F. E. Burstall, M. Kilian
We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.
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We prove uniqueness of equivariant harmonic maps into irreducible symmetric spaces of non-compact type and Euclidean buildings associated to isometric actions by Zariski dense subgroups.
We prove that when assuming suitable non-degeneracy conditions equivariant harmonic maps into symmetric spaces of non-compact type depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in \cite{EellsLemaire} can be applied.
Einstein Metrics and Equivariant Harmonic Maps: The Einstein Detection Principle
本文针对紧致同质性为1的爱因斯坦度量的局部形变理论，构建结合爱因斯坦边值问题内在重述与等变调和映射辅助理论的分析框架，提出爱因斯坦检测原理，辅助调和映射构建框架且不引入额外自由度。
On the equivariant stability of harmonic self-maps of cohomogeneity one manifolds
本文研究余齐性一流形上调和自映射的等变稳定性，分析相应Jacobi方程，并在球面、特殊正交群及SU(3)情形显式求解，证明球面上特定作用下恒等映射等变稳定。
Equivariant primitive harmonic maps into $k-$symmetric spaces, with applications to Willmore surfaces
本文研究了等变本原调和映入 $k-$对称空间的构造，并应用于 Willmore 曲面，推广了 Lawson 极小 Klein 瓶得到了 $S^3$ 中的 Willmore Moebius 带。
For covering spaces and properly discontinuous actions with compatible diffusion operators, we discuss Lyons-Sullivan discretizations of the associated diffusions and harmonic functions of bounded growth.
Reflection Symmetry, APS Boundary Conditions, and Equivariant Spectral Flow on a Warped Cylinder
研究扭曲狄拉克算符在有限扭曲圆柱上的反演对称性和APS边界条件，证明当2A为整数时，反演可提升为扭曲狄拉克设置的酉对称性，并探讨参数依赖模型的等变谱流分解。
In this paper we give an explicit parametrisation of the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere. As Hitchin proved, a harmonic map of a 2-torus is described by its spectral data, which consists of a hyperelliptic curve together with a pair of differentials and a line bundle. The space of spectral data is naturally a fibre bundle over the space of spectral curves. For homogeneous tori the space of spectral curves is a disc and the bundle is trivial. For tori with a one-dimensional invariance group, we enumerate the path connected components of the space of spec…
We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explicit convergence rate. We also examine center of mass methods, after showing a generalized mean value property for harmonic maps. We feature a concrete illustration of these methods with Harmony, a computer software that we developed in C++, whose main functionality is to numerical…
We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüller harmonic map flow exist for all times. Furthermore, for solutions for which a three-point-condition does not degenerate as $t\to\infty$, we show convergence along a sequence $t_i\to\infty$ to a critical point of the area given either by a minimal cylinder or by two minimal dis…
Equivariant discretizations of diffusions, random walks, and harmonic functions
本文研究了扩散、随机游走和调和函数的协变离散化方法，探讨了Lyons-Sullivan离散化与相关函数理论。
Explicit harmonic self-maps of complex projective spaces
本文构造了复射影空间上两个显式不可数调和自映射族，分别由全纯和混合映射给出，并证明其等变弱稳定性及计算等变谱。
Construction of harmonic maps between cohomogeneity one manifolds
本文提出了一种构造cohomogeneity one manifolds之间等变调和映射的方法。
We characterize the harmonic forms on a flag manifold $K/T$ defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are ``Poisson harmonic" with respect to the so-called Bruhat Poisson structure on $K/T$. This enables us to give Poisson geometrical proofs of many of the special properties of these harmonic forms. In particular, we construct explicit representatives for the Schubert basis of the $S^1$-equivariant cohomology of $K/T$, where the $S^1$-action is defined by $ρ$. Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmoni…
We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left(\frac{N^2 - m^2}{2} + N\right) $. Hence if $\frac{m}{N}= c $ for fixed $0 < c < 1$, the genus has order $N^3$.
Higher-Rank Orthogonal Twists, APS Boundary Conditions, and $O(2)$-Equivariant Spectral Flow on a Warped Cylinder
研究在翘曲有限圆柱上带有固定可容许正则化APS边界条件的Dirac算子的$O(2)$-等变谱流，通过将扭曲丛复化并正交对角化，得到实$RO(O(2))$类的块状公式，细化整数值谱流并揭示维数映射的信息损失。
This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.
