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Publisher: Springer-Verlag (Total: 2351 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  [2351 journals]
• Selberg and Ruelle zeta functions for non-unitary twists
• Authors: Polyxeni Spilioti
Pages: 151 - 203
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: 2018-03-01
DOI: 10.1007/s10455-017-9571-3
Issue No: Vol. 53, No. 2 (2018)

• Isoparametric submanifolds in two-dimensional complex space forms
• Authors: José Carlos Díaz-Ramos; Miguel Domínguez-Vázquez; Cristina Vidal-Castiñeira
Pages: 205 - 216
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: 2018-03-01
DOI: 10.1007/s10455-017-9572-2
Issue No: Vol. 53, No. 2 (2018)

• The existence of J -holomorphic curves in almost Hermitian manifolds
• Authors: Qiang Tan
Pages: 217 - 231
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: 2018-03-01
DOI: 10.1007/s10455-017-9573-1
Issue No: Vol. 53, No. 2 (2018)

• Eigenvalues of the complex Laplacian on compact non-Kähler manifolds
• Authors: Gabriel J. H. Khan
Pages: 233 - 249
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: 2018-03-01
DOI: 10.1007/s10455-017-9574-0
Issue No: Vol. 53, No. 2 (2018)

• Slope instability of projective spaces blown up along a line
• Authors: Yoshinori Hashimoto
Pages: 251 - 264
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: 2018-03-01
DOI: 10.1007/s10455-017-9576-y
Issue No: Vol. 53, No. 2 (2018)

• Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$ M × e ψ N
• Authors: Isabel M. C. Salavessa
Pages: 265 - 281
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: 2018-03-01
DOI: 10.1007/s10455-017-9577-x
Issue No: Vol. 53, No. 2 (2018)

• Correction to: The heat flow for the full bosonic string
• Authors: Volker Branding
Pages: 283 - 286
Abstract: We clarify several items in the original paper.
PubDate: 2018-03-01
DOI: 10.1007/s10455-017-9591-z
Issue No: Vol. 53, No. 2 (2018)

• On some $${\mathcal {AC}}^{\perp }$$ AC ⊥ manifolds
• Authors: Włodzimierz Jelonek
Pages: 1 - 10
Abstract: We give a description of all Gray AC manifolds (M, g) whose Ricci tensor has two eigenvalues of multiplicity 1 and $$\dim M-1$$ .
PubDate: 2018-01-01
DOI: 10.1007/s10455-017-9564-2
Issue No: Vol. 53, No. 1 (2018)

• Mean curvature flow of area decreasing maps between Riemann surfaces
• Authors: Andreas Savas-Halilaj; Knut Smoczyk
Pages: 11 - 37
Abstract: In this article, we give a complete description of the evolution of an area decreasing map $$f{: }M\rightarrow N$$ , induced by the mean curvature of their graph, in the situation where M and N are complete Riemann surfaces with bounded geometry, M being compact, for which their sectional curvatures $$\sigma _M$$ and $$\sigma _N$$ satisfy $$\min \sigma _M\ge \sup \sigma _N$$ .
PubDate: 2018-01-01
DOI: 10.1007/s10455-017-9566-0
Issue No: Vol. 53, No. 1 (2018)

• Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$ S 3 × S 3
• Authors: Bart Dioos; Luc Vrancken; Xianfeng Wang
Pages: 39 - 66
Abstract: In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in $$\mathbb {S}^3 \times \mathbb {S}^3$$ . We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler $$\mathbb {S}^3 \times \mathbb {S}^3$$ can only be $$\frac{2}{\sqrt{3}}$$ or $$\frac{4}{\sqrt{3}}$$ .
PubDate: 2018-01-01
DOI: 10.1007/s10455-017-9567-z
Issue No: Vol. 53, No. 1 (2018)

• 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
Pages: 97 - 119
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: 2018-01-01
DOI: 10.1007/s10455-017-9569-x
Issue No: Vol. 53, No. 1 (2018)

