A. Balmus
We construct a new class of biharmonic maps, which are the critical points for the bienergy functional, by deforming conformally the codomain metric of harmonic Riemannian submersions such that they become nonharmonic but biharmonic.
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This note reviews some of the recent work on biharmonic conformal maps (see \cite{OC}, Chapter 11, for a detailed survey). It will be focused on biharmonic conformal immersions and biharmonic conformal maps between manifolds of the same dimension and their links to isoparametric functions and Yamabe type equations, though biharmonic morphisms (maps that preserve solutions of bi-Laplace equations), generalized harmonic morphisms (maps that pull back germs of harmonic functions to germs of biharmonic functions), and biharmonic conformal and Riemannian submersions will also be touched.
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。
Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particular, explicits the conditions required for a conformal map in dimension four to preserve biharmonicity and helps producing the first example of a biharmonic morphism which is not a special type of harmonic morphism. Then, we compute the bitension field of horizontally weakly confo…
Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answer the question, "what surfaces can be biharmonically conformally immersed into Euclidean 3-space?" We prove that a circular cylinder is the only CMC surface that can be biharmonically conformally immersed into a Euclidean 3-space; We obtain a classification of biharmonic conforma…
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。
In this paper, we present some new properties for p-biharmonic hypersurfaces in Riemannian manifold. We also characterize the p-biharmonic submanifolds in an Einstein space. We construct a new example of proper p-biharmonic hypersurfaces. We present some open problems.
On biharmonic conformal hypersurfaces
本文研究了共形超曲面的双调和性质，证明了在特定条件下共形因子必须为等参数函数，并展示了某些超曲面在特定空间形式中的双调和浸入性。
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.
We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a Möbius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of Möbius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.
We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian manifolds. We are able to characterize harmonic maps and minimal submanifolds by using the concept of f-biharmonic maps and prove that the set of all f-biharmonic maps from 2-dimensional domain is invariant under the conformal change of the metric on the domain. We give an improved equation for f-biharmonic hypersurfaces and use it to prove some rigidity theorems a…
We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions.
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。
This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter family of non-minimal conformal biharmonic immersions of cylinder into $\r^3$ and some examples of conformal biharmonic immersions of 4-dimensional Euclidean space into sphere and hyperbolic space thus provide many simple examples of proper biharmonic maps with rich geometric mea…
Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details). In this paper, we study generalized harmonic morphisms which are defined to be maps between Riemannian manifolds that pull back harmonic functions to biharmonic functions. We obtain some characterizations of generalized harmonic morphisms into a Euclidean space and give two met…
For the reduction ordinary differential equation due to Baird and Kamissoko \cite{BK} for biharmonic maps from a Riemannian manifold $(M^m,g)$ into another one $(N^n,h)$, we show that this ODE has no global positive solution for every $m\geq 5$. On the contrary, we show that there exist global positive solutions in the case $m=3$. As applications, for the the Riemannian product $(M^3,g)$ of the line and a Riemann surface, we construct the new metric $\widetilde{g}$ on $M^3$ conformal to $g$ such that every nontrivial product harmonic map from $M^3$ with respect to the original metric $g$ must …
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。
f-Biharmonic maps are the extrema of the f-bienergy functional. f-biharmonic submanifolds are submanifolds whose defining isometric immersions are f-biharmonic maps. In this paper, we prove that an f-biharmonic map from a compact Riemannian manifold into a non-positively curved manifold with constant f-bienergy density is a harmonic map; any f-biharmonic function on a compact manifold is constant, and that the inversions about $S^m$ for $m\ge 3$ are proper f-biharmonic conformal diffeomorphisms. We derive f-biharmonic submanifolds equations and prove that a surface in a manifold $(N^n, h)$ is …
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 …
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…
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
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。
Biharmonic Maps Between Conformally Compact Manifolds
研究共形紧致流形间双调和映射的非存在性，证明其在特定条件下为调和映射。
We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.
In this note, we generalize biharmonic equation for rotationally symmetric maps ([4], [16], [10]) to equivariant maps between model spaces and use it to give a complete classification of rotationally symmetric conformal biharmonic maps from a $4$-dimensional space form into a $4$-dimensional model space. We also give an improved second variation formula for biharmonic maps into a space form and use it to prove that there exists no stable proper biharmonic maps with constant square norm of tension field from a compact Riemannian manifold without boundary into a space form of positive sectional …
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.
The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are alway…
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…
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 $\mathbb{S}^n$.
