Cellular harmonic maps which are not diffeomorphisms
F. T. Farrell, P. Ontaneda
我们给出了负弯曲流形之间的调和胞腔映射的例子，这些流形不是同伦同伦的。
We give examples of harmonic cellular maps between negatively curved manifolds which are not diffeomorphisms but are homotopic to diffeomorphisms.
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In this paper, we will prove a result of nonexistence on harmonic diffeomorphisms between punctured spaces. In particular, we will given an elementary proof to the nonexistence of rotationally symmetric harmonic diffeomorphisms from the punctured Euclidean space onto the punctured hyperbolic space.
In this paper, we show that the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.
In this paper, we prove an existence and uniqueness theorem for orientation-reversing harmonic diffeomorphisms from $\mathbb{H}_*^n$ to $\mathbb{R}_*^n$ with rotational symmetry, which is a generalization of the corresponding result for dimension $2$.
We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs in the Lorentzian product $M\times\mathbb{R}_1,$ where $M$ is an arbitrary $n$-dimensional compact Riemannian manifold, $n\geq 2.$ In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the once punctured sphere and no harmonic…
In this paper, we introduce a Liouville action for a harmonic diffeomorphism from a compact Riemann surface to a compact hyperbolic Riemann surface of genus $g\ge 2$. We derive the variational formula of this Liouville action for harmonic diffeomorphisms when the source Riemann surfaces vary with a fixed target Riemann surface.
Minimal surfaces and harmonic diffeomorphisms from the complex plane onto a Hadamard surface
我们构造了从复平面$C$到任何Hadamard曲面$M$的调和微分同胚，该曲面的曲率由负常数限制。为此，我们证明了$M$中被理想测地多边形包围的整环上的$M\times R$中的极小图的Jenkins-Serrin型定理，并证明了在$C$的共形结构下，存在一个收敛于$M\times R$中的整极小图的多边形整环上的极小图序列.
Construction of harmonic diffeomorphisms and minimal graphs
本文研究了H_xR中的完全极小图，它在H_xR中的理想内接多边形的交错边上取渐近边界值正负无穷大。我们给出了保证这种图存在的多边形（和中的所有内接多边形）的边的“长度”的充要条件。然后，我们应用这一点来构造HxR中的整体极小图，这些图共形地是复平面C。这样一个图的垂直投影产生了从C到H的调和同构，从而反驳了Rick Schoen的一个猜想。
We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $Σ$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a harmonic diffeomorphism from $S$ onto $Σ$. The proof uses the theory of divergence lines to construct minimal graphs. We also generalize a theorem of R. Schoen. Let $g_1$ and $g_2$ be two complete metrics on a orientable surface $S$ with compact boundary and suppose $$\int_{S_r^2}K_{g_2}^-dσ_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all $r>0$. If there is a harmonic diffe…
In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefore is a harmonic function. The real part of the logarithm of the Beltrami function satisfies an elliptic nonlinear differential equation, which in the case of constant curvature is an elliptic sinh-Gordon equation. Solutions are calculated for the constant curvature case in a unif…
Harmonic diffeomorphisms between pseudo-riemannian surfaces
本文在统一形式化框架下研究黎曼曲面与洛伦兹曲面间的调和映射，通过广义 sine-Gordon 方程分类并构造单孤子解，并利用 Bäcklund 变换建立不同方程解之间的联系。
Diffeomorphismes harmoniques du plan hyperbolique
Sampson和Schoen-Yau在1978年的一个经典结果指出，紧致双曲Riemann曲面之间的每一个复同态都同伦于调和复同态。Schoen在1993年提出并由Wan在1992年和Tam-Wan在1995年部分证明了这个定理，本文证明了这个定理推广到非紧情形：圆的每个拟对称同胚扩张到双曲平面的拟等距调和同胚.这使得参数化的通用Teichmuller空间的有界全纯二次微分的双曲平面。
We prove that a harmonic quasi-isometric map between pinched Hadamard surfaces is a quasi-conformal diffeomorphism.
We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$, from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray, to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of …
Let $N=(Ω,σ)$ and $M=(Ω^*,ρ)$ be doubly connected Riemann surfaces and assume that $ρ$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms between $Ω$ and $Ω^*$ that minimize the Dirichlet energy. In the class of all homeomorphisms $f \colon Ω\onto Ω^\ast$ between doubly connected domains such that $\Mod Ω\le \Mod Ω^\ast$ there exists, unique up to conformal authomorphisms of $Ω$, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec, Koh…
Let $f$ be a harmonic map from a Riemann surface to a Riemannian $n$-manifold. We prove that if there is a holomorphic diffeomorphism $h$ between open subsets of the surface such that $f\circ h = f$, then $f$ factors through a holomorphic map onto another Riemann surface. If such $h$ is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver-Osserman-Royden. Our proof relies on various geometric properties of the Hopf differential.
Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to other PDEs. We prove a versatile sufficient condition that shows, in particular, that any (possibly disconnected) algebraic noncompact hypersurface can be transformed onto a union of components of the zero set of a harmonic function via a diffeomorphism of $\RR^n$. The technique we use, which is a significant improvement of the basic strategy we recently ap…
Back to harmonic mappings of compact Riemannian manifolds
本文探讨了黎曼流形间调和映射理论的几个关联问题，提出了验证调和性的解析准则，分析了拉回度量的L2-正交分解，并研究了调和对称双线性形式在恒等调和映射中的作用。
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harm…
Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric identities which relate the scalar curvature of Riemannian 3-manifolds to global invariants in terms of harmonic functions. These quantitative formulas are useful for stability results and show promise for more applications of this type. In this paper, we analyze harmonic maps to flat model spaces in order to address conjectures concerning the geometric stability of the positive mass theorem and the Geroch conjecture. By imposing integral Ricci curvature and isoperimetric bounds, we leverage the previously mentioned f…
Stability of Euclidean space under Ricci flow
研究了R^n上欧氏度量的C^0小扰动初始度量的里奇流.在这个度量是渐近欧几里德的情况下，我们证明了一个Ricci调和映射热流存在的所有时间，并一致收敛到欧几里德度量的时间接近无穷大。在证明这个稳定性的结果，我们引入了一个单调的积分量的措施，从欧几里德度量的演变度量的偏差。我们还研究了Ricci调和映射热流与里奇流之间的关系。
Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics
本文构造了从带孔黎曼曲面到双曲平面的调和映射，在孔洞处指定Scherk渐近性，并涵盖水平悬链面的调和映射。
In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics $(u,g)$ into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where $u$ maps from a fixed closed surface $M$ with metric $g$ to a general target manifold $N$. It arises naturally as a gradient flow for the Dirichlet energy functional viewed as acting on equivalence classes of such pairs, obtained from the invariance under diffeomorphisms and conformal changes of the domain metric. In the construction of a suitable inner produc…
For a harmonic diffeomorphism between the Poincaré disks, Wan showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to $r$-differetials. We study the relationship between bounded holomorphic $r$-differentials and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space $SL(r,\mathbb R)/SO(r)$ arising from cyclic/subcyclic harmonic Higgs bundles. Also, we show the equivalences between the boundedness of holomorphic differentials and havi…
On a natural map between twistor spaces
本文研究了光滑流形上两个 Riemannian 度量的 twistor 空间之间保持纤维的微分同胚，证明了其全纯性与度量的共形或同位关系相关，并探讨了其为调和映射的条件。
Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps …
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…
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