Luis J. Alias, Paolo Piccione
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.
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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 comment on full discretized isothermic tori in Euclidean spaces
本文基于周期性离散正交系统，利用达布变换及其可交换性，构建了全离散的k维等温 тор ，并探讨其在n维欧几里得空间中的性质。
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, …
We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.
Gromov's tori are optimal
本文证明大维数欧氏球中浸入 n-环面的法曲率存在最优界，从而表明 Gromov 环面在该意义下达到最优。
This is the second part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, I_F. (See math.DG/0111313 for part I). Having constructed I_F and outlined a proof of its invariance based on bifurcation analysis in part I, in this part we prove a series of gluing theorems to confirm the bifurcation behavior predicted in part I. These gluing theorems are different from (and much harder than) the more conventional versions in that they deal with broken trajectories or broken orbits connected at degenerate rest points. The issues of orientation and signs are also se…
We give three lower bounds for the Morse index of a constant mean curvature torus in Euclidean 3-space in terms of its spectral genus g. The first two lower bounds grow linearly in g and are stronger for smaller values of g, while the third grows quadratically in g but is weaker for smaller values of g.
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 this paper the generic bifurcations of the Minkowski symmetry set for 1-parameter families of plane curves are classified and the necessary and sufficient geometric criteria for each type are given. The Minkowski symmetry set is an analogue of the standard Euclidean symmetry set, and is defined to be the locus of centres of all its bitangent pseudo-circles. It is shown that the list of possible bifurcation types are different to those that occur in the list of possible types for the Euclidean symmetry set.
The Willmore conjecture states that any immersion F:T^2 -> R^n of a 2-torus into flat euclidean space satisfies $\int_{T^2} H^2\geq 2π^2$. We prove it under the condition that the L^p-norm of the Gaussian curvature is sufficiently small.
We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not contain the rotational vector field in its tangent space, then mean curvature flow exists for all time and converges to a flat cross-section as $t \rightarrow \infty$.
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。
The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend the results on special Lagrangian submanifolds on $\mathbb C^n$ to the toric Calabi-Yau cones over Sasaki-Einstein manifolds.
We prove that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere is either of "finite type", that is, has a spectral curve of finite genus, or is of "holomorphic type" which means that it is super conformal or Euclidean minimal with planar ends. This implies that all constrained Willmore tori in the 4-sphere can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin's approach to the study of harmonic tori in the 3-sphere.
Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surface…
An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
通过基本矩阵特征值，建立了欧氏球面中紧致极小子流形的最优刚性定理，不限制余维数，并给出广义克利福德环面和韦罗内塞流形的新刻画。
Extrinsically Symmetric Spaces, Submanifolds of Clifford Type and a Theorem of Harish-Chandra
本文证明欧氏空间中紧致内在对称子流形为外在对称当且仅当其极大环面为Clifford环面，并以此给出Harish-Chandra定理的几何证明。
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。
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})…
Isothermic tori with one family of planar curvature lines and area constrained hyperbolic elastica
本文分类了具有一族平面曲率线的等温环面，给出显式theta函数公式，并揭示其与面积约束双曲弹性曲线的联系，为构造紧致Bonnet对提供基础。
We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and ${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real $n$-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.
Compact Bonnet Pairs: isometric tori with the same curvatures
我们构造了第一对紧致 Bonnet 环面，它们由保持平均曲率的等距变换联系，解决了度量与平均曲率函数是否唯一决定紧致浸入的长期问题，并证明了实解析性。构造基于等温曲面与共形变换，源于离散微分几何的计算研究。
Constant mean curvature (CMC) tori in Euclidean 3-space are described by an algebraic curve, called the spectral curve, together with a line bundle on this curve and a point on $ S ^ 1 $, called the Sym point. For a given spectral curve the possible choices of line bundle and Sym point are easily described. The space of spectral curves of tori is totally disconnected. Hence to characterise the "moduli space" of CMC tori one should, for each genus $g$, determine the closure $\overline{\mathcal{P}^g}$ of spectral curves of CMC tori within the spectral curves of CMC planes having spectral genus $…
We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-γ}}$ for $|X|$ large, when $A<0$ and $γ\in(0,2)$. Such surfaces are close to sections of unduloids with small necksize, folded along circumferences centered at the origin and with larger and larger radii. The construction involves a deep study of the corresponding Jacobi operators, an application of the Lyapunov-Schmidt reduction method and some variational argument.
The rigidity of minimal Legendrian submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
本文利用基本矩阵特征值与 Lu 不等式，建立单位球中各维紧致极小 Legendrian 子流形的最优夹挤与刚性定理，并刻画 Calabi 环面。
A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8π$. Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution …
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