Marco Freibert, Lothar Schiemanowski, Hartmut Weiss
We study the spinor flow on homogeneous spin manifolds. After providing the general setup we discuss the homogeneous spinor flow in dimension 3 and on almost abelian Lie groups in detail. As a further example the flag manifold in dimension 6 is treated.
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On the spinor structure of the homogeneous and isotropic universe in closed model
考虑了封闭的均匀各向同性宇宙。这个宇宙的外尔和狄拉克旋量的束明确描述。给出了赤平坐标系和球面坐标系中基本场和连接分量的显式表达式。
Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds
利用一个特殊的旋量（称为广义Killing旋量）的存在性，给出了等距浸入曲面到具有四维等距群的三维齐次流形中的旋量刻画.这推广了T. Friedrich为$\R^3$和B. Morel用于$\Ss^3$和$\HH^3$。本文的主要论点是将广义Killing旋量的能量动量张量解释为依赖于周围空间结构的张量的第二基本形式
Homogeneous Sasakian and 3-Sasakian Structures from the Spinorial Viewpoint
本文从旋量视角构造任意维数的Sasakian与3-Sasakian结构，完整刻画齐性3-Sasakian空间上的不变旋量空间及其基，并确定能诱导齐性3-Sasakian结构的Killing旋量。
$\mathcal{H}$-Killing Spinors and Spinorial Duality for Homogeneous 3-$(α,δ)$-Sasaki Manifolds
本文提出并研究推广 Killing 旋量的 $\mathcal{H}$-Killing 方程，证明其在齐性 3-$(α,δ)$-Sasaki 空间对偶对上的一一对应，并涵盖 7 维 $\text{G}_2$ 几何中的已知特殊旋量。
We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension $3.$ We in particular obtain a representation theorem for surfaces in the $\mathbb{L}(κ,τ)$ spaces. We then recover the Calabi correspondence between minimal surfaces in $\mathbb{R}^3$ and maximal surfaces in $\mathbb{R}_1^3$, and obtain a new Lawson type correspondence between CMC surfaces in $\mathbb{R}_1^3$ and in the 3-dimensional pseudo-hyperbolic space $\mathbb{H}_1^{3}.$
Invariant Spinors on Homogeneous Spheres
本文利用外形式刻画旋量表示，完整分类了球面九个齐性实现上的不变旋量，确定其维数并显式描述，恢复已知结果并首次给出四特征值广义 Killing 旋量例子。
We show stability of pairs of Ricci flat metrics and parallel spinor fields with respect to the spinor flow, i.e. we show that the spinor flow with initial conditions near such pairs converges to a critical point with exponential speed. Moreover, we show stability of certain volume constrained critical points of the spinorial energy.
Homogeneous $\mathbb Z/2$-Harmonic Forms and Spinors on $\R^4$ from Regular 4-Polytopes
研究四维空间中齐次Z/2调和1形式、自对偶2形式和旋量的局部奇异性模型，基于正则四维正多面体的1骨架锥结构。
Geometric aspects of transversal Killing spinors on Riemannian flows
研究了自旋黎曼流上的Killing旋量型方程。我们证明了可积性条件和部分分类的情况下，$M$是一个局部黎曼积，一个Sasakian流形或三维的方程的非平凡的解决方案，这些黎曼流$M$。
In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in \cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in \cite{Sc2018} for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling b…
We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Weiß, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volume-normalized standard 3-sphere together with a Killing spinor is a stable critical point of the volume-normalized version of the flow. Our results also include an example of a critical point of the volume-normalized flow on the 3-sphere, which is not a Killing spinor.