• Rigidity of Riemannian manifolds with positive scalar curvature
• Authors: Guangyue Huang
Abstract: For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity theorem involving the Weyl curvature and the traceless Ricci curvature. Moreover, we provide a similar rigidity result with respect to the $$L^{\frac{n}{2}}$$ -norm of the Weyl curvature, the traceless Ricci curvature, and the Yamabe invariant. In particular, we also obtain rigidity results in terms of the Euler–Poincaré characteristic.
PubDate: 2018-02-15
DOI: 10.1007/s10455-018-9600-x

• Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of
revolution
• Authors: Rainer Mandel
Abstract: In this paper, we prove explicit formulas for all Willmore surfaces of revolution and demonstrate their use in the discussion of the associated Dirichlet boundary value problems. It is shown by an explicit example that symmetric Dirichlet boundary conditions do in general not entail the symmetry of the surface. In addition, we prove a symmetry result for a subclass of Willmore surfaces satisfying symmetric Dirichlet boundary data.
PubDate: 2018-02-09
DOI: 10.1007/s10455-018-9598-0

• Improved Moser–Trudinger inequality of Tintarev type in dimension n and
the existence of its extremal functions
• Authors: Van Hoang Nguyen
Abstract: Let $$\Omega$$ be a smooth bounded domain in $$\mathbb R^n$$ with $$n\ge 2$$ , $$W^{1,n}_0(\Omega )$$ be the usual Sobolev space on $$\Omega$$ and define $$\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega \nabla u ^n \mathrm{d}x}{\int _\Omega u ^n \mathrm{d}x}$$ . Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type \begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega \nabla u ^n \mathrm{{d}}x-\alpha \int _\Omega u ^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} u ^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned} for any $$0 \le \alpha < \lambda _1(\Omega )$$ , where $$\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}$$ with $$\omega _{n-1}$$ being the surface area of the unit sphere in $$\mathbb R^n$$ . This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any $$0< \alpha < \lambda _{1}(\Omega )$$ . (The case $$\alpha =0$$ corresponding to the Moser–Trudinger inequality is well known.)
PubDate: 2018-02-08
DOI: 10.1007/s10455-018-9599-z

• A classification of totally geodesic and totally umbilical Legendrian
submanifolds of $$(\kappa ,\mu )$$ ( κ , μ ) -spaces
• Authors: Alfonso Carriazo; Verónica Martín-Molina; Luc Vrancken
Abstract: We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of $$(\kappa ,\mu )$$ -spaces with Boeckx invariant $$I \le -1$$ . In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.
PubDate: 2018-01-31
DOI: 10.1007/s10455-018-9597-1

• Variations of the total mixed scalar curvature of a distribution
Abstract: We examine the total mixed scalar curvature of a smooth manifold endowed with a distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution and use this key and technical result to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g., on the distribution). For each of those cases, we obtain the Euler–Lagrange equation and its several solutions. Examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows, codimension-one foliations, and distributions of interesting geometric properties (e.g., totally umbilical and minimal).
PubDate: 2018-01-18
DOI: 10.1007/s10455-018-9594-4

• The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds
• Authors: Timothy Buttsworth
Abstract: Let $$G{/}H$$ be a compact homogeneous space, and let $$\hat{g}_0$$ and $$\hat{g}_1$$ be G-invariant Riemannian metrics on $$G/H$$ . We consider the problem of finding a G-invariant Einstein metric g on the manifold $$G/H\times [0,1]$$ subject to the constraint that g restricted to $$G{/}H\times \{0\}$$ and $$G/H\times \{1\}$$ coincides with $$\hat{g}_0$$ and $$\hat{g}_1$$ , respectively. By assuming that the isotropy representation of $$G/H$$ consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.
PubDate: 2018-01-16
DOI: 10.1007/s10455-018-9596-2

