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 [2468 journals] 
 Some regularity of submetries

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Abstract: Abstract We discuss regularity statements for equidistant decompositions of Riemannian manifolds and for the corresponding quotient spaces. We show that any stratum of the quotient space has curvature locally bounded from both sides.
PubDate: 20240221

 Subgraphs of BV functions on RCD spaces

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Abstract: Abstract In this work, we extend classical results for subgraphs of functions of bounded variation in \(\mathbb R^n\times \mathbb R\) to the setting of \({\textsf{X}}\times \mathbb R\) , where \({\textsf{X}}\) is an \({\textrm{RCD}}(K,N)\) metric measure space. In particular, we give the precise expression of the pushforward onto \({\textsf{X}}\) of the perimeter measure of the subgraph in \({\textsf{X}}\times \mathbb R\) of a \({\textrm{BV}}\) function on \({\textsf{X}}\) . Moreover, in properly chosen good coordinates, we write the precise expression of the normal to the boundary of the subgraph of a \({\textrm{BV}}\) function f with respect to the polar vector of f, and we prove changeofvariable formulas.
PubDate: 20240217

 Some remarks on almost Hermitian functionals

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Abstract: Abstract We study critical points of natural functionals on various spaces of almost Hermitian structures on a compact manifold \(M^{2n}\) . We present a general framework, introducing the notion of gradient of an almost Hermitian functional. As a consequence of the diffeomorphism invariance, we show that a Schur’s type theorem still holds for general almost Hermitian functionals, generalizing a known fact for Riemannian functionals. We present two concrete examples, the Gauduchon’s functional and a close relative of it. These functionals have been studied previously, but not in the most general setup as we do here, and we make some new observations about their critical points.
PubDate: 20240131

 On subelliptic harmonic maps with potential

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Abstract: Abstract Let \((M,H,g_H;g)\) be a subRiemannian manifold and (N, h) be a Riemannian manifold. For a smooth map \(u: M \rightarrow N\) , we consider the energy functional \(E_G(u) = \frac{1}{2} \int _M[ \textrm{d}u_\text {H} ^2  2\,G(u)] \textrm{d}V_M\) , where \(\textrm{d}u_\text {H}\) is the horizontal differential of u, \(G:N\rightarrow \mathbb {R}\) is a smooth function on N. The critical maps of \(E_G(u)\) are referred to as subelliptic harmonic maps with potential G. In this paper, we investigate the existence problem for subelliptic harmonic maps with potentials by a subelliptic heat flow. Assuming that the target Riemannian manifold has nonpositive sectional curvature and the potential G satisfies various suitable conditions, we prove some Eells–Sampsontype existence results when the source manifold is either a step2 subRiemannian manifold or a stepr subRiemannian manifold whose subRiemannian structure comes from a tense Riemannian foliation.
PubDate: 20240130

 Almost CR manifolds with contracting CR automorphism

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Abstract: Abstract In this paper, we deal with a strongly pseudoconvex almost CR manifold with a CR contraction. We will prove that the stable manifold of the CR contraction is CR equivalent to the Heisenberg group model.
PubDate: 20240123

 Instability of a family of examples of harmonic maps

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Abstract: Abstract The radial map u(x) \(=\) \(\frac{x}{\Vert x\Vert }\) is a wellknown example of a harmonic map from \({\mathbb {R}}^m\,\,\{0\}\) into the spheres \({\mathbb {S}}^{m1}\) with a point singularity at x \(=\) 0. In Nakauchi (Examples Counterexamples 3:100107, 2023), the author constructed recursively a family of harmonic maps \(u^{(n)}\) into \({\mathbb {S}}^{m^n1}\) with a point singularity at the origin \((n = 1,\,2,\ldots )\) , such that \(u^{(1)}\) is the above radial map. It is known that for m \(\ge \) 3, the radial map \(u^{(1)}\) is not only stable as a harmonic map but also a minimizer of the energy of harmonic maps. In this paper, we show that for n \(\ge \) 2, \(u^{(n)}\) may be unstable as a harmonic map. Indeed we prove that under the assumption n > \({\displaystyle \frac{\sqrt{3}1}{2}\,(m1)}\) \((m \ge 3\) , \(n \ge 2)\) , the map \(u^{(n)}\) is unstable as a harmonic map. It is remarkable that they are unstable and our result gives many examples of unstable harmonic maps into the spheres with a point singularity at the origin.
PubDate: 20240109

 Modular geodesics and wedge domains in noncompactly causal symmetric
spaces
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Abstract: Abstract We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3grading). Since any Euler element of a semisimple Lie algebra specifies a canonical noncompactly causal symmetric space \(M = G/H\) , we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the socalled observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic. It can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for Gtranslates of open Horbits in the minimal flag manifold specified by the 3grading.
PubDate: 20231231

