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Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in ({text {CAT}}(1)) spaces (1)空间中从强可直空间到正则球的调和映射的Dirichlet问题
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-09-16 DOI: 10.1007/s10455-023-09924-x
Yohei Sakurai

In this note, we study the Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in ({text {CAT}}(1)) space. Under the setting, we prove that the Korevaar–Schoen energy admits a unique minimizer.

在本文中,我们研究了在({text{CAT}}(1))空间中从强可直空间到正则球的调和映射的Dirichlet问题。在这种背景下,我们证明了Korevaar–Schoen能源允许一个独特的极小值。
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引用次数: 0
Modified conformal extensions 改进的共形扩展
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-09-13 DOI: 10.1007/s10455-023-09918-9
Matthias Hammerl, Katja Sagerschnig, Josef Šilhan, Vojtěch Žádník

We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive a complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show the vanishing of the Q-curvature.

我们给出了具有可积核的扭曲旋量的2n维分裂特征共形结构的几何构造和特征。该构造被认为是对n维投影流形的保角Patterson–Walker度量构造的改进。用扭旋子和保角Weyl曲率上的可积条件给出了其特征。我们进一步得到了爱因斯坦度量和无穷小共形对称性在适当投影数据方面的完整描述。最后,我们得到了一个显式几何构造的Fefferman–Graham环境度量,并展示了Q曲率的消失。
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引用次数: 0
On the intrinsic and extrinsic boundary for metric measure spaces with lower curvature bounds 关于曲率下界度量测度空间的内外边界
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-31 DOI: 10.1007/s10455-023-09920-1
Vitali Kapovitch, Xingyu Zhu

We show that if an Alexandrov space X has an Alexandrov subspace ({bar{Omega }}) of the same dimension disjoint from the boundary of X, then the topological boundary of ({bar{Omega }}) coincides with its Alexandrov boundary. Similarly, if a noncollapsed ({{,textrm{RCD},}}(K,N)) space X has a noncollapsed ({{,textrm{RCD},}}(K,N)) subspace ({bar{Omega }}) disjoint from boundary of X and with mild boundary condition, then the topological boundary of ({bar{Omega }}) coincides with its De Philippis–Gigli boundary. We then discuss some consequences about convexity of such type of equivalence.

我们证明了如果一个Alexandrov空间X有一个同维的Alexandrov子空间({bar{Omega}})与X的边界不相交,那么({bar{{Omega})的拓扑边界与其Alexandrov边界重合。类似地,如果一个非collapsed({{,textrm{RCD},}}(K,N))空间X有一个与X的边界不相交且具有温和边界条件的非collapsed ({,textrm{RCD}、}}}(K,N))子空间({bar{Omega}),则({bar{Omega}})的拓扑边界与其De Philippis–Gigli边界重合。然后我们讨论了这类等价的凸性的一些结果。
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引用次数: 0
Hadamard expansions for powers of causal Green’s operators and “resolvents” 因果格林算子幂的Hadamard展开式与“解”
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-29 DOI: 10.1007/s10455-023-09921-0
Lennart Ronge

The Hadamard expansion describes the singularity structure of Green’s operators associated with a normally hyperbolic operator P in terms of Riesz distributions (fundamental solutions on Minkowski space, transported to the manifold via the exponential map) and Hadamard coefficients (smooth sections in two variables, corresponding to the heat Kernel coefficients in the Riemannian case). In this paper, we derive an asymptotic expansion analogous to the Hadamard expansion for powers of advanced/retarded Green’s operators associated with P, as well as expansions for advanced/retarded Green’s operators associated with (P-z) for (zin mathbb {C}). These expansions involve the same Hadamard coefficients as the original Hadamard expansion, as well as the same or analogous (with built-in z-dependence) Riesz distributions.

Hadamard展开描述了与正双曲算子P相关的Green算子的奇异性结构,用Riesz分布(Minkowski空间上的基本解,通过指数映射传输到流形)和Hadamard系数(两个变量中的光滑部分,对应于黎曼情况下的热核系数)表示。在本文中,我们导出了与P相关的高级/延迟Green算子的幂的类似于Hadamard展开式的渐近展开式,以及与(P-z)相关的高级或延迟Green算子对(zInmathbb{C})的展开式。这些展开涉及与原始Hadamard展开相同的Hadamard系数,以及相同或类似的(具有内置的z依赖性)Riesz分布。
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引用次数: 0
Explicit harmonic morphisms and p-harmonic functions from the complex and quaternionic Grassmannians 复格拉斯曼和四元数格拉斯曼的显调和态射和p-调和函数
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-24 DOI: 10.1007/s10455-023-09919-8
Elsa Ghandour, Sigmundur Gudmundsson

We construct explicit complex-valued p-harmonic functions and harmonic morphisms on the classical compact symmetric complex and quaternionic Grassmannians. The ingredients for our construction method are joint eigenfunctions of the classical Laplace–Beltrami and the so-called conformality operator. A known duality principle implies that these p-harmonic functions and harmonic morphisms also induce such solutions on the Riemannian symmetric non-compact dual spaces.

