Unique continuation theorems for biharmonic maps

Quantitative stratification and higher regularity for biharmonic maps

In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from $Ω\subset \mathbb{R}^5$, we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps …

四维球面上一类近似双调和映射的能量恒等式

Energy identity for a class of approximate biharmonic maps into sphere in dimension four

本文证明了四维弱收敛的近似双调和方程列在球面上，其双张力场有界于$L^p$中，对某些$p>1$ .我们证明了一个能量恒等式，它用$\mathbb R^4$上的100个非平凡双调和映射的Hessian能量之和来解释Hessian能量的损失.

Compactness Results for Sequences of Approximate Biharmonic Maps

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.

更多从球到球面的弱双调和映射

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。

通过定量分层方法获得双调和映射的最优高阶正则性

Optimal higher regularity for biharmonic maps via quantitative stratification

本文利用定量分层方法改进Breiner-Lamm关于双调和映射的几乎最优正则性结果，并得到极小化双调和映射的最优正则性。

Biharmonic wave maps into spheres

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.

欧几里得球面中的稳定真双调和映射

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。

Quantization for biharmonic maps from non-collapsed degenerating Einstein 4-manifolds

For a sequence of extrinsic or intrinsic biharmonic maps $u_j: M_j\rightarrow N$ from a sequence of non-collapsed degenerating closed Einstein 4-manifolds $(M_j,g_j)$ with bounded Einstein constants, bounded diameters and bounded $L^2$ curvature energy into a compact Riemannian manifold $(N,h)$ with uniformly bounded biharmonic energy, we establish a compactness theory modular finitely many bubbles, which are finite energy biharmonic maps from $\mathbb{R}^4$, or from $\mathbb{R}^4 / Γ$ for some nontrivial finite group $Γ\subset SO(4)$, or from some complete, noncompact, Ricci flat, non-flat AL…

A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the $L^p$-norm of the tension field is bounded and the $n$-energy of the map is sufficiently small then every biharmonic map must be harmonic, where \(2<p<n\).

Biharmonic wave maps: Local wellposedness in high regularity

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.

Biharmonic and f-biharmonic maps from a 2-sphere

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…

A note on equivariant biharmonic maps and stable biharmonic maps

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 …

f-Biharmonic maps and f-biharmonic submanifolds II

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…

CR rigidity of pseudo harmonic maps and pseudo biharmonic maps

The CR analogue of B.-Y. Chen's conjecture on pseudo biharmonic maps will be shown. Pseudo biharmonic, but not pseudo harmonic, isometric immersions with parallel pseudo mean curvature vector fields, will be characterized. Several examples of pseudo biharmonic maps will be given.

临界维数中球面映射的能量恒等式

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。

Some recent work on biharmonic conformal maps

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.

关于f-双调和映射与f-双调和子流形

On f-biharmonic maps and f-biharmonic submanifolds

f-双调和映射是f-双能量泛函的极值。f-双调和子流形是定义等距浸入为f-双调和映射的子流形。本文证明了从紧致黎曼流形到非正曲流形的具有常数f-双能量密度的f-双调和映射是调和映射，紧致流形上的任何f-双调和函数都是常数，以及关于$m\ge 3$的$S^m$的逆是真f-双调和共形同态.我们导出了f-双调和子流形方程，证明了流形$(N^n, h)$中的曲面是f-双调和曲面当且仅当它能双调和共形浸入$(N^n,h)$中.我们还给出了三维欧几里得空间中f-双调和曲线的一个完全分类。给出了许多真f-双调和映射和f-双调和曲面与曲线的例子。

Harmonic maps and biharmonic Riemanian submersions

Characterizations for Riemannian submersions to be harmonic or biharmonic are shown. Examples of biharmonic but not harmonic Riemannian submersions are shown.

2-球面上的双调和映射

Biharmonic maps from a 2-sphere

受2-球面上调和映射丰富理论的启发，本文研究了2-球面上的重调和映射。首先导出了旋转对称2-流形之间的旋转对称映射的双调和方程。然后，我们应用该方程，以获得一个家庭的旋转对称映射之间的2-球的双调和映射的分类。我们还发现了许多在2-球面上局部定义的真双调和映射的例子。我们的结果似乎表明，任何双调和映射$S^2\longrightarrow (N^n, h)$是一个弱共形浸入。

临界维数下双调和映射的Morse指数稳定性

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。

关于黎曼流形间的双调和映射的一个小议

A short survey on biharmonic maps between Riemannian manifolds

在这个简短的调查，我们报告的理论，双调和映射之间的黎曼流形。

二维双调和映射

Biharmonic maps in two dimensions

研究了曲面间的双调和映射。在复坐标系中，我们计算了具有共形度量的曲面间映射的双张力场。作为应用，我们证明了如果共形因子$σ$是双解析的，则从欧几里得平面到$(\mathbb{R}^2, σ^2dwd\bar w)$的线性映射总是双调和的;我们构造了一类这样的$ σ$，并给出了$2$ -球面之间线性双调和映射的分类。我们还研究了曲面之间的双调和映射与翘曲的产品度量。这包括双曲平面之间的线性双调和映射的分类和许多到圆锥或螺旋面的真双调和映射的一些构造。

