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Publisher: Springer-Verlag (Total: 2352 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  [2352 journals]
• Asymptotic expansions and conformal covariance of the mass of conformal
differential operators
• Authors: Matthias Ludewig
Pages: 237 - 268
Abstract: Abstract We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns out to be given by the local zeta function of L. In particular, the constant term in the asymptotic expansion of the Green’s function $$L^{-1}$$ is often called the mass of L, which (in case that L is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant m-Laplace operators L (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of M is odd and that $$\ker L = 0$$ , and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied.
PubDate: 2017-10-01
DOI: 10.1007/s10455-017-9556-2
Issue No: Vol. 52, No. 3 (2017)

• p -Fundamental tone estimates of submanifolds with bounded mean curvature
• Authors: Israel Evangelista; Keomkyo Seo
Pages: 269 - 287
Abstract: Abstract In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan–Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one $$C^2$$ -foliations of open subsets $$\Omega$$ of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of $$\Omega$$ .
PubDate: 2017-10-01
DOI: 10.1007/s10455-017-9557-1
Issue No: Vol. 52, No. 3 (2017)

• On the structure of geodesic orbit Riemannian spaces
Pages: 289 - 311
Abstract: Abstract The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterized by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic orbit Riemannian space, we describe the structure of the nilradical and the radical of the Lie algebra of the isometry group. In the final part, we discuss some new tools to study geodesic orbit Riemannian spaces, related to compact Lie group representations with non-trivial principal isotropy algebras. We discuss also some new examples of geodesic orbit Riemannian spaces, new methods to obtain such examples, and some unsolved questions.
PubDate: 2017-10-01
DOI: 10.1007/s10455-017-9558-0
Issue No: Vol. 52, No. 3 (2017)

• Szegő kernel expansion and equivariant embedding of CR manifolds with
circle action
• Authors: Hendrik Herrmann; Chin-Yu Hsiao; Xiaoshan Li
Pages: 313 - 340
Abstract: Abstract Let X be a compact strongly pseudoconvex CR manifold with a transversal CR $$S^1$$ -action. In this paper, we establish the asymptotic expansion of Szegő kernels of positive Fourier components, and by using the asymptotics, we show that X can be equivariant CR embedded into some $$\mathbb {C}^N$$ equipped with a simple $$S^1$$ -action. An equivariant embedding of quasi-regular Sasakian manifold is also derived.
PubDate: 2017-10-01
DOI: 10.1007/s10455-017-9559-z
Issue No: Vol. 52, No. 3 (2017)

• Non-degenerate para-complex structures in 6D with large symmetry groups
• Authors: B. S. Kruglikov; H. Winther
Pages: 341 - 362
Abstract: Abstract For an almost product structure J on a manifold M of dimension 6 with non-degenerate Nijenhuis tensor $$N_J$$ , we show that the automorphism group $$G=\mathrm{Aut}(M,J)$$ has dimension at most 14. In the case of equality G is the exceptional Lie group $$G_2^*$$ . The next possible symmetry dimension is proved to be equal to 10, and G has Lie algebra $$\mathfrak {sp}(4,{\mathbb R})$$ . Both maximal and submaximal symmetric structures are globally homogeneous and strictly nearly para-Kähler. We also demonstrate that whenever the symmetry dimension is at least 9, then the automorphism algebra acts locally transitively.
PubDate: 2017-10-01
DOI: 10.1007/s10455-017-9561-5
Issue No: Vol. 52, No. 3 (2017)

• Neck analysis of extrinsic polyharmonic maps
• Authors: Wanjun Ai; Hao Yin
Pages: 129 - 156
Abstract: Abstract We prove the energy identity and the no neck property for a sequence of smooth extrinsic polyharmonic maps with bounded total energy.
PubDate: 2017-09-01
DOI: 10.1007/s10455-017-9551-7
Issue No: Vol. 52, No. 2 (2017)

• Basic Kirwan surjectivity for K -contact manifolds
• Authors: Lana Casselmann
Pages: 157 - 185
Abstract: Abstract We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free $$S^1$$ -action, the $$S^1$$ -quotient is a symplectic manifold, and our result reproduces Kirwan’s surjectivity for these symplectic manifolds. We further prove a Tolman–Weitsman type description of the kernel of the basic Kirwan map for $$S^1$$ -actions and show that torus actions on a K-contact manifold that preserve the contact form and admit 0 as a regular value of the contact moment map are equivariantly formal in the basic setting.
PubDate: 2017-09-01
DOI: 10.1007/s10455-017-9552-6
Issue No: Vol. 52, No. 2 (2017)

