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Publisher: Springer-Verlag   (Total: 2335 journals)

 Annals of Global Analysis and Geometry   [SJR: 1.136]   [H-I: 23]   [1 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1572-9060 - ISSN (Online) 0232-704X    Published by Springer-Verlag  [2335 journals]
• Metrics on 4-dimensional unimodular Lie groups
• Authors: Scott Van Thuong
Pages: 109 - 128
Abstract: We classify left invariant metrics on the 4-dimensional, simply connected, unimodular Lie groups up to automorphism. When the corresponding Lie algebra is of type (R), this is equivalent to classifying the left invariant metrics up to isometry, but in general the classification up to automorphism is finer than that up to isometry. In the abelian case, all left invariant metrics are isometric. In the nilpotent case, the space of metrics can have dimension 1 or 3. In the solvable case, the dimension can be 2, 4, or 5. There are two non-solvable 4-dimensional unimodular groups, and the space of metrics has dimension 6 in both of these cases.
PubDate: 2017-03-01
DOI: 10.1007/s10455-016-9527-z
Issue No: Vol. 51, No. 2 (2017)

• On vector bundle manifolds with spherically symmetric metrics
• Authors: R. Albuquerque
Pages: 129 - 154
Abstract: We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $$E\longrightarrow M$$ , over a Riemannian manifold M, when E is endowed with a metric connection. The tangent bundle of E admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of E; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant–Salamon type $${\mathrm {G}_{2}}$$ manifolds.
PubDate: 2017-03-01
DOI: 10.1007/s10455-016-9528-y
Issue No: Vol. 51, No. 2 (2017)

• Generalized deformations and holomorphic Poisson cohomology of
solvmanifolds
• Authors: Hisashi Kasuya
Pages: 155 - 177
Abstract: We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explicit finite-dimensional cochain complexes which computes the holomorphic Poisson cohomology of nilmanifolds and solvmanifolds.
PubDate: 2017-03-01
DOI: 10.1007/s10455-016-9529-x
Issue No: Vol. 51, No. 2 (2017)

• Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes
• Authors: Giuseppe Pipoli; Carlo Sinestrari
Pages: 179 - 188
Abstract: We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.
PubDate: 2017-03-01
DOI: 10.1007/s10455-016-9530-4
Issue No: Vol. 51, No. 2 (2017)

• The slab theorem for minimal surfaces in $$\mathbb {E}(-1,\tau )$$ E ( - 1
, τ )
• Authors: Vanderson Lima
Pages: 189 - 208
Abstract: Unlike $$\mathbb {R}^{3}$$ , the homogeneous spaces $$\mathbb {E}(-1,\tau )$$ have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantee that a minimal surface in $$\mathbb {E}(-1,\tau )$$ is such a graph. More specifically, we introduce the definition of a generalized slab in $$\mathbb {E}(-1,\tau )$$ and prove that a properly immersed minimal surface of finite topology inside such a slab region has multi-graph ends. Moreover, when the surface is embedded, the ends are graphs. When the surface is embedded and simply connected, it is an entire graph.
PubDate: 2017-03-01
DOI: 10.1007/s10455-016-9531-3
Issue No: Vol. 51, No. 2 (2017)

• The first positive eigenvalue of the sub-Laplacian on CR spheres
• Authors: Amine Aribi; Ahmad El Soufi
Pages: 1 - 9
Abstract: We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudo-convex pseudo-Hermitian structure $$\theta$$ on the CR sphere $$\mathbb {S}^{2n+1}\subset \mathbb {C}^{n+1}$$ , achieves its maximum when $$\theta$$ is the standard contact form.
PubDate: 2017-01-01
DOI: 10.1007/s10455-016-9519-z
Issue No: Vol. 51, No. 1 (2017)

• Twistor spaces of Riemannian manifolds with even Clifford structures
• Authors: Gerardo Arizmendi; Charles Hadfield
Pages: 11 - 20
Abstract: In this paper, we introduce the twistor space of a Riemannian manifold with an even Clifford structure. This notion generalizes the twistor space of quaternion-Hermitian manifolds and weak- $$\mathrm {Spin}(9)$$ structures. We also construct almost complex structures on the twistor space for parallel even Clifford structures and check their integrability. Moreover, we prove that in some cases one can give Kähler and nearly Kähler metrics to these spaces.
PubDate: 2017-01-01
DOI: 10.1007/s10455-016-9520-6
Issue No: Vol. 51, No. 1 (2017)

• The log-term of the Bergman kernel of the disc bundle over a homogeneous
Hodge manifold
• Authors: Andrea Loi; Roberto Mossa; Fabio Zuddas
Pages: 35 - 51
Abstract: We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kähler–Einstein manifold of classical type. Our results extends that in (Engliš and Zhang, Math Z 264(4):901–912, 2010) for the case of Hermitian symmetric spaces of compact type.
PubDate: 2017-01-01
DOI: 10.1007/s10455-016-9522-4
Issue No: Vol. 51, No. 1 (2017)