• Witten’s perturbation on strata with general adapted metrics
• Authors: Jesús A. Álvarez López; Manuel Calaza; Carlos Franco
Abstract: Let M be a stratum of a compact stratified space A. It is equipped with a general adapted metric g, which is slightly more general than the adapted metrics of Nagase and Brasselet–Hector–Saralegi. In particular, g has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then, g is called good. We consider the maximum/minimum ideal boundary condition, $$d_{\mathrm{max/min}}$$ , of the compactly supported de Rham complex on M, in the sense of Brüning–Lesch. Let $$H^*_{\mathrm{max/min}}(M)$$ and $$\Delta _{\mathrm{max/min}}$$ denote the cohomology and Laplacian of $$d_{\mathrm{max/min}}$$ . The first main theorem states that $$\Delta _{\mathrm{max/min}}$$ has a discrete spectrum satisfying a weak form of the Weyl’s asymptotic formula. The second main theorem is a version of Morse inequalities using $$H_{\mathrm{max/min}}^*(M)$$ and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for $$d_{\mathrm{max/min}}$$ of the Witten’s perturbation of the de Rham complex. Another ingredient is certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory. The condition on g to be good is general enough in the following sense. Assume that A is a stratified pseudomanifold, and consider its intersection homology $$I^{\bar{p}}H_*(A)$$ with perversity $$\bar{p}$$ ; in particular, the lower and upper middle perversities are denoted by $$\bar{m}$$ and $$\bar{n}$$ , respectively. Then, for any perversity $$\bar{p}\le \bar{m}$$ , there is an associated good adapted metric on M satisfying the Nagase isomorphism $$H^r_{\mathrm{max}}(M)\cong I^{\bar{p}}H_r(A)^*$$ ( $$r\in \mathbb {N}$$ ). If M is oriented and $$\bar{p}\ge \bar{n}$$ , we also get $$H^r_{\mathrm{min}}(M)\cong I^{\bar{p}}H_r(A)$$ . Thus our version of the Morse inequalities can be described in terms of $$I^{\bar{p}}H_*(A)$$ .
PubDate: 2018-01-13
DOI: 10.1007/s10455-017-9592-y

• Kähler structures on spaces of framed curves
• Authors: Tom Needham
Abstract: We consider the space $${\mathcal {M}}$$ of Euclidean similarity classes of framed loops in $${\mathbb {R}}^3$$ . Framed loop space is shown to be an infinite-dimensional Kähler manifold by identifying it with a complex Grassmannian. We show that the space of isometrically immersed loops studied by Millson and Zombro is realized as the symplectic reduction of $${\mathcal {M}}$$ by the action of the based loop group of the circle, giving a smooth version of a result of Hausmann and Knutson on polygon space. The identification with a Grassmannian allows us to describe the geodesics of $$\mathcal {M}$$ explicitly. Using this description, we show that $${\mathcal {M}}$$ and its quotient by the reparameterization group are nonnegatively curved. We also show that the planar loop space studied by Younes, Michor, Shah and Mumford in the context of computer vision embeds in $${\mathcal {M}}$$ as a totally geodesic, Lagrangian submanifold. The action of the reparameterization group on $${\mathcal {M}}$$ is shown to be Hamiltonian, and this is used to characterize the critical points of the weighted total twist functional.
PubDate: 2018-01-13
DOI: 10.1007/s10455-018-9595-3

• Deformation of the $$\sigma _2$$ σ 2 -curvature
• Authors: Almir Silva Santos; Maria Andrade
Abstract: Our main goal in this work is to deal with results concern to the $$\sigma _2$$ -curvature. First we find a symmetric 2-tensor canonically associated to the $$\sigma _2$$ -curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of $$\sigma _2$$ -singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and $$\sigma _2$$ -curvature. With a suitable condition on the $$\sigma _2$$ -curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative $$\sigma _2$$ -curvature unless it is flat.
PubDate: 2018-01-10
DOI: 10.1007/s10455-018-9593-5

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