Liouville type theorems for transversally harmonic and biharmonic maps

We study the Liouville type theorems for transversally harmonic and biharmonic maps on foliated Riemannian manifolds

关于来自梯度Ricci孤立子的调和映射与双调和映射

On harmonic and biharmonic maps from gradient Ricci solitons

该研究聚焦梯度Ricci孤立子上的调和与双调和映射，推导了使调和映射为常值、双调和映射退化为调和映射的解析与几何条件，并证明二维cigar孤立子的有限能量双调和映射必为调和映射。

球面上双调和映射的指数

The index of biharmonic maps in spheres

双调和映射是双能量泛函和广义调和映射的临界点。本文研究了一类由极小黎曼浸入到球面中导出的双调和映射的指数。本研究的动机是由三个家庭的例子：全测地包含领域，Veronese地图和Clifford环面。

次椭圆双调和映射

Subelliptic biharmonic maps

研究了次椭圆双调和映射，即从紧严格伪凸CR流形M到黎曼流形N的光滑映射，其中N是某个双能量泛函的临界点。证明了一个映射是次椭圆双调和映射当且仅当它到标准圆丛的（全空间）的垂直提升是关于February man度量的双调和映射。

到Sol和Nil空间的双调和映射

Biharmonic maps into Sol and Nil spaces

本文研究了三维Thurston几何的两个模型空间Sol和Nil空间中的双调和映射。我们刻画了Sol空间中的非测地双调和曲线，证明了Sol空间中不存在非测地双调和螺旋线。证明了从欧几里得空间到Sol或Nil空间的线性映射是双调和映射的充要条件是它是调和映射，并给出了这类映射的一个完全分类.

双翘曲积流形之间的双调和映射

Biharmonic maps between doubly warped product manifolds

本文研究了双曲积流形之间的双调和映射。证明了黎曼流形$B$和$F$到双曲积$_{f}B\times_{b}F$中的包含映射不可能是真的双调和映射.并分别分析了投影$_{f}B\times_{b}F\to B$和$_{f}B\times_{b}F\to F$的双调和性条件。利用调和映射与翘曲度量的乘积，给出了非调和双调和映射的一些刻划。特别地，在$f=1$的情况下，得到了\cite{Balmus-mont}中翘曲积的结果。

双调和映射与共形映射的态射

Biharmonic maps and morphisms from conformal mappings

受二维域上调和映射的共形不变性的启发，研究了双调和性与共形性之间的关系。我们首先给出了一个特征的双调和态射，类似的调和态射的Fuglede和石原，其中，特别是，解释了所需的条件，为保双调和性的四维共形映射，并帮助生产的第一个例子的双调和态射，这不是一个特殊类型的调和态射。然后，我们计算了水平弱共形映射的双张力场，其中包括共形映射。这就引出了几个真（即非调和）双调和共形映射的例子，其中四维起着关键作用。我们还构造了一族黎曼浸没，它们是真双调和映射。

球面间双调和多项式映射的新构造

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。

Finsler空间的双调和映射

Biharmonic maps from Finsler spaces

将光滑映射的双能量和黎曼流形间的双调和映射的概念推广到区域为Finslerian的情形。我们确定了双能泛函的第一变分和二阶变分，Finsler到Riemann双调和映射的方程和一些具体的例子。在黎曼几何中关于非调和双调和映射不存在的两个显著结果在我们的情形中得到了推广。

调和映射与双调和映射的可去奇点与冒泡

Removable singularities and bubbling of harmonic maps and biharmonic maps

利用Moser的迭代技巧，给出了到非正曲率流形上的双调和映射的张力场的若干可去奇点定理，以及双调和映射和调和映射的冒泡定理.

环面到2-球面的双调和映射

Biharmonic maps from tori into a 2-sphere

双调和映射是调和映射的推广。Eells和Wood关于曲面间调和映射的一个著名结果表明，在Brower度为$\pm 1$的同伦映射类中，不存在从环面到球面的调和映射（无论选择何种度量）.如果知道在那个同伦类映射中是否存在任何双调和映射，那将是很有趣的。本文给出了环面到球面的双调和映射的分类，其中环面具有平坦度量或一类非平坦度量，而球面具有标准度量.结果表明，在从环面到球面的映射大族中，不存在次数为$\pm 1$的真双调和映射。

等变双调和映射的一般方法

A general approach to equivariant biharmonic maps

本文给出了等变双调和映射解析构造的一维变分方法。我们的目标是提供一个直接法，使分析人员能够直接计算的分析条件，保证在适当的对称性的存在下的双调和性。在我们工作的第二部分中，我们举例说明并讨论了一些例子。特别地，我们得到了一个一维稳定性结果，同时也证明了双调和映射不满足经典的调和映射的最大值原理。

关于 $p$-双调和与双 $p$-调和映射的推广

On the generalized of $p$-biharmonic and bi-$p$-harmonic maps

本文提出并研究了 $p$-双调和与双 $p$-调和映射的定义及其性质。

对称空间的双调和映射与可积系统

Biharmonic maps into symmetric spaces and integrable systems

本文给出了由$G$上的双不变黎曼度量$h$导出的紧致黎曼流形到黎曼对称空间$(G/K,h)$的所有光滑映射的双调和映射方程的Maurer-Cartan形式的描述.利用这个公式，确定了对称空间中的所有双调和曲线，以及${\Bbb R}^2$的开域在标准黎曼度量下到$(G/K,h)$中的所有双调和映射.

