Pascal Redou
Let ${\mathcal F}_λ(\mathbb{S}^n)$ be the space of tensor densities on $\mathbb{S}^n$ of degree $λ$. We consider this space as an induced module of the nonunitary spherical series of the group $\mathrm{SO}_0(n+1,1)$ and classify $(\mathrm{so}(n+1,1),\mathrm{SO}(n+1))$-sim$unitary submodules of ${\mathcal F}_λ(\mathbb{S}^n)$ as a function of $λ$.
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Conserved quantities of distinguished curves on conformal sphere
本文研究了共形球面上共形圆的两种推广，分别基于特殊平行tractor和共形Mercator方程，通过Lagrangian形式主义计算守恒量，并揭示了两者守恒量之间的关系。
In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space.
We give a complete classification of conformally covariant differential operators between the spaces of $i$-forms on the sphere $S^n$ and $j$-forms on the totally geodesic hypersphere $S^{n-1}$. Moreover, we find explicit formulæ for these new matrix-valued operators in the flat coordinates in terms of basic operators in differential geometry and classical orthogonal polynomials. We also establish matrix-valued factorization identities among all possible combinations of conformally covariant differential operators. The main machinery of the proof is the "F-method" based on the "algebraic Fouri…
For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.
In this paper we give bounds for the first eigenvalue of the conformal Laplacian and the Yamabe invariant of a compact Riemannian manifold, by using conditions on the Ricci curvature and the diameter and deduce certain conditions on the manifold to be conformal to a sphere.
We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.
For a surface in 3-sphere, by identifying the conformal round 3-sphere as the projectivized positive light cone in Minkowski 5-spacetime, we use the conformal Gauss map and the conformal transform to construct the associate homogeneous 4-surface in Minkowski 5-spacetime. We then derive the local fundamental theorem for a surface in conformal round 3-sphere from that of the associate 4-surface in Minkowski 5-spacetime. More importantly, following the idea of Fefferman and Graham, we construct local scalar invariants for a surface in conformal round 3-sphere. One distinct feature of our construc…
Conformal stereographic projections of sphere quotients are Majumdar-Papapetrou manifolds
本文通过计算球面商空间标准度量的共形球极投影，证明其结果为 Majumdar-Papapetrou 度量，揭示了这类流形之间的具体几何联系。
Singular Solutions for the Conformal Dirac-Einstein Problem on the Sphere
本文研究了共形 Dirac-Einstein 系统的奇异解存在性，利用其共形不变性，在三维球面上构造了一族恰好有两个奇点的奇异解。
We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $σ\in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math. 187, no. 2, 395-506, 2012) for the scalar curvature flow on the standard unit sphere.
On the stability of minimal submanifolds in conformal spheres
本文证明了在共形且δ-夹挤的n维球面中，不存在维数在2到n-δ^{-1}之间的闭稳定极小子流形。
This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into quaternions, which extends the classical complex function theory on Riemann surfaces. The first results along these lines were presented at the ICM 98 in Berlin. An important new invariant of the quaternionic holomorphic theory is the Willmore energy. For quaternionic holomorphic curves in the …
In this paper we relate the geometric Poisson brackets on the Grassmannian of 2-planes in R^4 and on the (2,2) Moebius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Moebius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results into either a decoupled system or a complexly coupled system of KdV…
In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constrain on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the sphere which are invariant by a $k-$parameter subgroup of conformal diffeomorphisms of the sphere, giving a bound on its maximum dimension. Moreover, we classify conformal metrics on the sphere whose eigenvalues of the Shouten tensor are all constant (we call them \emph{isoparametric …
In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a priori estimates in dimension 5 and 6. On the five dimensionl sphere these a priori estimates combined with the perturbation result of Part I allow us to obtain some existence results. On the six dimensional sphere we prove the existence of at least one solution when an associated index is different from zero.
In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree.
Let ${\cal F}_λ$ be the space of tensor densities on ${\bf R}^n$ of degree $λ$ (or, equivalently, of conformal densities of degree $-λn$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\cal F}_λ\otimes{\cal F}_μ$ to~${\cal F}_ν$. The classification of linear $so(p+1,q+1)$-invariant differential operators from ${\cal F}_λ$ to ${\cal F}_μ$ already known in the literature is obtained in a different manner.