We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition forces the domain to stretch out as a boundary bubble forms. Our main result then establishes that for prescribed boundary curves that satisfy Douglas' separation condition, these boundary bubbles will not only be harmonic but will themselves be branched minimal immersions. Toget…
Asymptotic analysis for approximate harmonic maps from degenerating cylinders and applications to minimal surfaces
本文研究了从退化环面或退化圆柱到紧致黎曼流形的映射序列的吹胀分析和定量行为，建立了广义能量恒等式，并证明了在无气泡区域，渐进行为要么是流形上的测地线，要么是子流形上的测地线状曲线。同时，研究了寻找具有自由边界条件的最小圆柱体的演化系统，得出了存在性结果，并揭示了颈部可能收敛到子流形上的测地线状曲线的新现象。
The purpose of the article is to study a foliation associated to a lattice-equivariant harmonic map of small rank from a complex ball to another. The result is related to rigidity of some complex ball quotients.
We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isomet…
For the 1-equivariant harmonic map flow from $ R^2$ into $S^2$ \begin{equation*} \left\{ \begin{aligned} &v_t=v_{rr}+\frac{v_r}{r} - \frac{\sin(2v)}{2r^2} , ~\quad(r,t)\in R_+\times (t_0,+\infty),\\ &v(r,t_0)=v_0, \qquad\qquad\qquad\quad r\in R_+, \end{aligned} \right. \end{equation*} we construct global growing, bounded and decaying solutions with the initial data $v_0(r)$ satisfying $$v_0(0)=π~\mbox{ and }~ v_0(r)\sim r^{1-γ} ~\mbox{ as }~ r\to+\infty, \quad γ>1.$$ These global solutions exhibit the following trichotomy long-time asymptotic behavior \begin{equation*} \| v_r(\cdot,t) \|_{L^\i…
A Harmonic-Map Proof of Mostow Rigidity
该研究针对维数不小于3的紧双曲流形基本群同构，通过建立等变调和提升的精确逐点梯度估计，给出了Mostow刚性的调和映射证明。
$\mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we study the asymptotics of $\mathcal{H}-$holomorphic curves defined on a sequence of degenerating cylinders.
We consider the question of whether solutions of variants of Teichmüller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least $2$, as well as the flow from cylinders, we prove that such a finite-time degeneration must occur in situations where the image of thin collars is `stretching out' at a rate of at least $\text{inj}(M,g)^{-(\frac14+δ)}$, and we construct targets in which the flow from cylinders must indeed degenerate in finite time. For the rescaled Teichmüller harmonic map flow, the condition t…
We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each, of SO(10) with degree -7, and of SO(14) with degree -11 by exhibiting linear solutions to non-linear singular boundary value problems.
We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible $SL_2({\mathbb C})$ representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ${\mathbb R}$-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.
The Initial Value Problem for Harmonic maps of Cohomogeneity One manifolds
本文研究了余维数为一的调和映射的初值问题，证明了在奇点轨道邻域内调和映射的局部存在性，并探讨了正则-奇点系统的一阶理论。
We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-)Riemannian geodesics, and constant mean curvature hypersurfaces.
Let $(S,g_0)$ be a hyperbolic surface, $ρ$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $ρ$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\equiv 1$ and $ρ$ is the base $n$-Fuchsian representation of $(S,g_0)$. In particular, we show given a Hitchin representation $ρ$ for $PSL(n,\mathbb R)$, every $ρ$-equivariant minimal immersion $f$ from a hyperbolic plane $\mathbb H^2$ into the corresponding symmetric space $X$ is d…
We examine the question of uniqueness for the equivariant reduction of the harmonic map heat flow in the energy supercritical dimension. It is shown that, generically, singular data can give rise to two distinct solutions which are both stable, and satisfy the local energy inequality. We also discuss how uniqueness can be retrieved.
We consider a compact, oriented, smooth Riemannian manifold $M$ (with or without boundary) and we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $d_{X_M} = d+ι_{X_M}: Ω_G^\pm \toΩ_G^\mp$ (even/odd invariant forms on $M$) and its adjoint $δ_{X_M}$. In the 1980s Witten showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $Δ_{X_M} = (d_{X_M}+δ_{X_M})^2$), and the cohomology groups are isomorphic to the ordinar…
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