我们在经典紧致对称复形和四元数Grassmann上构造了显式复值p-调和函数和调和态射。我们构造方法的成分是经典拉普拉斯-贝尔特拉米算子和所谓的保形算子的联合本征函数。一个已知的对偶原理意味着这些p-调和函数和调和态射也在黎曼对称非紧对偶空间上导出了这样的解。
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引用次数: 2
Quantitative version of Weyl’s law 魏尔定律的定量版本
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-20 DOI: 10.1007/s10455-023-09922-z
Nikhil Savale

We prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy (Comm Partial Differential Equations 15:1509–1563, 1990; Ann Global Anal Geom 8:127–136, 1990). Our result particularly improves Weyl remainder exponents for compact Lie groups and surfaces of revolution. And gives a quantitative estimate for Bérard’s Weyl remainder in terms of the maximal expansion rate and topological entropy of the geodesic flow.

我们用主符号Hamilton流的递推集的体积证明了半经典拟微分算子的Weyl余数的一般估计。这量化了Volovoy的早期结果(Comm偏微分方程15:1509-15631990;Ann Global Anal Geom 8:127-1361990)。我们的结果特别改进了紧致李群和公转曲面的Weyl余数指数。并根据测地流的最大展开率和拓扑熵,给出了Bérard的Weyl余数的定量估计。
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引用次数: 0
Almost contact metric manifolds with certain condition 具有一定条件的几乎接触度量流形
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-08 DOI: 10.1007/s10455-023-09917-w
Benaoumeur Bayour, Gherici Beldjilali, Moulay Larbi Sinacer

The object of this article is to study a new class of almost contact metric structures which are integrable but non normal. Secondly, we explain a method of construction for normal manifold starting from a non-normal but integrable manifold. Illustrative examples are given.

本文的目的是研究一类新的几乎接触度量结构,它是可积的但又是非正规的。其次,我们从一个非正规但可积的流形出发,解释了正规流形的一种构造方法。举例说明。
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引用次数: 0
Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one Picard数1的投影球面流形的最大Ricci下界
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-08 DOI: 10.1007/s10455-023-09915-y
DongSeon Hwang, Shin-young Kim, Kyeong-Dong Park

A horospherical variety is a normal G-variety such that a connected reductive algebraic group G acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The projective horospherical manifolds of Picard number one are classified by Pasquier, and it turned out that the automorphism groups of all nonhomogeneous ones are non-reductive, which implies that they admit no Kähler–Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be Kähler–Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat–Heckman measure based on a recent work of Delcroix and Hultgren. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian (text {SGr}(n,2n+1)) can be arbitrarily close to zero as n grows.

星形球簇是一个正规的G-簇,使得连通的约化代数群G与同构于有理齐次流形上的环面丛的开轨道作用。Pasquier对Picard数为1的投影星形球面流形进行了分类,证明了所有非齐次流形的自同构群都是非约化的,这意味着它们不允许Kähler–Einstein度量。作为Fano流形接近Kähler–Einstein程度的数值测度,我们根据Delcroix和Hultgren最近的一项工作,利用每个矩多面体相对于Duistermaat–Heckman测度的重心,计算了Picard一号投影球面流形的最大Ricci下界。特别地,随着n的增长,奇辛Grassmannian(text{SGr}(n,2n+1))的最大Ricci下界可以任意地接近于零。
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引用次数: 0
Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces 非紧化类空柯西超曲面的规定平均曲率流
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-02 DOI: 10.1007/s10455-023-09914-z
Giuseppe Gentile, Boris Vertman

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson–Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by considering globally hyperbolic Lorentzian space-times equipped with a specific class of warped product metrics.

在本文中,我们考虑了广义Robertson–Walker时空中有界几何的非紧类柯西超曲面的规定平均曲率流。我们证明了流动保持了空间相似性条件,并且存在于无限长的时间内。我们还证明了具有边界的流形集的收敛性。我们的讨论概括了Ecker、Huisken、Gerhardt和其他人以前关于一个关键方面的工作:我们在有界几何的假设下考虑任何非紧Cauchy超曲面。此外,我们通过考虑配备有一类特定翘曲积度量的全局双曲洛伦兹时空来专门化上述工作。
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引用次数: 2
The metric completion of the space of vector-valued one-forms 向量值一空间的度量完备形式
IF 0.7 3区 数学 Q3 MATHEMATICS Pub Date : 2023-08-01 DOI: 10.1007/s10455-023-09916-x
Nicola Cavallucci, Zhe Su

The space of full-ranked one-forms on a smooth, orientable, compact manifold (possibly with boundary) is metrically incomplete with respect to the induced geodesic distance of the generalized Ebin metric. We show a distance equality between the induced geodesic distances of the generalized Ebin metric on the space of full-ranked one-forms and the corresponding Riemannian metric defined on each fiber. Using this result, we immediately have a concrete description of the metric completion of the space of full-ranked one-forms. Additionally, we study the relationship between the space of full-ranked one-forms and the space of all Riemannian metrics, leading to quotient structures for the space of Riemannian metrics and its completion.

关于广义Ebin度量的诱导测地距离,光滑、可定向、紧致流形(可能有边界)上的全秩一形式的空间是度量不完备的。我们证明了全秩一形式空间上广义Ebin度量的诱导测地距离与每个纤维上定义的相应黎曼度量之间的距离相等。利用这个结果,我们立即得到了全秩一形式空间的度量完备的具体描述。此外,我们还研究了全秩一形式的空间与所有黎曼度量的空间之间的关系,得到了黎曼度量空间的商结构及其完备性。
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引用次数: 1
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Annals of Global Analysis and Geometry
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