 Immersions of Sasaki–Ricci solitons into homogeneous Sasakian
manifolds
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Abstract: Abstract We discuss local Sasakian immersion of Sasaki–Ricci solitons (SRS) into fiber products of homogeneous Sasakian manifolds. In particular, we prove that SRS locally induced by a large class of fiber products of homogeneous Sasakian manifolds are, in fact, \(\eta \) Einstein. The results are stronger for immersions into Sasakian space forms. Moreover, we show an example of a Kähler–Ricci soliton on \(\mathbb C^n\) which admits no local holomorphic isometry into products of homogeneous bounded domains with flat Kähler manifolds and generalized flag manifolds.
PubDate: 20231214

 Optimal transport approach to Michael–Simon–Sobolev inequalities in
manifolds with intermediate Ricci curvature lower bounds
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Abstract: Abstract We generalize McCann’s theorem of optimal transport to a submanifold setting and use it to prove Michael–Simon–Sobolev inequalities for submanifolds in manifolds with lower bounds on intermediate Ricci curvatures. The results include a variant of the sharp Michael–Simon–Sobolev inequality in Brendle’s (arXiv:2009.13717) when the intermediate Ricci curvatures are nonnegative.
PubDate: 20231213

 From complex contact structures to real almost contact 3structures

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Abstract: Abstract We prove that every complex contact structure gives rise to a distinguished type of almost contact metric 3structure. As an application, we provide several new examples of manifolds which admit taut contact circles, taut and round almost cosymplectic 2spheres, and almost hypercontact (metric) structures. These examples generalize the wellknown examples of contact circles defined by the LiouvilleCartan forms on the unit cotangent bundle of Riemann surfaces. Further, we provide sufficient conditions for a compact complex contact manifold to be the twistor space of a positive quaternionic Kähler manifold.
PubDate: 20231212

 Families of degenerating Poincaré–Einstein metrics on $$\mathbb
{R}^4$$
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Abstract: Abstract We provide the first example of continuous families of Poincaré–Einstein metrics developing cusps on the trivial topology \(\mathbb {R}^4\) . We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Plebański–Demiański. We additionally indicate how to construct similar examples on more complicated topologies.
PubDate: 20231206

 Commutativity of quantization with conic reduction for torus actions on
compact CR manifolds
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Abstract: Abstract We define conic reductions \(X^{\textrm{red}}_{\nu }\) for torus actions on the boundary X of a strictly pseudoconvex domain and for a given weight \(\nu \) labeling a unitary irreducible representation. There is a natural residual circle action on \(X^{\textrm{red}}_{\nu }\) . We have two natural decompositions of the corresponding Hardy spaces H(X) and \(H(X^{\textrm{red}}_{\nu })\) . The first one is given by the ladder of isotypes \(H(X)_{k\nu }\) , \(k\in {\mathbb {Z}}\) ; the second one is given by the kth Fourier components \(H(X^{\textrm{red}}_{\nu })_k\) induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for k sufficiently large. The result is given for spaces of (0, q)forms with \(L^2\) coefficient when X is a CR manifold with nondegenerate Levi form.
PubDate: 20231129
DOI: 10.1007/s1045502309931y

 Sasaki–Einstein 7manifolds and Orlik’s conjecture

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Abstract: Abstract We study the homology groups of certain 2connected 7manifolds admitting quasiregular Sasaki–Einstein metrics, among them, we found 52 new examples of Sasaki–Einstein rational homology 7spheres, extending the list given by Boyer et al. (Ann Inst Fourier 52(5):1569–1584, 2002). As a consequence, we exhibit new families of positive Sasakian homotopy 9spheres given as cyclic branched covers, determine their diffeomorphism types and find out which elements do not admit extremal Sasaki metrics. We also improve previous results given by Boyer (Note Mat 28:63–105, 2008) showing new examples of Sasaki–Einstein 2connected 7manifolds homeomorphic to connected sums of \(S^3\times S^4\) . Actually, we show that manifolds of the form \(\#k\left( S^{3} \times S^{4}\right) \) admit Sasaki–Einstein metrics for 22 different values of k. All these links arise as Thom–Sebastiani sums of chain type singularities and cycle type singularities where Orlik’s conjecture holds due to a recent result by Hertling and Mase (J Algebra Number Theory 16(4):955–1024, 2022).
PubDate: 20231116
DOI: 10.1007/s1045502309930z

 Boundary properties for a MongeAmpère equation of prescribed affine
Gauss curvature
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Abstract: Abstract Considering a MongeAmpère equation with prescribed affine Gauss curvature, we first show the completeness of centroaffine metric on the convex domain and derive a gradient estimate of the convex solution and then give different orders of two eigenvalues of the Hessian with respect to the distance function. We also show that the curvature of level sets of the convex solution is uniformly bounded, and show that there exist a class of Euclideancomplete hyperbolic surfaces with prescribed affine Gauss curvature and with bounded affine principal curvatures.
PubDate: 20231116
DOI: 10.1007/s1045502309933w