Harmonic spinors on homogeneous spaces
设G是紧的单李群，H是极大秩约化子群. G的不可约表示可以构造为关于齐性空间G/H上的狄拉克算子的调和旋量空间，该空间由与H的不可约（可能是射影）表示相关的丛扭曲。在这里，我们给出了一个快速证明这一结果，计算指数和内核的扭曲狄拉克算子使用齐次版本的外尔字符公式注意到格罗斯，Kostant，拉蒙德，和斯滕伯格，以及最近的工作Kostant关于代数版本的狄拉克算子。
We determine the holonomy of generalized Killing spinor covariant derivatives of the form $D= \nabla + Ω$ on pseudo-Riemannian reductive homogeneous spaces in a purely algebraic and algorithmic way, where $Ω: TM \rightarrow Λ^*(TM)$ is a left-invariant homomorphism. This is essentially an application of the theory of invariant principal bundle connections defined over homogeneous principal bundles. Moreover, the algorithm allows a computation of the associated Killing superalgebra in certain cases. The procedure is demonstrated by determining the supersymmetries of certain homogeneous M2 duals…
On pseudo-Riemannian manifolds with many Killing spinors
设$M$是一个维数为$n$，签名为$s$的伪黎曼自旋流形，用$N$表示实旋量丛的秩。我们证明了$M$是局部齐性的，如果它允许多个${3/4}N$独立的Killing旋量具有相同的Killing数，除非$n\equiv 1 \pmod 4$和$s\equiv 3 \pmod 4$。我们还证明了$M$是局部齐次的，如果它允许$k_+$独立的Killing旋量与Killing数$λ$和$k_-$独立的Killing旋量与Killing数$-λ$，使得$k_++k_->{3/2}N$，除非$n\equiv s\equiv 3\pmod 4$。类似地，具有多于一个${3/4}N$独立\emph{conformal} Killing旋量的伪黎曼流形是\emph{conformally}局部齐次的。对于（正或负）定度量，上述结果中的界${3/4}N$和${3/2}N$可以分别放宽为${1/2}N$和$N$.进一步证明了具有多个${3/4}N$平行旋量的伪黎曼自旋流形是平坦的，并且如果度量是确定的，则${1/4}N$平行旋量是足够的.类似地，一个黎曼自旋流形有多于一个${3/8}N$ Killing旋量，且Killing数为$λ\in \bR$，则它有常曲率$4λ^2$。对于洛伦兹度规或负定度规，在有界${1/2}N$中也是如此。最后，我们给出了允许Killing旋量的（不一定完备的）黎曼流形的一…
A class of generalised Killing spinors determined by the Ricci tensor and the metric
本文引入仿射Killing旋量（AKS），在调和曲率和局部共形平坦条件下分类允许AKS的黎曼自旋流形，刻画带平行1-形式的流形，并证明三维曲率齐次流形的局部齐次性及Bianchi度量的完整分类。
A Spinorial Perelman's Functional: Critical Points and Gradient Flow
本文提出一种统一Perelman泛函、Baldauf-Ouzch泛函和Dirichlet能量的旋量能量泛函，研究其临界点与梯度流的性质。
Parallel spinor flows on three-dimensional Cauchy hypersurfaces
该研究探讨三维柯西超曲面上的平行旋量流，证明其保持真空约束，给出里奇平坦洛伦兹流形平行旋量及pp波的初始数据刻画，求解左不变情形的永生条件并构造相关几何结构与曲率流解。
Spinor fields which are covariantly constant with respect to a connection with flux are of particular interest in unified string theories and supergravity theories, as their existence is required by supersymmetry. In this paper, flows of spinor fields are introduced which are parabolic and whose stationary points are such covariantly constant spinors.
On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, ϕ) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor ϕ. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.
Generalised Spin$^r$ Structures on Homogeneous Spaces
本文在配备 Lie 群作用的流形上引入并研究了 spin$^r$ 结构的 $G$-不变性，并基于迷向表示对齐性空间上的不变结构进行了分类，同时以球面的齐性实现为例进行了分析。
We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff-Wallach spaces $N_{k, l}={\rm SU}(3)/i_{k, l}(S^{1})$ (which are all nearly ${\rm G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability of most of the simply connected non-symmetric compact homogeneous Einstein spaces of dimensions $5, 6, $ and $7$, including the strict nearly Kähler ones (except ${\rm G}_2/{\rm SU}(3)$).
We present a novel approach to the classification of conformally equivariant differential operators on spinors in the case of homogeneous conformal geometry. It is based on the classification of solutions for a vector-valued system of partial differential equations, associated to $\mathcal{D}$-modules for the homogeneous conformal structure and controlled by the spin Howe duality for the orthogonal Lie algebras.