Biharmonic Conformal Surfaces and Dirac Factorization I: Exact Spinorial Encoding and Scalar--Chiral Rigidity
本文研究双调和共形曲面，通过构造Laplace型算子将Ou方程组等价为单个Dirac方程，分类标量-手性分解，证明其刚性由Dirac判别式控制，为更广构造提供基线。
$f$-Biharmonic submanifolds in space forms and $f$-biharmonic Riemannian submersions from 3-manifolds
本文研究了空间形式中的$f$-双和曲线与可展曲面等距浸入的分类，并利用可积性数据探讨了源自3-空间形式的黎曼浸没的$f$-双和性，给出了相关实例。
In this paper, we prove that the class of bi-f-harmonic maps and that of f-biharmonic maps from a conformal manifold of dimension not equal to 2 are the same (Theorem 1.1). We also give several results on nonexistence of proper bi-f-harmonic maps and f-biharmonic maps from complete Riemannian manifolds into nonpositively curved Riemannian manifolds. These include: any bi-f-harmonic map from a compact manifold into a non-positively curved manifold is f-harmonic (Theorem 1.6), and any f-biharmonic (respectively, bi-f-harmonic) map with bounded f and bounded f-bienrgy (respectively, bi-f-energy) …
Extrinsic bi-Conformal Heat Flow and its smoothness
本文提出双共形热流并证明其全局光滑性及无有限时间奇异性。
We consider the energy and bienergy functionals as variational problems on the set of Riemannian metrics and present a study of the biharmonic stress-energy tensor. This approach is then applied to characterise weak conformality of the Gauss map of a submanifold. Finally, working at the level of functionals, we recover a result of Weiner linking Willmore surfaces and pseudo-umbilicity.
For $n \geq 5$, we consider positive solutions $u$ of the biharmonic equation \[ Δ^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\} \] with a non-removable singularity at the origin. We show that $|x|^{\frac{n-4}{2}} u$ is a periodic function of $\ln |x|$ and we classify all periodic functions obtained in this way. This result is relevant for the description of the asymptotic behavior near singularities and for the $Q$-curvature problem in conformal geometry.
This paper, in which we develop ideas introduced in \cite{MR}, focuses on \emph{reduction methods} (basically, group actions or, more generally, simmetries) for the bienergy. This type of techniques enable us to produce examples of critical points of the bienergy by reducing the study of the relevant fourth order PDE's system to ODE's. In particular, we shall study rotationally symmetric biharmonic conformal diffeomorphisms between \emph{models}. Next, we will adapt the reduction method to study an ample class of $G-$invariant immersions into the Euclidean space. At present, the known instance…
In the year 1984 Shibata investigated the theory of a change which is called a $ β$-change of a Finsler metric. On the other hand in 1985 a systematic study of geometry of hypersurfaces in Finsler spaces was given by Matsumoto. In the present paper is to devoted to the study of a condition for a Randers conformal change to be projective and find out when a totally geodesic hypersurface $ F^{n-1} $ remains to be a totally geodesic hypersurface $ F^{n-1} $ under the projective Randers conformal change. Further obatined the condition under which a Finslerian hypersurfaces given by the projective …
A conformal change of $TM\oplus T^*M$ is a morphism of the form $(X,α)\mapsto(X,e^τα)$ $(X\in TM,α\in T^*M,τ\in C^\infty(M))$. We characterize the generalized almost complex and almost Hermitian structures that are locally conformal to integrable and to generalized Kähler structures, respectively, and give examples of such structures.
Special anisotropic conformal changes of conic pseudo-Finsler surfaces
本文提出了圆锥伪-Finsler曲面的多种特殊各向异性共形变换，研究了$ϕT$-条件及其等价性，并将理论应用于具有Finslerian球对称性的Schwarzschild-de Sitter解。
The notion of a Douglas space of second kind of a Finsler space with $(α, β)$-metric was introduced by I. Y. Lee [9]. Since then, so many geometers have studied this topic e. g., [14]. In this paper, we prove that a Douglas space of second kind with special $% (α, β)$-metric $α+εβ+ k \frac{β^2}{α}$ is conformally transformed to a Douglas space of second kind. Further, we obtain some results which prove that a Douglas space of second kind with certain $(α, β)$-metrics such as Randers metric, Kropina metric, first approximate Matsumoto metric and Finsler space with square metric is conformally t…
Anisotropic conformal change of conic pseudo-Finsler surfaces, II
本文研究锥伪芬斯勒曲面的各向异性共形变换，建立 Berwald、Landsberg、Douglas 张量及 T-张量的对应关系，并给出黎曼曲面变换为各类特殊曲面及保持 T-条件的充要条件。
Anisotropic conformal change of conic pseudo-Finsler surfaces, I
本文研究圆锥伪Finsler曲面的各向异性共形变换，利用修改的Berwald标架推导出逆度量张量分量的张量形式变换，并探讨了变换后保持伪Finsler性质的条件及几何不变量。
We investigate what we call a conformal $β$ - change in Finsler spaces, namely $$ L(x,y)\to ~^{\ast}L(x,y)=e^{σ(x)}L(x,y)+β(x,y)$$ where~$σ~$ is a function of $x~ only ~and ~ β(x, y)$ is a given 1- form. This change generalizes various types of changes: conformal changes, Randers changes and $β$ - changes. Under this change, we obtain the relationships between some tensors associated with $(M,L)$ and the corresponding tensors associated with $(M,{^\ast}L)$. We investigate some $σ$- invariant tensors . This investigation allows us to give an answer to the question: Are the properties of C-reduc…
In this paper, we introduce and investigate a general transformation or change of Finsler metrics, which is referred to as a generalized $β$-conformal change: $$L(x,y) \longrightarrow\overline{L}(x,y) = f(e^{σ(x)}L(x,y),β(x,y)).$$ This transformation combines both $β$-change and conformal change in a general setting. The change, under this transformation, of the fundamental Finsler connections, together with their associated geometric objects, are obtained. Some invariants and various special Finsler spaces are investigated under this change. The most important changes of Finsler metrics exist…
The aim of the present paper is to investigate conformal changes in absolute parallelism geometry. We find out some new conformal invariants in terms of the Weitzenböck connection and the Levi-Civita connection of an absolute parallelism space.