• Generic irreducibilty of Laplace eigenspaces on certain compact Lie groups
• Authors: Dorothee Schueth
Pages: 187 - 200
Abstract: Abstract If G is a compact Lie group endowed with a left invariant metric g, then G acts via pullback by isometries on each eigenspace of the associated Laplace operator  $$\Delta _g$$ . We establish algebraic criteria for the existence of left invariant metrics g on G such that each eigenspace of  $$\Delta _g$$ , regarded as the real vector space of the corresponding real eigenfunctions, is irreducible under the action of G. We prove that generic left invariant metrics on the Lie groups $$G={ SU}(2)\times \cdots \times { SU}(2)\times T$$ , where T is a (possibly trivial) torus, have the property just described. The same holds for quotients of such groups G by discrete central subgroups. In particular, it also holds for $${ SO}(3)$$ , $${ U}(2)$$ , $${ SO}(4)$$ .
PubDate: 2017-09-01
DOI: 10.1007/s10455-017-9553-5
Issue No: Vol. 52, No. 2 (2017)

• Weakly horospherically convex hypersurfaces in hyperbolic space
• Authors: Vincent Bonini; Jie Qing; Jingyong Zhu
Pages: 201 - 212
Abstract: Abstract In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces $$\phi :M^n \rightarrow \mathbb {H}^{n+1}$$ and a class of conformal metrics on domains of the round sphere $$\mathbb {S}^n$$ . Some of the key aspects of the correspondence and its consequences have dimensional restrictions $$n\ge 3$$ due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of $$\mathbb {S}^n$$ . In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions $$n\ge 2$$ in a unified way. In the case of a single point boundary $$\partial _{\infty }\phi (M)=\{x\} \subset \mathbb {S}^n$$ , we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in $$\mathbb {H}^{3}$$ .
PubDate: 2017-09-01
DOI: 10.1007/s10455-017-9554-4
Issue No: Vol. 52, No. 2 (2017)

• Taut contact circles and bi-contact metric structures on three-manifolds
• Authors: D. Perrone
Pages: 213 - 235
Abstract: Abstract Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), $${\widetilde{SL}}(2,{\mathbb {R}})$$ , $${\widetilde{E}}(2)$$ , E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure $$(\eta _1,\eta _2)$$ with the so-called Webster function $${\mathcal {W}}$$ constant along the flow of $$\xi _1$$ (equivalently $$\xi _2$$ ). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).
PubDate: 2017-09-01
DOI: 10.1007/s10455-017-9555-3
Issue No: Vol. 52, No. 2 (2017)

• Eigenvalues of the complex Laplacian on compact non-Kähler manifolds
• Authors: Gabriel J. H. Khan
Abstract: Abstract We consider $$\lambda$$ is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that $$\lambda \ge C$$ where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.
PubDate: 2017-10-13
DOI: 10.1007/s10455-017-9574-0

• Slope instability of projective spaces blown up along a line
• Authors: Yoshinori Hashimoto
Abstract: Abstract Let $$\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n$$ be a Kähler manifold obtained by blowing up a complex projective space $$\mathbb {P}^n$$ along a line $$\mathbb {P}^1$$ . We prove that $$\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n$$ is slope unstable with respect to any polarisation, and hence, it does not admit constant scalar curvature Kähler metrics in any rational Kähler class.
PubDate: 2017-10-03
DOI: 10.1007/s10455-017-9576-y

• Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$ M × e ψ N
• Authors: Isabel M. C. Salavessa
Abstract: Abstract If a graph submanifold (x, f(x)) of a Riemannian warped product space $$(M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)$$ is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, $$m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}$$ holds, where $$A_{\psi }(\partial D)$$ and $$V_{\psi }(D)$$ are the $${\psi }$$ -weighted area and volume, respectively. In particular, $$H=0$$ if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases $$n=1$$ or $$\psi =0$$ . We also conclude minimality using a closed calibration, assuming $$(M,g_*)$$ is complete where $$g_*=g+e^{2\psi }f^*h$$ , and for some constants $$\alpha \ge \delta \ge 0$$ , $$C_1>0$$ and $$\beta \in [0,1)$$ , $$\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta$$ , $$\mathrm {Ricci}_{\psi ,g_*}\ge \alpha$$ , and $${\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }$$ holds when $$r\rightarrow +\infty$$ , where r(x) is the distance function on $$(M,g_*)$$ from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.
PubDate: 2017-09-30
DOI: 10.1007/s10455-017-9577-x