We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results.
A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic contact manifold is locally quaternionic contact conformal to the standard flat quaternionic contact structure on the quaternionic Heisenberg group, or equivalently, to the standard 3-sasakian structure o…
We consider the conformal class of the Riemannian product $g_0 + g$, where $g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect to the square root of the scalar curvature of $g$. This is obtained by studying radial solutions of the equation $Δu -λu + λu^p =0$ on $S^m$, and the number of solutions in terms of $λ$.
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。
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。
In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. Our results extend the classifications of Escobar in \cite{E0} when $n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.
The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in $\mathbb R^{n+2}$, isotropic surfaces in $S^4$ and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in $S^6$ without dual surfaces is also presented.
We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type operators parameterized over the complexified dual torus. The kernel line bundle of this family over the spectral curve describes the generalized Darboux transforms of the conformally immersed torus. If the spectral curve has finite genus the kernel bundle can be extended to the compa…
The conformal anomalies and functional determinants of the Branson--GJMS operators, P_{2k}, on the d-dimensional sphere are evaluated in explicit terms for any d and k such that k < d/2+1 (if d is even). The determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly, which vanishes for odd d. Taking the mode system on the sphere as the union of Neumann and Dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non--existence of `super--critical' operators, 2k>d for even d. Significant use is made of the …
In this paper we extend some well-known rigidity results for conformal changes of Einstein metrics to the class of generalized quasi-Einstein (GQE) metrics, which includes gradient Ricci solitons. In order to do so, we introduce the notions of conformal diffeomorphisms and vector fields that preserve a GQE structure. We show that a complete GQE metric admits a structure-preserving, non-homothetic complete conformal vector field if and only if it is a round sphere. We also classify the structure-preserving conformal diffeomorphisms. In the compact case, if a GQE metric admits a structure-preser…
A note on the conformal geometry of the 2-sphere
研究二维球面共形几何，通过特定对称无迹张量导出线性二阶偏微分方程，其解空间有闵可夫斯基结构，存在无穷多相关空间，且二维球面能规范等距嵌入闵可夫斯基空间。
In this note we prove that a fourth order conformal invariant on the product of a circle with an (n-1)-dimensional sphere can be arbitrarily close to that of the n-dimensional sphere, generalizing a result of Schoen about the classical Yamabe invariant.
Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions
该研究证明3至10维中接近球面圆度量的共形类里，共形拉普拉斯第二共形特征值无极小值子，与11维及以上的存在性结果形成对比，还推广了低维Yamabe方程变号解能量下界的结论。
We construct pairs of conformally equivalent isospectral Riemannian metrics $ϕ_1 g$ and $ϕ_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres and balls, the metric $g$ is not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds $(M,g)$ we also show that the functions $ϕ_1$ and $ϕ_2$ are isospectral potentials for the Schrödinger operator $\hbar^2Δ+ϕ$. To our knowledge, thes…
Geometry of conformal minimal two-spheres immersed in $G(2,6;\mathbb{R})$ is studied in this paper by harmonic maps. We construct a non-homogeneous constant curved minimal two-sphere in $G(2,6;\mathbb{R})$, and give a classification theorem of linearly full conformal minimal immersions of constant curvature from $S^2$ to $G(2,6;\mathbb{R})$, or equivalently, a complex hyperquadric $Q_{4}$, which illustrates minimal two-spheres of constant curvature in $Q_{4}$ are in general not congruent.
We will show that in the conformal class of the standard metric $g_{S^n}$ on $S^n$, the scaling invariant functional $(μ_g(S^n))^{\frac{2m-n}{n}}\int_{S^n}Q_{2m,g}dμ_g$ maximizes at $g_{S^n}$ when $n$ is odd and $m=\frac{n+1}{2}$ or $\frac{n+3}{2}$. For $n$ odd and $m\geq\frac{n+5}{2}$, $g_{S^n}$ is not stable and the functional has no local maximizer. Here $Q_{2m,g}$ is the $2m$th order $Q $-curvature.
