E. Loubeau, C. Oniciuc
Biharmonic maps are the critical points of the bienergy functional and generalise harmonic maps. We investigate the index of a class of biharmonic maps, derived from minimal Riemannian immersions into spheres. This study is motivated by three families of examples: the totally geodesic inclusion of spheres, the Veronese map and the Clifford torus.
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In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.
We study biharmonic maps and f-biharmonic maps from a round sphere $(S^2, g_0)$, the latter maps are equivalent to biharmonic maps from Riemann spheres $(S^2, f^{-1}g_0)$. We proved that for rotationally symmetric maps between rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to a 2nd order linear ordinary differential equation. As applications, we give a method to produce biharmonic maps and f-biharmonic maps from given biharmonic maps and we construct many examples of biharmonic and f-biharmonic maps from a round sphere $S^2$ and between two round spheres. Our exam…
We study properties of non-minimal biharmonic hypersurfaces of spheres. The main result is a CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of rotationally symmetric maps between 2-spheres. We also find many examples of proper biharmonic maps defined locally on a 2-sphere. Our results seem to suggest that any biharmonic map $S^2\longrightarrow (N^n, h)$ be a weakly conformal immersion.
We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.
We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.
In this note we improve a gap result concerning the range of the mean curvature of complete $CMC$ proper-biharmonic hypersurfaces in unit Euclidean spheres.
In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.
New constructions of biharmonic polynomial maps between spheres
本文研究球面间由两个齐次多项式映射构成的对角映射，通过计算双张力场，提出了利用不同次数调和齐次多项式映射构造真双调和映射及乘积映射的新方法。
We consider proper-biharmonic flat tori with constant mean curvature (CMC) in spheres and find necessary and sufficient conditions for certain rectangular tori and square tori to admit full CMC proper-biharmonic immersions in $\mathbb{S}^n$, as well as the explicit expressions of some of these immersions.
In this note, we classify biharmonic submanifolds in a sphere defined by bi-eigenmaps ($Δ^2 ϕ=λϕ$) or buckling eigenmaps ($Δ^2 ϕ=-μΔϕ$). We then classify biharmonic bi-eigenmaps and buckling eigenmaps into spheres with constant energy density. The results can be viewed as generalizations of Takahashi's characterization of minimal submanifolds in a sphere by eigenmaps.
We obtain the parametric equations of all biharmonic Legendre curves and Hopf cylinders in the 3-dimensional unit sphere endowed with the modified Sasakian structure defined by Tanno.
We construct new explicit proper r-harmonic functions on the standard n-dimensional sphere S^n and hyperbolic space H^n for any r\ge 1 and n\ge 2.
Gap results for biharmonic submanifolds in spheres
本文研究了球面中具有平行平均曲率向量场的真双调和子流形，确定了平均曲率的更大间隙，并在边界情形下得到了分裂结果。
CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in $\sn^5$ with $|H|=h$, while a necessary and sufficient condition on $h$ is found for the existence of CMC proper-biharmonic tori in $\sn^5$.
Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree $\pm 1$. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. In this paper, we obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. Ou…
In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat $2$-dimensional torus $\mathbb{T}$ into the $3$-dimensional unit Euclidean sphere $\mathbb{S}^3$. Then, for the special case of maps between spheres whose components are given by homogeneous polynomials of the same degree, we find a more specific form for their bitension field. Further, we apply this formula to the case when the degree is $2$, and we obtain the classification of all proper biharmonic quadratic forms from $…
We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces $\mathbb S^{n+1}$ are either the hypersphere $\mathbb S^n(1/\sqrt2)$ or the Clifford hypersurface $\mathbb S^{n_1}(1/\sqrt2)\times\mathbb S^{n_2}(1/\sqrt2)$ with $n_1+n_2=n$ and $n_1\neq n_2$. Moreover, we also show that there does not exi…
The energy density of biharmonic quadratic maps between spheres
本文研究球面间二次映射，证明非调和双调和等价于恒定能量密度 (m+1)/2，并解决其结构公开问题，进而得到相应分类结果。
We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in $\mathbb{S}^n$. Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in $\mathbb{S}^n$, their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic subma…
In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen's conjecture on biharmonic submanifolds.
Biharmonic conformal immersions into a 3-dimensional conformally flat space
本文研究曲面到共形平坦3-空间的双调和共形浸入，给出全脐曲面的刻画与构造方法，分类一族共形浸入中的双调和映射，并构造包括Hopf柱面在内的大量新例子。
We classify the biharmonic Legendre curves in a Sasakian space form, and obtain their explicit parametric equations in the $(2n+1)$-dimensional unit sphere endowed with the canonical and deformed Sasakian structures defined by Tanno. Then, composing with the flow of the Reeb vector field, we transform a biharmonic integral submanifold into a biharmonic anti-invariant submanifold. Using this method we obtain new examples of biharmonic submanifolds in spheres and, in particular, in $\mathbb{S}^{7}$.