Weitzenböck Remainder Spectrum on Rational Homogeneous Varieties
本文研究有理齐性流形上闭不变(1,1)形式的谱，推导Weitzenböck余算子的谱并给出最小特征值公式，同时通过Atiyah-Singer定理和Borel-Weil-Bott定理分类Spin^c结构。
Torsion parallel spinors on Lorentzian four-manifolds and supersymmetric evolution flows on bundle gerbes
本文利用旋量多重形式理论研究洛伦兹四流形上的挠率平行旋量，并探讨由四维超对称NS-NS系统整体双曲解导出的演化流，将斜对称挠率解释为阿贝尔丛格上联络的曲率。
A Spinorial Heat Flow Framework for Geometric Degeneration on $3$-Manifolds
本文提出了一种基于旋量场驱动的 3 维流形几何退化分析框架，揭示了旋量节点行为与诱导度量退化之间的关系，并探讨了退化抛物理论中的开放问题。
We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Hull-Strominger system. Our main result investigates a method to produce new solutions of the Ricci flow and the Killing spinor equations. For this, we consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrat…
This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
This paper studies the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exoti…
Simply connected 3-dimensional homogeneous manifolds $E(κ, τ)$, with 4-dimensional isometry group, have a canonical Spin$^c$ structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into $E(κ, τ)$. As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in $E(κ, τ)$. Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spin$^c$ spinors.
A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$. An associative submanifold in $Y$ is a minimal 3-submanifold whose cone is Cayley. We study its deformations, namely, Cayley cone deformations, explicitly when it is homogeneous in the 7-sphere $S^{7}$.
Almost contact manifolds, connections with torsion and parallel spinors
我们分类局部齐次拟Sasakian流形在5维，承认一个平行旋量$ψ$的代数型$F \cdot ψ= 0$关于唯一的连接$\nabla$保持准Sasakian结构和完全反对称扭转。我们引入了几乎切触度量流形的一种共形变换，并讨论了它们与5维弦理论中的伸缩函数之间的联系。我们发现自然条件暗示共形不变性的平行旋量。我们提出了紧致情况下平行旋量存在的拓扑障碍。
In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow.
This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds…
We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.
Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)
本文研究了局部齐次三维流形上的正交叉曲率流。我们描述了这些流量的长期行为。我们结合这与早期的结果有关的负交叉曲率流的渐近行为来描述局部齐次3-流形上的交叉曲率流的最大解的双边行为。我们证明了局部齐次3-流形上的正交叉曲率流产生一个Heisenberg型次黎曼几何。
We show that for an immortal homogeneous Ricci flow solution any sequence of parabolic blow-downs subconverges to a homogeneous expanding Ricci soliton. This is established by constructing a new Lyapunov function based on curvature estimates which come from real geometric invariant theory.
Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly elliptic surfaces. We also construct expanding soliton solutions to the flow on the universal covers of these surfaces by taking blowdown limits of these homogeneous solutions.
In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.
Long-time behavior of awesome homogeneous Ricci flows
本文证明非紧流形上令人惊叹的齐性度量在里奇流下保持不变，并在万有覆盖不可缩时证实有限灭绝时间及动力学阿列克谢耶夫斯基猜想，同时分析了流的长时间极限。
We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we completely describe the behaviour of the homogeneous Ricci flow on this kind of spaces. Moreover, we investigate the existence of ancient solutions and relate this to the existence and non-existence of invariant Einstein metrics.
We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite exstinction time $T>0$ and we analyze its behaviour when $t\to T$. We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.
On the Isometry Group of Immortal Homogeneous Ricci Flows
本文研究非紧齐次流形上不朽齐次 Ricci 流解的等距群，并给出其结构性结论。
Supersymmetric Kundt four manifolds and their spinorial evolution flows
本文研究了配备实Killing旋量的四维洛伦兹流形的超对称Kundt配置，刻画了其局部等距的度量类，证明了旋量演化流等价于微分流方程并保持Einstein方程约束，同时分类了三维李群上的左不变演化流。
Ricci Flow Preserves Positive Sectional Curvature on Homogeneous Spheres
本文证明Ricci流在齐次球面和复射影空间上保持正截面曲率，结合先前结果完成了齐次空间中正曲率度量是否流出正曲率度量集的分类。
Motivated by the Hamilton's Ricci flow, we define the homogeneous flow of a parallelizable manifold and show the long time existence and uniqueness of its solutions on $[0,\infty).$ Using this flow, we outline a simple proof of the Poincare Conjecture.