The coefficient of the logarithmic term in the entropy on even spheres is re-computed by the local technique of integrating the finite temperature energy density up to the horizon on static d--dimensional de Sitter space and thence finding the entropy by thermodynamics. Numeric evaluation yields the known answer i.e. (minus) the conformal anomaly on the d-sphere. The de Sitter quantities are obtained by conformal transformation of the Rindler ones, themselves obtained, for convenience, from those around a cosmic string. The expressions are given in terms of generalised Bernoulli polynomials fo…
The conformal logarithmic Laplacian on the sphere: Yamabe-type problems and Sobolev spaces
该论文研究球面上的共形对数拉普拉斯算子，建立其与ℝ^N中对数拉普拉斯算子的联系，引入对应希尔伯特空间，衔接Yamabe型问题分类结果并推广至弱情形。
We are interested in the study of the space of $n$-ary differential operators denoted by $\mathfrak{D}_{\underlineł,μ}$ where $\underlineł=(ł_{1},...,ł_{n})$ acting on weighted densities from $\frak F_{ł_1}\otimes\frak F_{ł_2}\otimes...\otimes\frak F_{ł_n}$ to $\frak F_μ$ as a module over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$. As a consequence, we prove the existence and the uniqueness of a canonical conformally equivariant symbol map from $\mathfrak{D}_{\underlineλ,μ}^k$ to the corresponding space of symbols as well for the explicit expression of the associated quantization m…
A Willmore surface $y:M\rightarrow S^{n+2}$ has a natural harmonic oriented conformal Gauss map $Gr_y:M\rightarrow SO^{+}(1,n+3)/SO(1,3)\times SO(n)$, which maps each point $p\in M$ to its oriented mean curvature 2-sphere at $p$. An easy observation shows that all conformal Gauss maps of Willmore surfaces satisfy a restricted nilpotency condition which will be called "strongly conformally harmonic." The goal of this paper is to characterize those strongly conformally harmonic maps from a Riemann surface $M$ to $SO^+ (1, n + 3)/{SO^+(1, 3) \times SO(n) }$ which are the conformal Gauss maps of s…
In this paper we investigate the nature of stationary points of functionals on the space of Riemannian metrics on a smooth compact manifold. Special cases are spectral invariants associated with Laplace or Dirac operators such as functional determinants, and the total Q-curvature. When the functional is invariant under conformal changes of the metric, and the manifold is the standard n-sphere, we apply methods from representation theory to give a universal form of the Hessian of the functional at a stationary point. This reveals a very strong rigidity in the local structure of any such functio…
Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constant mean curvature (CMC) surface or an index one minimal surface. We prove that $Σ$ is homeomorphic either to a sphere or to a torus. Moreover, in case $Σ$ is homeomorphic to a torus, then it is embedded, minimal, conformal to a flat square torus and Ric$(N)=0$ where $N$ is a unit …
Some global properties of umbilic points of Willmore immersions in the $3$-sphere
该文从共形高斯映射视角研究3-球面中余维1的Willmore曲面脐点，刻画了脐曲线经共形变换为测地线的等价条件，证明含脐曲线长度的共形高斯映射Gauss–Bonnet公式，并将其作为各空间形式中共形极小曲面Willmore能量表达式的统一版本。
Legendrian submanifolds in the unit sphere with conformal Maslov form and constant sectional curvature
本文分类了单位球面中具有共形Maslov形式和常截面曲率的Legendrian子流形，并将Dillen-Vrancken夹逼定理从极小情形推广到共形Maslov类，证明在曲率界0≤sec≤1下仅有两类闭子流形。
It is shown that analytic conformal submersions of $S^3$ are given by intersections of (not necessary closed) complex surfaces with a quadratic real hyper-surface in $\mathbb{C}P^3.$ A new description of the space of circles in the 3-sphere in terms of a natural bilinear form on the tangent sphere bundle of $S^3$ is given. As an application it is shown that every conformal fibration of $S^3$ by circles is the Hopf fibration up to conformal transformations.
The classification of Willmore 2-spheres in the $n$-dimensional sphere $S^n$ is a long-standing problem, solved only when $n=3,4$ by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when $n=5$. There are three types of such surfaces up to Möbius transformations: (1) super-conformal surfaces in $S^4$; (2) minimal surfaces in $R^5$; (3) adjoint transforms of super-conformal minimal surfaces in $R^5$. In particular, Willmore surfaces in the third class are not S-Willmore (i.e., without a dual Willmore surface). To show the existence of Willmore 2-spheres in $…
Let $S$ be a surface with a metric $d$ satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into $(S,d)$ is a branched covering. As a consequence, if $(S,d)$ is homeomorphically equivalent to the 2-sphere $\mathbb S^2$, then it is conformally equivalent to $\mathbb S^2$.