A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded separating hypersurface that, in dimension three, is necessarily a Willmore energy minimiser or, in higher dimensions, satisfies a conformally invariant analog of the Willmore equation. In any case the zero locus is critical for a conformal functional that generalises the total Q-curvature by including extrinsic data. These observations lead to some interesting globa…
In this paper, we investigate the change of Finslr metrics $$L(x,y) \to\bar{L}(x,y) = f(e^{σ(x)}L(x,y),β(x,y)),$$ which we refer to as a generalized $β$-conformal change. Under this change, we study some special Finsler spaces, namely, quasi C-reducible, semi C-reducible, C-reducible, $C_2$-like, $S_3$-like and $S_4$-like Finsler spaces. We also obtain the transformation of the T-tensor under this change and study some interesting special cases. We then impose a certain condition on the generalized $β$-conformal change, which we call the b-condition, and investigate the geometric consequences …
The present paper is a continuation of a foregoing paper [Tensor, N. S., 69 (2008), 155-178]. The main aim is to establish \emph{an intrinsic investigation} of the conformal change of the most important special Finsler spaces, namely, $C^{h}$-recurrent, $C^{v}$-recurrent, $C^{0}$-recurrent, $C_{2}$-like, quasi-$C$-reducible, $C$-reducible, Berwald space, $S^{v}$-recurrent, $P^*$-Finsler manifold, $R_{3}$-like, $P$-symmetric, Finsler manifold of $p$-scalar curvature and Finsler manifold of $s$-$ps$-curvature. Necessary and sufficient conditions for such special Finsler manifolds to be invariant…
For a generic distribution of rank two on a manifold $M$ of dimension five, we introduce the notion of a generalized contact form. To such a form we associate a generalized Reeb field and a partial connection. From these data, we explicitly constructed a pseudo--Riemannian metric on $M$ of split signature. We prove that a change of the generalized contact form only leads to a conformal rescaling of this metric, so the corresponding conformal class is intrinsic to the distribution. In the second part of the article, we relate this conformal class to the canonical Cartan connection associated to…
On a Finsler manifold $(M,L)$, we consider the change $L\longrightarrow\bar{L}(x,y)=e^{σ(x)}L(x,y)+β(x,y)$, which we call a $β$-conformal change. This change generalizes various types of changes in Finsler geometry: conformal, $C$-conformal, $h$-conformal, Randers and generalized Randers changes. Under this change, we obtain an explicit expression relating the Cartan connection associated to $(M,L)$ and the transformed Cartan connection associated to $(M,\bar{L})$. We also express some of the fundamental geometric objects (canonical spray, nonlinear connection, torsion tensors, ...etc.) of $(M…
We show that the class of CAT(0) spaces is closed under suitable conformal changes. In particular, any CAT(0) space admits a large variety of non-trivial deformations.
Prescribing curvatures in the disk via conformal changes of the metric: the case of negative Gaussian curvature
本文通过共形变形在对称圆盘上规定负高斯曲率及边界测地曲率，采用变分方法结合单调性技巧、爆破分析和Morse指标证明存在性，并给出不存在性结果。
We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other hand, there is a one-to-one correspondence between infinitesimal conformal deformations in space and infinitesimal isometric deformations of the stereographic image on the sphere. The space of infinitesimal conformal deformations can be parametrized in terms of the change in dihed…
In this paper, we prove that lightlike geodesics of a pseudo-Finsler manifold and its focal points are preserved up to reparametrization by anisotropic conformal changes, using the Chern connection and the anisotropic calculus and the fact that geodesics are critical points of the energy functional and Jacobi fields, the kernel of its index form. This result has applications to the study of Finsler spacetimes.
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