Backward Ricci Flow on Locally Homogeneous Three-manifolds
本文研究了局部齐次三维流形上的向后里奇流。我们描述了长时间的行为，并表明，通常和适当的重新缩放后，有收敛到一个子黎曼几何。作者在交叉曲率流的情况下也观察到类似的行为。
Ricci flow on locally homogeneous closed 4-manifolds
讨论了齐次四维流形上的里奇流。在对这些流形进行分类之后，我们注意到存在初始度量族，使得我们可以将它们对角化，并且里奇流保持对角化。我们分析了这些家庭的长期行为。我们发现，如果一个解决方案存在的所有时间，那么流动表现出第三型奇异性的意义上的Hamilton。
On the existence of homogeneous solitons of gradient type for the G$_2$-Laplacian flow
本研究通过构造首个实例证明G₂-拉普拉斯流齐次梯度孤立子的存在性，指出G₂-拉普拉斯流是首个在一维扩张空间上存在这类孤立子的几何流。
This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; however, in a homogeneous manifold, the Ricci Flow reduces to an ODE. The behavior of Ricci Flow in two, three, and four dimensional homogenous manifolds has been calculated and is well understood. The work presented here will describe efforts to better understand the behavior of R…
The Geometry of Generalised Spin$^r$ Spinors on Projective Spaces
本文将外形式刻画自旋表示的方法推广到广义 spin$^r$ 情形，在复、四元数射影空间及 Cayley 平面的所有齐性实现上完整描述了最小非零 $r$ 对应的不变 spin$^r$ 旋量空间，并给出其几何意义。
In this paper we study backward Ricci flow of locally homogeneous geometries of $4$-manifolds which admit compact quotients. We describe the long-term behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian geometry after suitable rescaling.
We study the quasi-convergence equivalence of some families of metrics on locally homogeneous closed 4-manifolds with trivial isotropy group, and identify the dimension of each equivalence class under certain conditions.
We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a $\mathbb{Z}/2$ harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are $\mathbb{Z}/2$ harmonic 1-forms or spinors on $\mathbb{R}^3$ that are homogeneous with respect to rescaling of $\mathbb{R}^3$ with their zero locus consisting of four or more rays from the origin. The rays point from the origin to the vertices of a centered tetrahedron in one example; and they point from those of a centered octahedron and a centered icosahedron in two others.
On the long time behavior of ancient homogeneous Ricci flows
本文研究古老均匀里奇流的长时间行为，证明了其通过缩小序列收敛到梯度缩小里奇溶子的性质。
Finite extinction time of a family of homogeneous Ricci flows
本文证明了某些非紧致同质 Riemann 流形上 Ricci 流解具有有限时间灭绝，并展示了其在正标量曲率度量空间合同性上的应用。
Spectral action and heat kernel trace for Ricci flat manifolds from stochastic flow over second quantized $L^2$-differential forms
本文通过Fock空间上的量子随机微分方程的小时间流，在紧致Ricci平坦流形上计算连接拉普拉斯热核迹，并恢复平行旋量情形下的Dirac热核迹，从而将4维谱作用及爱因斯坦-希尔伯特作用实现为随机流。
Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows
本文利用黎曼群胚研究了三维交叉曲率流的齐次解的性质。黎曼广群技术，介绍了约翰洛特，使我们能够调查的长期行为崩溃的解决方案的流动，产生孤子解决方案的限制。我们还审查洛特的长期行为的三维齐次解的里奇流的结果，证明我们选择的坐标和审查广群技术。我们发现了Sol和Nil上的交叉曲率孤子度量，并证明了SL（2，R）的交叉曲率流仅限于Sol。
We show the existence of expanding solitons of the G$_2$-Laplacian flow on non-solvable Lie groups, and we give the first example of a steady soliton that is not an extremally Ricci pinched G$_2$-structure.
We prove that the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds is well posed. The proof is based on the derivation and analysis of suitable hyperbolic evolution equations given in terms of the Ricci tensor and other geometric objects. Moreover, we classify Riemannian manifolds satisfying the constraint conditions for this Cauchy problem. It is then possible to characterise certain holonomy reductions of globally hyperbolic manifolds with parallel null vector in terms of flow equations for Riemannian special holonomy metrics. For exceptional holonomy groups thes…
We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds $V_2\mathbb{R}^n$ and $V_{1+k_2}\mathbb{R}^n$, with $n = 1+k_2+k_3$. We use techniques from the theory of differential equations, in particular the Poincaré compactification.
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