Deformations of harmonic function germs
Yuki Yasuda
在右等价下研究了二元调和首项函数芽奇点的分类问题。我们观察到拉普拉斯算子的多重作用出现在这类函数芽的分类中。
We study the classification problem of singularities of function-germs with harmonic leading terms of two variables under the right-equivalence. We observe that the multiple actions of Laplacian appear for the classifications of such class of function-germs.
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Singularities of functions with harmonic leading terms of two variables
在右等价下研究了二元调和首项函数芽奇点的分类问题。我们研究了函数芽的阶数不超过7的分类问题。此外，我们还观察到拉普拉斯算子的多重作用出现在这类函数芽的分类中。
We consider harmonic sections of a bundle over the complement of a codimension 2 submanifold in a Riemannian manifold, which can be thought of as multivalued harmonic functions. We prove a result to the effect that these are stable under small deformations of the data. The proof is an application of a version of the Nash-Moser implicit function theorem.
In this paper we investigate the space of harmonic maps from a 2-torus to $\mathbb{S}^3$ using the spectral curve correspondence and Whitham deformations. In an open and dense subset of a parameter space we find that the space of harmonic maps is smooth and has dimension two. We also show that the points that correspond to minimal tori (conformal harmonic maps) are either smooth points of dimension two or singular.
Pseudo Harmonic Morphisms
研究了一个比水平弱共形性弱的几何条件（PHWC）。特别是，我们证明了调和映射满足这个条件，这将被称为{ \em伪调和态射}，包括调和态射，并可以被描述为拉回某些细菌某些其他细菌。最后，我们构造了一个与满足（PHWC）的每一个映射相关联的规范f-结构，并找到了这个f-结构上的条件，以确保映射的调和性。
The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach this bijection through the deformation to the normal cone that is associated to the embedding of M into V. The second purpose is to study generalizations to smooth manifolds equipped with Lie filtrations. Following in the footsteps of several others, we shall define a deformatio…
Metric perturbations and deformations of k-nondegenerate Z/2-harmonic 1-forms
研究退化Z/2调和1形式的度量扰动与变形理论，证明通过适当扰动度量可将形式变形为附近的非退化Z/2调和1形式，结合局部展开分析与量化Nash-Moser隐函数定理。
Pseudo Harmonic Morphisms on Riemannian Polyhedra
本文的目的是将Loubeau \cite {Lo}引入的伪调和态射的概念推广到源流形是容许黎曼多面体的情形。我们定义这些映射在Eells-Fuglede \cite {EF}意义下是调和的，在我们的意义下是伪水平弱共形的（见第3节）。我们通过Korevaar-Schoen \cite {KS}意义下的源多面体上的调和函数的芽和Kähler目标流形上的全纯函数的芽来刻画它们。
Simple tangential families and perestroikas of their envelopes
切向族是从平滑曲线切向发出的单参数射线族。我们分类切族芽左右等价：我们证明了有两个无穷级数和四个零星的简单奇异的切族芽（除了两个稳定的奇点）。我们给出了它们的标准形和极小切向变形（即，变形之间的切向家庭），我们描述了相应的信封perestroikas的小余维。我们还讨论了非简单切向族的包络奇点。
Exploring certain geometric and harmonic properties of the Berger-type metric conformal deformation on the Para-Kähler-Norden manifold
该研究在Para-Kähler-Norden流形上构造了由Berger型度量共形形变得到的新度量类，分析了其Levi-Civita联络与各类曲率，并研究了相关的一类调和映射。
Bi-Lipschitz triviality of function-germs on singular varieties
本文研究奇异簇上解析函数芽形变的双李普希茨平凡性，引入强有理双李普希茨平凡族概念并给出无穷小判别准则，进而得到齐次与加权齐次情形下的平凡性和刚性结果。
Harmonic spinors and local deformations of the metric
设（M，g）是紧致黎曼自旋流形. Atiyah-Singer指数定理给出了Dirac算子核维数的一个下界。证明了在任意小的开集上，通过改变黎曼度量g，可以得到这个界。
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.
Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called "Kossowski metrics". A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points(the definition of "non-parabolic singular point" is stated in the introduction here) can be realized as wave front germs (Kossowski's realization the…
A note on the spectral deformation of harmonic maps from the two-sphere into the unitary group
在[5]中，与J. C. Wood等，作者给出了从$2$ -球面到酉群$U(n)$的所有调和映射在$S^2$上的任意亚纯函数的完全显式公式.最简单的调和映射是迷向映射。Burstall和Guest [1]利用莫尔斯理论证明了调和映射是以各向同性映射为标记的簇。在这项工作中，使用上述调和映射的公式，我们明确地描述了这一过程，显示如何所有的调和映射可以建立从各向同性的。[1]F.~ E. Burstall和M.~ A. Guest，\textit{Harmonic two-spheres in compact symmetric spaces, revisited}，Math.Ann.309（1997）541- 572. [5]M.~ J. Ferreira，B.~ A Similes和J.~ C. Wood \emph{All harmonic $2$-spheres in the unitary group, completely explicitly}，Math. Z. { \bf 266}（2010），953- 978.
Deformations of harmonic maps with conical singularities
本文研究紧致流形间具有锥形奇点的调和映射的形变理论，在切映射满足特定指标和非退化条件下证明其在度量扰动下保持，并构造了满足条件的显式例子。
We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely A-determined map-germs from the plane to the plane can be represented by harmonic maps.