Remarks on constructing biharmonic and conformal biharmonic maps to spheres
本文提出一种几何算法，用于构造双调和与共形双调和映射到球面，并通过实例展示算法在不同域下的灵活性及方法的有效性。
We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non harmonic and indexed-biharmonic (i.e. proper indexed-biharmonic). We give some pseudo-umbilical and non pseudo-umbilical examples and we study the link between the proper indexed-biharmonic submanifolds of $\mathbb{S}^n$ and the proper conformal-harmonic submanifolds of $\mathbb{S}^n$.
We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb R^4$.
We consider closed biharmonic hypersurfaces in the Euclidean sphere and prove a rigidity result under a suitable condition on the scalar curvature. Moreover, we establish an integral formula involving the position vector for biharmonic hypersurfaces in space forms.
More weakly biharmonic maps from the ball to the sphere
本文证明了维数大于4的欧几里得球与适当维数的欧几里得球面间存在两个真双调和映射，且低维时这两个映射是双能量的不稳定临界点。
A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.
We give several construction methods and use them to produce many examples of proper biharmonic maps including biharmonic tori of any dimension in Euclidean spheres (Theorem 2.2, Corollaries 2.3, 2.4, and 2.6), biharmonic maps between spheres (Theorem 2.9) and into spheres (Theorem 2.10) via orthogonal multiplications and eigenmaps. We also study biharmonic graphs of maps, derive the equation for a function whose graph is a biharmonic hypersurface in a Euclidean space, and give an equivalent formulation of Chen's conjecture on biharmonic hypersurfaces by using the biahrmonic graph equation (Th…
Energy identity for stationary biharmonic mappings into spheres in supercritical dimensions
本文在超临界维数n≥5中，通过改进Lin-Rivière的方法，建立了驻定双调和映射到球面的能量恒等式。
Stable proper biharmonic maps in Euclidean spheres
本文构造了从单位球到欧几里得球面的首个显式稳定真弱双调和映射族，通过建立双能量第二变分公式并分析特定映射的稳定性实现。
Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $ψ:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $ϕ:\s^3\to \s^3$. We show $ϕ$ to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.
Index and nullity of proper biharmonic maps in spheres
本文研究真双调和映射的第二变分，计算若干已知例子的精确指数与零化度，证明非紧情形下一族映射的紧支集变分严格稳定性，并引入等变变分指数与零化度。
Biharmonic functions and bi-eigenfunctions on some model spaces
该研究针对球面上的双拉普拉斯算子给出简便公式与本征相关结果，推导了一类模型空间上双调和方程径向解的积分形式，证明其存在真双调和函数，给出空间形式上的显式例子，并完成穿孔欧氏空间上正拉普拉斯真双调和函数的分类。
On conformal biharmonic maps and hypersurfaces
本文引入共形双能量泛函，研究其临界点共形双调和映射，分类了球面中共形双调和超球面和广义Clifford环面，并发现双曲空间也存在共形双调和超曲面，最后计算了超球面的稳定性指标。
Conformal-biharmonic hypersurfaces in spheres and product spaces
研究了球面和乘积空间中具有常主曲率的共形双曲面的性质及分类
We prove that the problem of constructing biharmonic conformal maps on a $4$-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition, we characterize all solutions on Euclidean 4-space and show that there exists at least one non-constant proper biharmonic conformal map from any closed Einstein 4-manifold of negative Ricci curvature.
Biharmonic maps between surfaces are studied in this paper. We compute the bitension field of a map between surfaces with conformal metrics in complex coordinates. As applications, we show that a linear map from Euclidean plane into $(\mathbb{R}^2, σ^2dwd\bar w)$ is always biharmonic if the conformal factor $σ$ is bi-analytic; we construct a family of such $ σ$, and we give a classification of linear biharmonic maps between $2$-spheres. We also study biharmonic maps between surfaces with warped product metrics. This includes a classification of linear biharmonic maps between hyperbolic planes …
On the conformal-biharmonic stability of the identity map of Einstein manifolds
研究爱因斯坦流形恒等映射在共形双能功能下的稳定性，发现其与谐波稳定性指数相同，唯一体育四维欧几里得球面例外，该映射在能量功能下不稳定，在共形双能功能下稳定。
The Second Variational Formula for Statistical Biharmonic Maps
本文提出统计双能量泛函的第二变分公式与稳定性概念，推导了目标为Hessian流形时的曲率表示形式，给出相关重要实例并确定其稳定性。
Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a manifold of non positive curvature are given. Gauge field analogue is shown, and the isolation phenomena of bi-Yang-Mills fields are obtained.