• Quasi-classical generalized CRF structures
• Authors: Izu Vaisman
Pages: 53 - 71
Abstract: In an earlier paper, we studied manifolds M endowed with a generalized F structure $$\Phi \in \mathrm{End}(TM\oplus T^*M)$$ , skew-symmetric with respect to the pairing metric, such that $$\Phi ^3+\Phi =0$$ . Furthermore, if $$\Phi$$ is integrable (in some well-defined sense), $$\Phi$$ is a generalized CRF structure. In the present paper, we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields $$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$ , where $$A^3+A=0$$ and some relations between A and $$\pi$$ hold. We establish the integrability conditions in terms of $$(A,\pi )$$ . They include the facts that A is a classical CRF structure, $$\pi$$ is a Poisson bivector field and $$\mathrm{im}\,A$$ is a (non)holonomic Poisson submanifold of $$(M,\pi )$$ . We discuss the case where either $$\mathrm{ker}\,A$$ or $$\mathrm{im}\,A$$ is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of $$\mathrm{im}\,A$$ inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of $$\pi$$ , including an associated spectral sequence and a Dolbeault type grading.
PubDate: 2017-01-01
DOI: 10.1007/s10455-016-9523-3
Issue No: Vol. 51, No. 1 (2017)

• On f -non-parabolic ends for Ricci-harmonic metrics
• Authors: Lin Feng Wang
Pages: 91 - 107
Abstract: In this paper, we study gradient Ricci-harmonic soliton metrics and quasi Ricci-harmonic metrics (both metrics are called Ricci-harmonic). First, we prove that all ends of $$\tau$$ -quasi Ricci-harmonic metrics with $$\tau >1$$ should be f-non-parabolic if $$\lambda =0,\mu >0$$ , or $$\lambda <0, \mu \ge 0$$ . For the case that $$\lambda<0, \mu < 0$$ , we can also arrive at the f-non-parabolic property if we add a condition about the scalar curvature. Furthermore, we discuss the connectivity at infinity for quasi Ricci-harmonic metrics. We also conclude that all ends of steady or expanding gradient Ricci-harmonic solitons should be f-non-parabolic, based on which we establish structure theorems for these two solitons.
PubDate: 2017-01-01
DOI: 10.1007/s10455-016-9525-1
Issue No: Vol. 51, No. 1 (2017)

• Cohomogeneity one Kähler and Kähler–Einstein manifolds
with one singular orbit I
• Authors: Dmitri Alekseevsky; Fabio Zuddas
Abstract: Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit $$S_0 = G/H$$ . Then M is G-diffeomorphic to the total space $$G \times _H V$$ of the homogeneous vector bundle over $$S_0$$ defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction $$\mu : S = Gx = G/L \rightarrow F = G/K$$ of the moment map of M to a regular orbit $$S=G/L$$ is a holomorphic map of S with the induced CR structure onto a flag manifold $$F = G/K$$ , where $$K = N_G(L)$$ , endowed with an invariant complex structure $$J^F$$ . We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with $$(F = G/K,J^F)$$ and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.
PubDate: 2017-03-11
DOI: 10.1007/s10455-017-9550-8

• Homogeneous Weyl connections of non-positive curvature
• Authors: Gabriela Tereszkiewicz; Maciej P. Wojtkowski
Abstract: We study homogeneous Weyl connections with non-positive sectional curvatures. The Cartesian product $${\mathbb S}^1 \times M$$ carries canonical families of Weyl connections with such a property, for any compact Riemmanian manifold M. We prove that if a homogeneous Weyl connection on a manifold, modeled on a unimodular Lie group, is non-positive in a stronger sense (stretched non-positive), then it must be locally of the product type.
PubDate: 2017-02-27
DOI: 10.1007/s10455-016-9526-0

• Erratum to: Almost complex structures in 6D with non-degenerate Nijenhuis
tensors and large symmetry groups
• Authors: B. Kruglikov; H. Winther
Abstract: We correct an error in the second part of Theorem 3 of our original paper.
PubDate: 2017-02-23
DOI: 10.1007/s10455-017-9546-4

• Surfaces in a pseudo-sphere with harmonic or 1-type pseudo-spherical Gauss
map
• Authors: Burcu Bektaş; Joeri Van der Veken; Luc Vrancken
Abstract: We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification is obtained, while in other cases the solutions are described by an explicit system of partial differential equations.
PubDate: 2017-02-23
DOI: 10.1007/s10455-017-9548-2

