. These surfaces contain the Clifford torus, the Lawson's minimal surfaces, and seven new minimal surfaces with genera 9, 25, 49, 121, 121, 361 and 841. We will also discuss the relation between such surfaces and the maximal extendable group actions on subsurfaces of the three dimensional sphere.
A new family of minimal surfaces of even genus in the three-dimensional sphere
本研究在三维球面中发现一族含低亏格的新嵌入极小曲面，通过融合赤道球面与Clifford环面构建新型扫描并结合等变极小极大方法证明存在性，确定其对称群和Morse指数下界，证实其几何上异于已知曲面。
We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when the surface is orientable, and with Euler characteristic greater than or equal to -1, when the surface is nonorientable. Also we characterize RP^2 in CP^2 as the least possible index minimal Lagrangian compact nonorientable surface of CP^2.
Minimal Hypersurfaces with constant scalar curvature in $\mathbf{S}^6$
本文提出了一种假设，即在S^6中闭合的极小超曲面M^5的主曲率满足某些条件时，若函数S,f_3,f_4为常数，则该超曲面是等参曲面。该结果去除了Tang和Yan的非负标量曲率假设。作为刚性结果，如果M^5在S^6中有一个恰好有两个不同主曲率的点，则它必须是克利福德扭棱镜。
Free boundary and capillary minimal surfaces in spherical caps II: Low energy
本文研究球冠中自由边界和毛细极小曲面的几何性质，通过单调性公式揭示其能量最大化特性，并在半球中提出与克利福德 torus 相似的特征刻画。
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