 Berglund–Hübsch transpose rule and Sasakian geometry

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Abstract: Abstract We apply the Berglund–Hübsch transpose rule from BHK mirror symmetry to show that to an \(n1\) dimensional Calabi–Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension \(2n+1\) which are \(n1\) connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exist four sevendimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki–Einstein.
PubDate: 20231116
DOI: 10.1007/s1045502309932x

 The metric structure of compact rankone ECS manifolds

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Abstract: Abstract PseudoRiemannian manifolds with nonzero parallel Weyl tensor which are not locally symmetric are known as ECS manifolds. Every ECS manifold carries a distinguished null parallel distribution \(\mathcal {D}\) , the rank \(d\in \{1,2\}\) of which is referred to as the rank of the manifold itself. Under a natural genericity assumption on the Weyl tensor, we fully describe the universal coverings of compact rankone ECS manifolds. We then show that any generic compact rankone ECS manifold must be translational, in the sense that the holonomy group of the natural flat connection induced on \(\mathcal {D}\) is either trivial or isomorphic to \({\mathbb {Z}}_2\) . We also prove that all fourdimensional rankone ECS manifolds are noncompact, this time without having to assume genericity, as it is always the case in dimension four.
PubDate: 20231026
DOI: 10.1007/s10455023099296

 Harmonic flow of geometric structures

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Abstract: We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (Differ Geom Appl 19:193–210, 2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of analytic properties for this flow, such as uniqueness, smoothness, shorttime existence, and some sufficient conditions for longtime existence. This description potentially subsumes a large class of geometric PDE problems from different contexts. As applications, we recover and unify a number of results in the literature: for the isometric flow of \(\text {G}_2\) structures, by Grigorian (Adv Math 308:142–207, 2017; Calculas Variat Partial Differ Equ 58:157, 2019), Bagaglini (J Geom Anal, 2009), and DwivediGianniotisKarigiannis (J Geom Anal 31(2):18551933, 2021); and for harmonic almost complex structures, by He (Energy minimizing harmonic almost complex structures, 2019) and HeLi (Trans Am Math Soc 374(9):6179–6199, 2021). Our theory also establishes original properties regarding harmonic flows of parallelisms and almost contact structures.
PubDate: 20231017
DOI: 10.1007/s10455023099287

 On the index of a freeboundary minimal surface in Riemannian
SchwarzschildAdS
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Abstract: Abstract We consider the index of a certain noncompact freeboundary minimal surface with boundary on the rotationally symmetric minimal sphere in the SchwarzschildAdS geometry with \(m>0\) . As in the Schwarzschild case, we show that in dimensions \(n\ge 4\) , the surface is stable, whereas in dimension three, the stability depends on the value of the mass \(m>0\) and the cosmological constant \(\Lambda <0\) via the parameter \(\mu :=m\sqrt{\Lambda /3}\) . We show that while for \(\mu \ge \tfrac{5}{27}\) the surface is stable, there exist positive numbers \(\mu _0\) and \(\mu _1\) , with \(\mu _1<\tfrac{5}{27}\) , such that for \(0<\mu <\mu _0\) , the surface is unstable, while for all \(\mu \ge \mu _1\) , the index is at most one.
PubDate: 20231004
DOI: 10.1007/s1045502309925w

 Solution to the nbubble problem on $$\mathbb {R}^1$$ with logconcave
density
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Abstract: Abstract We study the nbubble problem on \(\mathbb {R}^1\) with a prescribed density function f that is even, radially increasing, and satisfies a logconcavity requirement. Under these conditions, we find that isoperimetric solutions can be identified for an arbitrary number of regions, and that these solutions have a wellunderstood and regular structure. This generalizes recent work done on the density function \( x ^p\) and stands in contrast to logconvex density functions which are known to have no such regular structure.
PubDate: 20230928
DOI: 10.1007/s10455023099278

 Compactness of harmonic maps of surfaces with regular nodes

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Abstract: Abstract In this paper, we formulate and prove a general compactness theorem for harmonic maps of Riemann surfaces using Deligne–Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex curves, there is a family of curves and a subsequence such that both the domains and the maps converge with the singular set consisting of only “nonregular” nodes. This provides a sufficient condition for a neck having zero energy and zero length. As a corollary, the following known fact can be proved: If all domains are diffeomorphic to \(S^2\) , both energy identity and zero distance bubbling hold.
PubDate: 20230925
DOI: 10.1007/s10455023099269
