Annals of Global Analysis and Geometry
Journal Prestige (SJR): 1.228 Citation Impact (citeScore): 1 Number of Followers: 2 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729060  ISSN (Online) 0232704X Published by SpringerVerlag [2467 journals] 
 On the Steklov spectrum of covering spaces and total spaces

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Abstract: Abstract We show the existence of a natural DirichlettoNeumann map on Riemannian manifolds with boundary and bounded geometry, such that the bottom of the Dirichlet spectrum is positive. This map regarded as a densely defined operator in the \(L^2\) space of the boundary admits Friedrichs extension. We focus on the spectrum of this operator on covering spaces and total spaces of Riemannian principal bundles over compact manifolds.
PubDate: 20230117

 Delorme’s intertwining conditions for sections of homogeneous vector
bundles on two and threedimensional hyperbolic spaces
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Abstract: Abstract The description of the Paley–Wiener space for compactly supported smooth functions \(C^\infty _c(G)\) on a semisimple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for \(G=\textbf{SL}(2,\mathbb {R})^d\) ( \(d\in \mathbb {N}\) ) and \(G=\textbf{SL}(2,\mathbb {C})\) . Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.
PubDate: 20221205

 On the eigenforms of compact stratified spaces

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Abstract: Abstract Let X be a compact Thom–Mather stratified pseudomanifold, and let M be the regular part of X endowed with an iterated metric. In this paper, we prove that if the curvature operator of M is bounded, then the \(L^2\) harmonic space of M is finite dimensional. Next we consider the absolute eigenvalue problems of the Hodge Laplacian of a sequence of compact domains \(\Omega _j\) converging to M. We prove that when the curvature operator of M is bounded, the eigenvalues of \(\Omega _j\) converge to eigenvalues of M, and the eigenforms of \(\Omega _j\) converge to eigenforms of M in the Sobolev norm. This generalizes Chavel and Feldman’s theorem in Chavel and Feldman (J Funct Anal 30:198222, 1978) from compact manifolds to compact pseudomanifolds and from functions to differential forms. Then, we apply our results to \(L^2\) chomology. We will give a correspondence between boundary cohomology and \(L^2\) cohomology.
PubDate: 20221124

 The effect of pinching conditions in prescribing $$ Q $$ curvature on
standard spheres
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Abstract: Abstract In this paper, we study the problem of prescribing a fourthorder conformal invariant on standard spheres. This problem is variational but it is noncompact due to the presence of nonconverging orbits of the gradient flow, the so called critical points at infinity. Following the method advised by Bahri we determine all such critical points at infinity and compute their contribution to the difference of topology between the level sets of the associated Euler–Lagrange functional. We then derive some existence results under pinching conditions.
PubDate: 20221107
DOI: 10.1007/s10455022098786

 Complexes, residues and obstructions for logsymplectic manifolds

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Abstract: Abstract We consider compact Kählerian manifolds X of even dimension 4 or more, endowed with a logsymplectic structure \(\Phi \) , a generically nondegenerate closed 2form with simple poles on a divisor D with local normal crossings. A simple linear inequality involving the iterated Poincaré residues of \(\Phi \) at components of the double locus of D ensures that the pair \((X, \Phi )\) has unobstructed deformations and that D deforms locally trivially.
PubDate: 20221107
DOI: 10.1007/s1045502209881x

 Compact geodesic orbit spaces with a simple isotropy group

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Abstract: Abstract Let \(M=G/H\) be a compact, simply connected, Riemannian homogeneous space, where G is (almost) effective and H is a simple Lie group. In this paper, we first classify all Gnaturally reductive metrics on M, and then all Ggeodesic orbit metrics on M.
PubDate: 20221107
DOI: 10.1007/s10455022098777

 Higherpower harmonic maps and sections

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Abstract: Abstract The variational theory of higherpower energy is developed for mappings between Riemannian manifolds, and more generally sections of submersions of Riemannian manifolds, and applied to sections of Riemannian vector bundles and their sphere subbundles. A complete classification is then given for leftinvariant vector fields on threedimensional unimodular Lie groups equipped with an arbitrary leftinvariant Riemannian metric.
PubDate: 20221107
DOI: 10.1007/s10455022098759