• The index of Callias-type operators with Atiyah–Patodi–Singer
boundary conditions
• Authors: Pengshuai Shi
Abstract: Abstract We compute the index of a Callias-type operator with APS boundary condition on a manifold with compact boundary in terms of combination of indexes of induced operators on a compact hypersurface. Our result generalizes the classical Callias-type index theorem to manifolds with compact boundary.
PubDate: 2017-09-06
DOI: 10.1007/s10455-017-9575-z

• The existence of J -holomorphic curves in almost Hermitian manifolds
• Authors: Qiang Tan
Abstract: Abstract In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) be an almost Hermitian manifold and $$f:\Sigma \rightarrow M$$ be an injective immersion. We prove that if the $$L_p$$ functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic.
PubDate: 2017-08-31
DOI: 10.1007/s10455-017-9573-1

• Selberg and Ruelle zeta functions for non-unitary twists
• Authors: Polyxeni Spilioti
Abstract: Abstract In this paper we study the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd-dimensional manifold. These are functions of a complex variable s in some right half-plane of $$\mathbb {C}$$ . Using the Selberg trace formula for arbitrary finite dimensional representations of the fundamental group of the manifold, we establish the meromorphic continuation of the dynamical zeta functions to the whole complex plane. We explicitly describe the singularities of the Selberg zeta function in terms of the spectrum of certain twisted Laplace and Dirac operators.
PubDate: 2017-08-30
DOI: 10.1007/s10455-017-9571-3

• Isoparametric submanifolds in two-dimensional complex space forms
• Authors: José Carlos Díaz-Ramos; Miguel Domínguez-Vázquez; Cristina Vidal-Castiñeira
Abstract: Abstract We show that an isoparametric submanifold of a complex hyperbolic plane, according to the definition of Heintze, Liu and Olmos’, is an open part of a principal orbit of a polar action. We also show that there exists a non-isoparametric submanifold of the complex hyperbolic plane that is isoparametric according to the definition of Terng’s. Finally, we classify Terng-isoparametric submanifolds of two-dimensional complex space forms.
PubDate: 2017-08-16
DOI: 10.1007/s10455-017-9572-2

• Short-time behavior of the heat kernel and Weyl’s law on
$${{\mathrm{RCD}}}^*(K,N)$$ RCD ∗ ( K , N ) spaces
• Authors: Luigi Ambrosio; Shouhei Honda; David Tewodrose
Abstract: Abstract In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of $${{\mathrm{RCD}}}^{*}(K,N)$$ -spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in $${{\mathrm{RCD}}}^*(K,N)$$ -spaces. We use then these results to initiate the study of Weyl’s law in the $${{\mathrm{RCD}}}$$ setting.
PubDate: 2017-08-15
DOI: 10.1007/s10455-017-9569-x

• Prescribed scalar curvature plus mean curvature flows in compact manifolds
with boundary of negative conformal invariant
• Authors: Xuezhang Chen; Pak Tung Ho; Liming Sun
Abstract: Abstract Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let $$(M,g_0)$$ be a smooth compact manifold of dimension $$n\ge 3$$ with boundary. Given any smooth functions f in M and h on $$\partial M$$ , does there exist a conformal metric of $$g_0$$ such that its scalar curvature equals f and boundary mean curvature equals h' Assume that f and h are negative and the conformal invariant $$Q(M,\partial M)$$ is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.
PubDate: 2017-08-10
DOI: 10.1007/s10455-017-9570-4

• Cohomologies of locally conformally symplectic manifolds and solvmanifolds
• Authors: Daniele Angella; Alexandra Otiman; Nicoletta Tardini
Abstract: Abstract We study the Morse–Novikov cohomology and its almost-symplectic counterpart on manifolds admitting locally conformally symplectic structures. More precisely, we introduce lcs cohomologies and we study elliptic Hodge theory, dualities, Hard Lefschetz condition. We consider solvmanifolds and Oeljeklaus–Toma manifolds. In particular, we prove that Oeljeklaus–Toma manifolds with precisely one complex place, and under an additional arithmetic condition, satisfy the Mostow property. This holds in particular for the Inoue surface of type $$S^0$$ .
PubDate: 2017-07-17
DOI: 10.1007/s10455-017-9568-y

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