• On the semiclassical functional calculus for h -dependent functions
• Authors: Benjamin Küster
Abstract: We study the functional calculus for operators of the form $$f_h(P(h))$$ within the theory of semiclassical pseudodifferential operators, where $$\{f_h\}_{h\in (0,1]}\subset \mathrm{C^\infty _c}({{\mathbb {R}}})$$ denotes a family of h-dependent functions satisfying some regularity conditions, and P(h) is either an appropriate self-adjoint semiclassical pseudodifferential operator in $$\mathrm{L}^2({{\mathbb {R}}}^n)$$ or a Schrödinger operator in $$\mathrm{L}^2(M), M$$ being a closed Riemannian manifold of dimension n. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of P(h) in spectral windows of width of order $$h^\delta$$ , where $$0\le \delta <\frac{1}{2}$$ .
PubDate: 2017-02-22
DOI: 10.1007/s10455-017-9549-1

• Regularity of maps between Sobolev spaces
• Authors: Martins Bruveris
Abstract: Let $$F : H^q \rightarrow H^q$$ be a $$C^k$$ -map between Sobolev spaces, either on $${{\mathbb {R}}}^d$$ or on a compact manifold. We show that equivariance of F under the diffeomorphism group allows to trade regularity of F as a nonlinear map for regularity in the image space: for $$0 \le l \le k$$ , the map $$F: H^{q+l} \rightarrow H^{q+l}$$ is well defined and of class $$C^{k-l}$$ . This result is used to study the regularity of the geodesic boundary value problem for Sobolev metrics on the diffeomorphism group and the space of curves.
PubDate: 2017-02-13
DOI: 10.1007/s10455-017-9544-6

• Liouville-type theorems for CC-harmonic maps from Riemannian manifolds to
pseudo-Hermitian manifolds
• Authors: Tian Chong; Yuxin Dong; Yibin Ren
Abstract: In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.
PubDate: 2017-02-06
DOI: 10.1007/s10455-017-9547-3

• On the classification of 4-dimensional $$(m,\rho )$$ ( m , ρ )
-quasi-Einstein manifolds with harmonic Weyl curvature
• Authors: Jinwoo Shin
Abstract: In this paper we study four-dimensional $$(m,\rho )$$ -quasi-Einstein manifolds with harmonic Weyl curvature when $$m\notin \{0,\pm 1,-2,\pm \infty \}$$ and $$\rho \notin \{\frac{1}{4},\frac{1}{6}\}$$ . We prove that a non-trivial $$(m,\rho )$$ -quasi-Einstein metric g (not necessarily complete) is locally isometric to one of the following: (i) $${\mathcal {B}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}$$ , where $${\mathcal {B}}^2_\frac{R}{2(m+2)}$$ is the northern hemisphere in the two-dimensional (2D) sphere $${\mathbb {S}}^2_\frac{R}{2(m+2)}$$ , $${\mathbb {N}}_\delta$$ is a 2D Riemannian manifold with constant curvature $$\delta$$ , and R is the constant scalar curvature of g. (ii) $${\mathcal {D}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}$$ , where $${\mathcal {D}}^2_\frac{R}{2(m+2)}$$ is half (cut by a hyperbolic line) of hyperbolic plane $${\mathbb {H}}^2_\frac{R}{2(m+2)}$$ . (iii) $${\mathbb {H}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}$$ . (iv) A certain singular metric with $$\rho =0$$ . (v) A locally conformal flat metric. By applying this local classification, we obtain a classification of the complete $$(m,\rho )$$ -quasi-Einstein manifolds given the condition of a harmonic Weyl curvature. Our result can be viewed as a local classification of gradient Einstein-type manifolds. A corollary of our result is the classification of $$(\lambda ,4+m)$$ -Einstein manifolds, which can be viewed as (m, 0)-quasi-Einstein manifolds.
PubDate: 2017-01-30
DOI: 10.1007/s10455-017-9542-8

• On toric locally conformally Kähler manifolds
• Authors: Farid Madani; Andrei Moroianu; Mihaela Pilca
Abstract: We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $$-\infty$$ , and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.
PubDate: 2017-01-25
DOI: 10.1007/s10455-017-9545-5

• On locally conformally flat critical metrics for quadratic functionals
• Authors: A. Barros; A. Da Silva
Abstract: The aim of this note is to present some results about critical metrics for quadratic functional $${\mathcal {F}}_{t,s}$$ defined on $${\mathcal {M}}_{1}=\{g\in {\mathcal {M}} \mathrm{Vol}(g)=1\}$$ , where $${\mathcal {M}}$$ is the space of smooth Riemannian metrics on a closed smooth Riemannian manifold $$M^n$$ . We know that space form metrics are critical point for $${\mathcal {F}}_{t,s}$$ . We investigate when the converse is true. In particular, we show that locally conformally flat critical metrics with some additional conditions are space form metrics.
PubDate: 2017-01-09
DOI: 10.1007/s10455-016-9541-1

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