 Umehara algebra and complex submanifolds of indefinite complex space forms

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Abstract: Abstract The Umehara algebra is studied with motivation on the problem of the nonexistence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the nonexistence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a complex manifold with a distinguished metric, such as homogeneous domains, the Hartogs triangle, the minimal ball, and the symmetrized polydisc, equipped with their intrinsic Bergman metrics, which generalizes more or less all existing results.
PubDate: 20221027
DOI: 10.1007/s10455022098768

 Nonexistence and rigidity of spacelike mean curvature flow solitons
immersed in a GRW spacetime
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Abstract: Abstract We study the nonexistence and rigidity of an important class of particular cases of trapped submanifolds, more precisely, ndimensional spacelike mean curvature flow solitons related to the closed conformal timelike vector field \(\mathcal K=f(t)\partial _t\) ( \(t\in I\subset \mathbb R\) ) which is globally defined on an \((n+p+1)\) dimensional generalized Robertson–Walker (GRW) spacetime \(I\times _fM^{n+p}\) with warping function \(f\in C^\infty (I)\) and Riemannian fiber \(M^{n+p}\) , via applications of suitable generalized maximum principles and under certain constraints on f and on the curvatures of \(M^{n+p}\) . In codimension 1, we also obtain new Calabi–Bernsteintype results concerning the spacelike mean curvature flow soliton equation in a GRW spacetime.
PubDate: 20221024
DOI: 10.1007/s10455022098795

 First $$\frac{2}{n}$$ stability eigenvalue of singular minimal
hypersurfaces in space forms
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Abstract: Abstract In this paper, we study the first \(\frac{2}{n}\) stability eigenvalue on singular minimal hypersurfaces in space forms. We provide a characterization of catenoids in space forms in terms of \(\frac{2}{n}\) stable eigenvalue. We emphasize that this result is even new in the regular setting.
PubDate: 20221021
DOI: 10.1007/s1045502209880y

 pKähler and balanced structures on nilmanifolds with nilpotent
complex structures
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Abstract: Abstract Let (X, J) be a nilmanifold with an invariant nilpotent complex structure. We study the existence of pKähler structures (which include Kähler and balanced metrics) on X. More precisely, we determine an optimal p such that there are no pKähler structures on X. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.
PubDate: 20220924
DOI: 10.1007/s10455022098679

 Nonhomogeneous expanding flows in hyperbolic spaces

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Abstract: Abstract In the present paper, we consider starshaped mean convex hypersurfaces of the real, complex and quaternionic hyperbolic space evolving by a class of nonhomogeneous expanding flows. For any choice of the ambient manifold, the initial conditions are preserved and the longtime existence of the flow is proved. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling, the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space; otherwise, it converges to a conformal multiple of the standard subRiemannian metric on the odddimensional sphere. Finally, in every case, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.
PubDate: 20220909
DOI: 10.1007/s1045502209873x

 On triangulations of orbifolds and formality

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Abstract: Abstract For an orbifold, there are two naturally associated differential graded algebras, one is the de Rham algebra of orbifold differential forms and the other one is the differential graded algebra of piecewise polynomial differential forms of a triangulation of the coarse space. In this paper, we prove that these two differential graded algebras are weakly equivalent; hence, the formality of these two differential graded algebras is consistent, when the triangulation is smooth. We show that global quotient orbifolds and global homogeneous isotropy orbifolds admit smooth triangulations; hence, the two kinds of formality coincide with each other for these orbifolds.
PubDate: 20220908
DOI: 10.1007/s1045502209874w

 On the second variation of the biharmonic Clifford torus in $$\mathbb
S^4$$ S 4
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Abstract: Abstract The flat torus \({{\mathbb T}}=\mathbb S^1\left( \frac{1}{2} \right) \times \mathbb S^1\left( \frac{1}{2} \right) \) admits a proper biharmonic isometric immersion into the unit 4dimensional sphere \(\mathbb S^4\) given by \(\Phi =i \circ \varphi \) , where \(\varphi :{{\mathbb T}}\rightarrow \mathbb S^3(\frac{1}{\sqrt{2}})\) is the minimal Clifford torus and \(i:\mathbb S^3(\frac{1}{\sqrt{2}}) \rightarrow \mathbb S^4\) is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion \(\Phi \) . After, we shall study in the detail the kernel of the generalised Jacobi operator \(I_2^\Phi \) . We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper, we shall analyse the specific contribution of \(\varphi \) to the biharmonic index and nullity of \(\Phi \) . In this context, we shall study a more general composition \({\tilde{\Phi }}=i \circ {\tilde{\varphi }}\) , where \({\tilde{\varphi }}: M^m \rightarrow \mathbb S^{n1}(\frac{1}{\sqrt{2}})\) , \( m \ge 1\) , \(n \ge {3}\) , is a minimal immersion and \(i:\mathbb S^{n1}(\frac{1}{\sqrt{2}}) \rightarrow \mathbb S^n\) is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of \({\tilde{\Phi }}\) is nonnegatively defined on \(\mathcal {C}\big ({\tilde{\varphi }}^{1}T\mathbb S^{n1}(\frac{1}{\sqrt{2}})\big )\) . Then, we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the pharmonic index and nullity of \(\varphi \) . In the final section, we compare our general results with those which can be deduced from the study of the equivariant second variation.
PubDate: 20220902
DOI: 10.1007/s10455022098697