In this article, we prove energy quantization for approximate (intrinsic and extrinsic) biharmonic maps into spheres where the approximate map is in $L \log L$. Moreover, we demonstrate that if the $L\log L$ norm of the approximate maps does not concentrate, the image of the bubbles are connected without necks.
The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two $m$-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case that $m=4$ and the models have constant sectional curvature. Then, by introducing the Hamiltonian associated to this problem, we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is ${\mathbb R}^4$. In the second part of t…
Morse Index Stability of Biharmonic Maps in Critical Dimension
本文将Da Lio-Gianocca-Rivière的Morse稳定性理论推广到临界维数下的内蕴与外蕴双调和映射，提出新方法证明几何分析中一类问题的强能量量子化，统一重现已有结果。
The Nullity of a Family of Proper Biharmonic Maps via Elliptic Curves
研究从平坦二维环面到二维球面的一族恰当双调和映射的零度猜想，通过揭示谱几何与算术几何联系，构造椭圆曲线同构转化问题，确定莫德尔 - 韦伊群，证实猜想并得出该族映射零度为5的结论。
J. Eells and L. Lemaire introduced k-harmonic maps, and Wang Shaobo showed the first variational formula. When, k=2, it is called biharmonic maps (2-harmonic maps). There have been extensive studies in the area. In this paper, We study k-harmonic immersion into a sphere, and get the rerationship between radious and "k" of k-harmonic. And we also consider k-harmonic submersion, and genelarize Oniciuc's results. Futhermore we construct non harmonic k-harmonic by Hopf map.
Making Surfaces Biharmonic by Conformal Reparametrization in Anti-de Sitter Three-Space
研究非退化曲面到三维反德西特空间的双调和共形浸入，通过特定符号约定和相关量表示双调和方程，证明类空曲面相关性质，推导解析系统并证明局部存在性，给出零坐标重构及具体初始数据，记录类时抛物约化。
In this note we investigate the $(E)$-index and the $(E)$-nullity of vacuum solutions from the two-torus into the two sphere.
The Morse index of constant curvature 2-spheres
计算了标准N-球面中所有具有常Gauss曲率的浸入极小2-球面的Morse指标和零化度，并给出了以Boruvka球面为链的R^7中结合锥的稳定性指标界。
We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with respect to the number of the surfaces' bulges, and the index of such tori can be arbitrarily large.
Given a closed manifold of dimension at least three, with non trivial homotopy group π_3(M) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bound one, with sum of their energies realizes a geometric invariant width.
It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface M\subset S^{n+1} with cmc cannot take the values 1,2,3... n.
Orientable minimal submanifolds of low index in the sphere
本文证明了球面中低指标可定向极小子流形的刚性：指标不超过$2m+1$的闭连通$(2m-2)$维极小子流形必为全测地球面。
Morse Index bound of simple closed geodesics on 2-spheres and strong Morse Inequalities
本文利用Morse理论刻画黎曼二维球面上的简单闭测地线，证明在通用度量下存在Morse指标为1、2、3的简单闭测地线，并对可定向黎曼曲面上的简单闭曲线空间建立了长度泛函的强Morse不等式。
The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_τ,\, 0<τ\leq 1$, are studied. Unlike the case of the standard sphere ($τ=1$), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if $τ^2\leq 1/2$. Moreover, there are no stable compact minimal $d$-dimensional submanifolds of $\mathbb{S}^{2n+1}_τ$ when $1 / (d+1) < τ^2 \leq 1$ and the stable ones are classified for $τ^2=1 / (d+1)$ when the submanifold is embedded. Finally, the compact orientable minimal surfaces of $\mathbb{S}^3_τ$ with index …
We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index - the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three-space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either t…
Index of Embedded Networks in the Sphere
本文计算了球面中平稳嵌入网络的Morse指标和零度，关键定理将整个网络的指标与小网络及新定义的Dirichlet-to-Neumann映射关联，并证明了$\mathbb{S}^2$中平稳三重结点网络特征值的性质。
For $n\geq 1$, we exhibit a lower bound for the volume of a unit vector field on $\mathbb{S}^{2n+1}\backslash\{\pm p\}$ depending on the absolute values of its Poincaré indices around $\pm p$. We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.
Suppose that $S^n$ is given a generic Riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures. This pinching condition is satisfied by manifolds whose real sectional curvatures $K_r(σ)$ satisfy $$1/2 < K_r(σ) \leq 1.$$ Then the number of minimal two spheres of Morse index $λ$, for $n-2 \leq λ\leq 2n-5$, is at least $p_{3}(λ-n+2)$, where $p_{3}(k)$ is the number of $k$-cells in the Schubert cell decomposition for $G_3({\mathbb R}^{n+1})$.
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