Harmonic deformations of hyperbolic 3-manifolds
本文给出了作者关于三维双曲锥流形的调和形变理论。我们讨论拓扑应用双曲Dehn手术以及最近的应用Kleinian群论。一个中心思想是，局部刚度的结果（固定锥角的变形）可以变成有效的控制变形的存在。这导致了具有短测地线的双曲结构接近于具有尖点的双曲结构的想法的精确解析和几何版本。本文还概述了一个新的谐波变形理论，适用于每当有一个足够大的嵌入管周围的奇异轨迹，删除以前的限制锥角最多2 π。
Symplectic Harmonic theory and the Federer-Fleming deformation theorem
在这篇文章中，我们提出了一个几何测度理论的方法辛霍奇理论。特别地，我们应用几何测度论中的中心结果之一，Federerer-Fleming形变定理，以及微分流形上的正规流的上同调理论，建立了辛调和形式的一个基本性质。本文证明了在闭辛流形上，每一个正度的实本原上同调类都有一个在整个流形上不支集的辛调和表示.作为应用，我们用它来研究Thom类的辛调和表示的支集，并给出Guillemin提出的一个公开问题的完整解.
Transformations of harmonic bundles and Willmore surfaces
Willmore曲面是Willmore泛函的极值（可能在共形结构的约束下）。随着表征Willmore表面的（可能扰动）调和性的中曲率球同余[Blaschke，Ejiri，Rigoli，Burstall-Calderbank]，一个零曲率公式如下[Burstall-Calderbank]。变形的水平上的调和映射证明引起变形的水平上的表面，与定义的频谱变形[Burstall-Pedit-Pinkall，Burstall-Calderbank]和Baecklund变换[Burstall-Quintino]的Willmore表面到新的，与一个比安奇置换之间的两个[Burstall-Quintino]。本文致力于一个独立的帐户的主题，从一个保形不变的观点，在达布的光锥模型的共形$n$ -球。
Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space
本文的目的是将可积系统与极小曲面理论联系起来。修整操作使用调和映射的相关平坦联络族来构造新的调和映射。由于三维空间中的极小曲面是Willmore曲面，它的共形高斯映射是调和的，并且可以定义共形高斯映射上的一个修饰。本文研究了极小曲面在最简单的单因子修整情况下的诱导变换，并证明了极小曲面的Lopez-Ros变形是这种诱导变换的一个特例.我们用极小曲面及其共轭曲面来表示单因子修整和Lopez-Ros变形。特别是，我们可以控制周期和结束行为的简单因素敷料。这允许构建新的例子双周期极小曲面所产生的简单因素敷料的Scherk的第一个表面。
Deformation of contracting maps under the harmonic map heat flow
本文受Lee-Wan工作启发，研究正曲率闭流形间收缩映射的刚性，分析曲率拼挤与含任意$k$个奇异值的收缩条件的关联。
Group actions and deformations for harmonic maps
本文得到了二维欧氏西格玛模型经典解的一些新结果。从早期的工作丁Zakrzewski，斯道拉，和许多微分几何学家，人们知道明确的解决方案的情况下，$S^n$ -模型，$CP^n$ -模型，和$U(n)$ -模型。然而，很少有人知道的“模空间“的解决方案本身。本文研究了这类空间的连通分支。在随后的一篇论文中（与M. Furuta和M. Kotani），我们计算的基本群，在$S^n$ -模型的情况下。
Deformations of harmonic mappings and variation of the energy
研究了扭曲调和映射$f$关于表示$ρ$的变形.在构造了一个连续的“万有”扭调和映射之后，利用Hodge理论给出了$f$的每一个一阶变形的构造，并将此结果应用于Kähler群的约化表示的模空间，证明了能量泛函$E$的临界点与Hodge结构的极化复变分的单值表示一致.然后，我们进行二阶变形，障碍物的出现，我们调查这种变形的存在，并给出了一种方法来构建它们，以及。将其应用于上述能量泛函，我们证明了（对于每一个群）能量泛函是“Betti”模空间的Kähler形式的势;进一步假设该群是Kähler，我们研究了$E$的Hessian在临界点的特征值。
New examples of Z/2 harmonic 1-forms and their deformations
本文通过构造 Z/2 调和 1-形式的新例子，系统揭示了其分支集可具有非平凡链环、多重覆盖、浸入极限及正维切锥族等几何特征，并确定了这些特征出现的最低维数。
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。
The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of immersed special Lagrangian submanifolds $\tilde{L}_t$, that are diffeomorphic to a branched covering of $L$ and $\tilde{L}_t$ convergence to $2L$ as current. This answers a question suggested by Donaldson. Combining with the work of Abouzaid and Imagi, we obtain constraints for the existence of nondegenerate $\mathbb{Z}_2$ harmonic 1-forms.
Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds
本文研究了3-流形上 $\mathbb Z_2$-调和旋量通用模空间的局部结构，利用 Nash-Moser 隐函数定理克服了无限维阻碍和正则性丧失问题，证明其投射为参数空间中的余维数1子流形。
Diffeological non-Abelian Hodge theory: relative harmonic metrics and deformation theory
研究紧致凯勒流形\(X\)上希格斯丛和平坦丛的微分几何模栈，通过构建相对解析理论，证明稳定希格斯族存在全局光滑调和度量，刻画扩张生成栈，分析热流数据损失并构造光滑霍奇\(\lambda\)族。
Inspired by a string duality, we construct a deformation family for $G_2$-orbifolds given as total spaces of coassociative fibrations by ADE singularities over a closed and oriented smooth three-manifold $Q$. The deformations are parametrized by sections of a fiber bundle on $Q$ that can be interpreted as spectral/cameral covers associated to certain Riemannian analogs of Higgs bundles. The spectral cover picture is a unifying framework for several different approaches to coassociative fibrations appearing in the literature. Our construction generalizes, to the context of $G_2$-geometry, a wel…
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