 First integrals for Finsler metrics with vanishing $$\chi $$ χ
curvature
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Abstract: Abstract We prove that in a Finsler manifold with vanishing \(\chi \) curvature (in particular with constant flag curvature) some nonRiemannian geometric structures are geodesically invariant and hence they induce a set of nonRiemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature.
PubDate: 20220902
DOI: 10.1007/s1045502209872y

 Graded hypoellipticity of BGG sequences

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Abstract: Abstract This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein–Gelfand–Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a noncanonical way) and show that the generalized BGG sequences are Rockland in a graded sense.
PubDate: 20220902
DOI: 10.1007/s10455022098700

 Constructions of helicoidal minimal surfaces and minimal annuli in
$$\widetilde{E(2)}$$ E ( 2 ) ~
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Abstract: Abstract In this article, we construct two oneparameter families of properly embedded minimal surfaces in a threedimensional Lie group \(\widetilde{E(2)}\) , which is the universal covering of the group of rigid motions of Euclidean plane endowed with a leftinvariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrasstype representation introduced by Meeks, Mira, Pérez and Ros for minimal surfaces in Lie groups of dimension three. In the end, we study the limit of the catenoidal minimal surfaces. As an application of this limit case, we get a new proof of a halfspace theorem for minimal surfaces in \(\widetilde{E(2)}\) .
PubDate: 20220823
DOI: 10.1007/s1045502209871z

 Special metrics and scales in parabolic geometry

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Abstract: Abstract Given a parabolic geometry, it is sometimes possible to find special metrics characterised by some invariant conditions. In conformal geometry, for example, one asks for an Einstein metric in the conformal class. Einstein metrics have the special property that their geodesics are distinguished, as unparameterised curves, in the sense of parabolic geometry. This property characterises the Einstein metrics. In this article, we initiate a study of corresponding phenomena for other parabolic geometries, in particular for the hypersurface CR and contact Legendrean cases.
PubDate: 20220728
DOI: 10.1007/s1045502209866w

 Pullback functors for reduced and unreduced $$L^{q,p}$$ L q , p
cohomology
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Abstract: Abstract In this paper we study the reduced and unreduced \(L^{q,p}\) cohomology groups of oriented manifolds of bounded geometry and their behavior under uniform maps. A uniform map is a uniformly continuous map such that the diameter of the preimage of a subset is bounded in terms of the diameter of the subset itself. In general, for each \(p,q \in [1, +\infty )\) , the pullback map along a uniform map does not induce a morphism between the spaces of pintegrable forms or even in \(L^{q,p}\) cohomology. Then our goal is to introduce, for each p in \([1, +\infty )\) and for each uniform map f between manifolds of bounded geometry, an \({\mathcal {L}}^p\) bounded operator \(T_f\) , such that it does induce in a functorial way the appropriate morphism in reduced and unreduced \(L^{q,p}\) cohomology.
PubDate: 20220711
DOI: 10.1007/s10455022098599

 A note on the moduli spaces of holomorphic and logarithmic connections
over a compact Riemann surface
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Abstract: Abstract Let X be a compact Riemann surface of genus \(g \ge 3\) . We consider the moduli space of holomorphic connections over X and the moduli space of logarithmic connections singular over a finite subset of X with fixed residues. We determine the Chow group of these moduli spaces. We compute the global sections of the sheaves of differential operators on ample line bundles and their symmetric powers over these moduli spaces and show that they are constant under certain conditions. We show the Torellitype theorem for the moduli space of logarithmic connections. We also describe the rational connectedness of these moduli spaces.
PubDate: 20220711
DOI: 10.1007/s1045502209864y