We consider a closed orientable Riemannian 3-manifold $(M,g)$ and a vector field $X$ with unit norm whose integral curves are geodesics of $g$. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of $g$. We study when this 2-plane bundle remains invariant under two natural almost complex structures. We also provide a geometric condition that ensures that $X$ is the Reeb vector field of the 1-form $λ$ obtained by contracting $g$ with $X$. We apply these results to the case of great circle flows on the 3-sphere with two obj…
We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold $(M,\rg)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on $T^*M$ and of differential operators on tensor densities over $M$, both viewed as modules over the Lie algebra $\so(p+1,q+1)$ where $p+q=\dim(M)$. This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In t…
A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishe…
For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of $\mathbb{C}^{n+1}$. In particular, the twistor space $\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic…
How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as the conformal class changes? In this paper, we investigate these and other related questions, focusing on the context of Simon-Smith min-max theory.
Fast ellipsoidal conformal and quasi-conformal parameterization of genus-0 closed surfaces
本文提出一种新框架，通过将共形变换与拟共形映射结合，计算亏格0闭曲面的椭球共形与拟共形参数化，以减少球面映射的几何畸变，并利用拟共形理论保证双射性。
We consider the space of tensor densities on the n-dimensional sphere with degree lambda (or, equivalently, of conformal densities with degree lambda). This space is a module over the group of diffeomorphisms, and consequently over the Lie algebra of vector fields, on the sphere, and we first prove that as a module over the conformal group, it is infinitesimally equivalent to an induced module from the principal series, precisely from the nonunitary spherical series on the space of smooth functions on the sphere. In function of lambda,(g,K)-simple and unitary submodules of the space of tensor …
Let M be a compact Riemannian manifold and let $μ$,d be the associated measure and distance on M. Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M -> M pushing forward $μ$ to a measure $ν$: each S factors uniquely a.e. into the composition S = T \circ U, where U : M -> M is volume preserving and T : M -> M is the optimal map transporting $μ$ to $ν$ with respect to the cost function d^2/2. In this article we study the polar factorization of conformal and projective maps of the sphere S^n. For conformal maps, which m…
A Conformal Quasi Einstein Characterization Of The Round Sphere
本文研究闭 m-拟 Einstein 流形上共形向量场与 Killing 向量场的关系，补全 m=-2 情形并证明非 Killing 共形场可将流形刻画为球面，同时给出 Bourguignon-Ezin 恒等式的直接积分证明。
In this paper we present some extensions of the celebrated finite point conformal compactification theorem of Huber \cite{Hu57} for complete open surfaces to general dimensions based on the n-Laplace equations in conformal geometry. We are able to conclude a domain in the round sphere has to be the sphere deleted finitely many points if it can be endowed with a complete conformal metric with the negative part of the smallest Ricci curvature satisfying some integrable conditions. Our proof is based on the strengthened version of the Arsove-Huber's type theorem on n-superharmonic functions in ou…
We define two transforms between non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between non-conformal harmonic maps into the 3-sphere, $H$-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold $S^3\times S^3$. As a consequence we can construct sequences of $H$-surfaces and almost complex surfaces.
We prove that a minimizer of the Yamabe functional does not exist for a sphere $\mathbb{S}^n$ of dimension $n \geq 3$, endowed with a standard edge-cone spherical metric of cone angle greater than or equal to $4π$, along a great circle of codimension two. When the cone angle along the singularity is smaller than $2π$, the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.
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.
We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local rigidity type phenomena are obtained for 1-parameter families of U(n+1), Sp(n+1) and Spin(9)-homogeneous metrics.
Conformal properties of spheres
本文将流形度量、共形类分别对应到高维标准球面的等距嵌入与共形同痕嵌入类，证明相关丛的同伦提升性质，结合Simons间隙定理研究球面近复结构，计算多类流形的共形不变量并做分类整理。
The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\R^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\R^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\R^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\R^{1,n+3}$, contains a fixed lightlik…
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