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For closed odd-dimensional manifolds with sectional curvature less than or equal to − 1, we establish the concept of minimal surface entropy that counts the number of surface subgroups. Additionally, we demonstrate that the minimal surface entropy attains the minimum value if and only if the metric is hyperbolic. [ABSTRACT FROM AUTHOR]

Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs of points on compact Riemann surfaces is surprisingly unknown, unless the surface is a sphere or a torus. This is because on higher-genus surfaces, the distance formula involves an infimum over infinitely many terms, so it cannot be evaluated in practice. Here we derive a computable distance formula for a broad class of Riemann surfaces. The formula reduces the infimum to a minimum over an explicit set consisting of finitely many terms. We also develop a distance computation algorithm, which cannot be expressed as a formula, but which is more computationally efficient on surfaces with high genuses. We illustrate both the formula and the algorithm in application to generalized Bolza surfaces, which are a particular class of highly symmetric compact Riemann surfaces of any genus greater than 1. [ABSTRACT FROM AUTHOR]

On negatively curved compact manifolds, it is possible to associate to every closed form a bounded cocycle – hence a bounded cohomology class – via integration over straight simplices. The kernel of this map is contained in the space of exact forms. We show that in degree 2 this kernel is trivial, in contrast with higher degree. In other words, exact non-zero 2-forms define non-trivial bounded cohomology classes. This result is the higher dimensional version of a classical theorem by Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] for surfaces. As a consequence, one gets that the second bounded cohomology of negatively curved manifolds contains an infinite dimensional space, whose classes are explicitly described by integration of forms. This also showcases that some recent results by Marasco [Proc. Amer. Math. Soc. 151 (2023), pp. 2707–2715] can be applied in higher dimension to obtain new non-trivial results on the vanishing of certain cup products and Massey products. Some other applications are discussed. [ABSTRACT FROM AUTHOR]

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$ -manifold. Our estimates are polynomials in the tight areas and Bowen–Margulis–Sullivan densities of geodesic planes, with degree given by the modified critical exponents. [ABSTRACT FROM AUTHOR]

A group G is subgroup conjugacy distinguished (resp. subgroup into conjugacy separable) if whenever an element y (resp. a finitely generated subgroup K) is not conjugate to an element (to a subgroup) of a finitely generated subgroup H of G there exists a finite quotient G/N of G where yN (resp. KN/N) is not conjugate to an element (to a subgroup) of HN/N. We prove that the fundamental group of a hyperbolic 3-manifold is subgroup conjugacy distinguished and subgroup into conjugacy separable. [ABSTRACT FROM AUTHOR]

We show that the simplicial volume of a contractible 3-manifold not homeomorphic to R3 is infinite. As a consequence, the Euclidean space may be characterized as the unique contractible 3-manifold with vanishing minimal volume, or as the unique contractible 3-manifold supporting a complete finite-volume Riemannian metric with Ricci curvature uniformly bounded from below. In contrast, we show that in every dimension n ≥ 4 there exists a contractible n-manifold with vanishing simplicial volume not homeomorphic to Rn. We also compute the spectrum of the simplicial volume of irreducible open 3-manifolds. [ABSTRACT FROM AUTHOR]

We show that large classes of non-arithmetic hyperbolic n-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds of dimension at least 2 that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient n-manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature. [ABSTRACT FROM AUTHOR]

We describe the phase space structures related to the semi-major axis of Molniya-like satellites subject to tesseral and lunisolar resonances. In particular, the questions answered in this contribution are: (1) we study the indirect interplay of the critical inclination resonance on the semi-geosynchronous resonance using a hierarchy of more realistic dynamical systems, thus discussing the dynamics beyond the integrable approximation. By introducing ad hoc tractable models averaged over fast angles, (2) we numerically demarcate the hyperbolic structures organising the long-term dynamics via fast Lyapunov indicators cartography. Based on the publicly available two-line elements space orbital data, (3) we identify two satellites, namely Molniya 1-69 and Molniya 1-87, displaying fingerprints consistent with the dynamics associated to the hyperbolic set. Finally, (4) the computations of their associated dynamical maps highlight that the spacecraft are trapped within the hyperbolic tangle. This research therefore reports evidence of actual artificial satellites in the near-Earth environment whose dynamics are ruled by hyperbolic manifolds and resonant mechanisms. The tools, formalism and methodologies we present are exportable to other region of space subject to similar commensurabilities as the geosynchronous region. [ABSTRACT FROM AUTHOR]

We propose a new method to approximate curves that interpolate a given set of time-labeled data on Riemannian symmetric spaces. First, we present our new formulation on the Euclidean space as a result of minimizing the mean square acceleration. This motivates its generalization on some Riemannian symmetric manifolds. Indeed, we generalize the proposed solution to the the special orthogonal group, the manifold of symmetric positive definite matrices, and Riemannian n-manifolds with constant negative curvature. By means of this generalization, we are able to prove that the approximates enjoy a number of nice properties: The solution exists and is optimal in many common situations. Several examples are provided together with some applications and graphical representations. [ABSTRACT FROM AUTHOR]

Using the Gallagher–Koyama approach, we reduce the exponent in the error term of the prime geodesic theorem for real hyperbolic manifolds with cusps. [ABSTRACT FROM AUTHOR]

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms M with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball B3 ⊂ ℝ3 as mappings equivariant with the standard conformal action of uniform hyperbolic lattices Γ ⊂ Isom H3 in the unit 3-ball and with its discrete representation G = ρ(Γ) ⊂ Isom H4. Here, G is the fundamental group of our non-trivial hyperbolic 4-cobordism M = (H4 ∪ Ω(G))/G, and the kernel of the homomorphism ρ: Γ → G is a free group F3 on three generators. [ABSTRACT FROM AUTHOR]