Some remarks on bi-f-harmonic maps and f-biharmonic maps

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) …

Proper biharmonic maps and $(2,1)$-harmonic morphisms from some wild geometries

In this work we construct a variety of new complex-valued proper biharmonic maps and (2,1)-harmonic morphisms on Riemannian manifolds with non-trivial geometry. These are solutions to a non-linear system of partial differential equations depending on the geometric data of the manifolds involved.

到非正曲率黎曼流形的双调和映射

Biharmonic maps into a Riemannian manifold of non-positive curvature

研究了有限能量和有限双能量的黎曼流形之间的双调和映射。我们证明了，如果区域是完全的，目标的非正曲率，那么这样的映射是调和的。然后，我们给等距浸入和水平共形浸入的应用程序。

Biharmonic conformal maps in dimension four and equations of Yamabe-type

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.

关于叶状结构上双调和映射的广义Chen猜想

Generalized Chen's conjecture for biharmonic maps on foliations

本文证明了叶状结构上(F,F')-双调和映射的广义Chen猜想，该映射为横截双能量泛函的临界点。

双调和映射的应力-能量张量

The stress-energy tensor for biharmonic maps

利用希尔伯特判据，我们考虑了与双能泛函相关的应力-能量张量。我们证明了它来源于度量上的一个变分问题，并表现出四维的特殊性。首先，我们使用这个张量构造新的例子的双调和映射，然后分类地图消失或平行的应力-能量张量和黎曼浸入的应力-能量张量是成比例的度量。

Some classifications of conformal biharmonic and k-polyharmonic maps

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.

切丛上的真双调和映射

Proper biharmonic maps on tangent bundle

本文在切丛上通过非共形变形Sasaki度量定义Mus-Gradient度量，研究其几何并刻画一类新的真双调和映射，且在欧氏空间因子情形给出实例。

统计双调和映射的第二变分公式

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。

超临界维数下到齐性空间的内禀稳态双调和映射的能量恒等式

Energy identity for Intrinsic Stationary Biharmonic Mappings into Homogeneous Spaces in Supercritical Dimensions

本文扩展了Hornung-Moser的临界维数能量恒等式结果，采用Lin-Rivière的策略，通过守恒律推导内禀稳态双调和映射的高阶正则性，研究超临界维数下到齐性空间的此类映射的能量恒等式。

双调和全纯映射与共形Kahler几何

Biharmonic holomorphic maps and conformally Kahler geometry

给出了几乎Hermitian流形的Lee向量场的条件，使得从该流形到（1，2）-辛流形的全纯映射必须满足四阶双调和条件，从而推广了调和映射的Lichnerowicz定理.这些三阶非线性条件大大简化了l.c.K.流形和建设方法和例子，在所有的维度。

Biharmonic homogeneous polynomial maps between spheres

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 $…

球面间双调和二次映射的能量密度

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。

双调和黎曼映射

Biharmonic Riemannian maps

给出了黎曼映射是双调和映射的充要条件。我们还定义了伪脐黎曼映射作为伪脐子流形的推广，并证明了这样的黎曼映射对基流形施加了一些限制。

完备黎曼流形到非正曲流形的双调和映射

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

考虑从完备黎曼流形到具有非正截面曲率的黎曼流形的双调和映射$ϕ:(M,g)\rightarrow (N,h)$.假设$α$满足$1<α<\infty$。如果对这样的$α$，$\int_M|τ(ϕ)|^αdv_g<\infty$和$\int_M|dϕ|^2dv_g<\infty,$，其中$τ(ϕ)$是$ϕ$的张力场，那么我们证明了$ϕ$是调和的.对于双调和子流形，我们得到了上述假设$\int_M|dϕ|^2dv_g<\infty$是不必要的。这些结果对广义Chen猜想的整体形式给出了部分肯定的回答。

关于共形双调和映射与超曲面

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。

Nonexistence of proper $p$-biharmonic maps and Liouville type theorems I: case of $p\geq2$

Let $u: (M, g)\to (N, h)$ be a map between Riemannian manifolds $(M, g)$ and $(N, h)$. The $p$-bienergy of $u$ is defined by $E_p(u)=\int_M|τ(u)|^pdν_g$, where $τ(u)$ is the tension field of $u$ and $p>1$. Critical points of $E_p(\cdot)$ are called $p$-biharmonic maps. In this paper we will prove nonexistence result of proper $p$-biharmonic maps when $p\geq2$. In particular when $M=\mathbb{R}^m$, we get Liouville type results under proper integral conditions , which extend the related results of Baird, Fardoun and Ouakkas \cite{BFO}.

Generalized harmonic morphisms and horizontally weakly conformal biharmonic maps

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…