In this paper we construct arbitrarily large families of smooth projective varieties and closed Riemannian manifolds that share many algebraic and analytic invariants. For instance, every non-arithmetic, closed hyperbolic 3-manifold admits arbitrarily large collections of non-isometric finite covers which are strongly isospectral, length isospectral, and have isomorphic integral cohomology where the isomorphisms commute with restriction and co-restriction. We can also construct arbitrarily large collections of pairwise non-isomorphic smooth projective surfaces where these isomorphisms in cohomology are natural with respect to Hodge structure or as Galois modules. In particular, the projective varieties have isomorphic Picard and Albanese varieties, and they also have isomorphic effective Chow motives. Our construction employs an integral refinement of the Gassman–Sunada construction that has recently been utilized by D. Prasad. One application of our work shows the non-injectivity of the map from the Grothendieck group of varieties over Q ¯ to the Grothendieck group of the category of effective Chow motives. We also answer a question of D. Prasad. [ABSTRACT FROM AUTHOR]

Abstract For a compact, oriented, hyperbolic n -manifold (M , g) , realised as M = Γ \ H n where Γ is a torsion-free cocompact subgroup of S O (n , 1) , we establish and study a relationship between differential geometric cohomology on M and algebraic invariants of the group Γ. In particular for F an irreducible S O (n , 1) -module, we show that the group cohomology with coefficients H • (Γ , F) arises from the cohomology of an appropriate projective BGG complex on M. This yields the geometric interpretation that H • (Γ , F) parameterises solutions to certain distinguished natural PDEs of Riemannian geometry, modulo the range of suitable differential coboundary operators. Viewed in another direction, the construction shows one way that non-trivial cohomology can arise in a BGG complex, and sheds considerable light on its geometric meaning. We also use the tools developed to give a new proof that H 1 (Γ , S 0 k R n + 1) ≠ 0 whenever M contains a compact, orientable, totally geodesic hypersurface. All constructions use another result that we establish, namely that the canonical flat connection on a hyperbolic manifold coincides with the tractor connection of projective differential geometry. [ABSTRACT FROM AUTHOR]

We give a new proof of the best presently-known error term in the prime geodesic theorem for compact hyperbolic surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived, yielding to a further reduction of the error term outside a set of finite logarithmic measure. [ABSTRACT FROM AUTHOR]

Around 1973, Macbeath provided a formula for the number of fixed points of an element in a group G of conformal automorphisms of a closed Riemann surface S of genus at least two. Such a formula was initially used to obtain the character of the representation associated to the induced action of G on the first homology group of S, and later turned out to be extremely useful in many other contexts. By using a simple counting procedure, we provide a similar formula for the number of connected components of an element in a finite group of isometries of a hyperbolic manifold. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that for a 3-dimensional small cover M over a polytope P, there are only 2-torsions in H ( M; Z). Moreover, the mod 2 Betti number growth of finite covers of M is studied. [ABSTRACT FROM AUTHOR]

We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds that can be covered by a bounded domain is a finite set. [ABSTRACT FROM AUTHOR]

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism M whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary ∂ M. In addition to the standard conformal ergodic action of a uniformhyperbolic lattice on the round sphere S and its quasiconformal deformations in S , we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed n-ball into S ), on non-trivial ( n − 1)-knots in S , as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from 'non-standard' components of its varieties of representations (faithful or with large kernels of defining homomorphisms). [ABSTRACT FROM AUTHOR]

We correct the exponent in the error term of the prime geodesic theorem for hyperbolic 3‐manifolds and in Park's theorem for higher dimensions [, ]. [ABSTRACT FROM AUTHOR]

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We then extend the linear upper bounds to hyperbolic manifolds with a finite cover that admits such an immersion. Since the quantifications are given in terms of geodesic length, we define the geodesic residual finiteness growth and show that this growth is equivalent to the usual residual finiteness growth defined in terms of word length. This equivalence implies that our results recover the quantification of residual finiteness from Bou-Rabee et al. (Math Z, [math.GR]) for hyperbolic manifolds that virtually immerse into a compact reflection orbifold. [ABSTRACT FROM AUTHOR]

This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved metrics on a manifold. [ABSTRACT FROM AUTHOR]

A surface F in a 3-manifold M is called cylindrical if M cut open along F admits an essential annulus A. If, in addition, (A, ∂A) is embedded in (M,F), then we say that F is strongly cylindrical. Let M be a connected 3-manifold that admits a triangulation using t tetrahedra and F a two-sided connected essential closed surface of genus g(F). We show that if g(F) is at least 38t, then F is strongly cylindrical. As a corollary, we give an alternative proof of the assertion that every closed hyperbolic 3-manifold admits only finitely many fibrations over the circle with connected fiber whose translation distance is not one, which was originally proved by Saul Schleimer. [ABSTRACT FROM AUTHOR]

For a compact (without boundary) hyperbolic n -manifold M with n ≥ 4 , we show that there exists a finite bound B such that for any homeomorphism f : M → M and any fixed point class F of f , the index | ind ( f , F ) | ≤ B , which is a partial positive answer of a question given by Jiang in [3] . Moreover, when M is a compact hyperbolic 4-manifold, or a compact hyperbolic n -manifold ( n ≥ 5 ) with isometry group Isom ( M ) a p -group, we give some explicit descriptions of the bound B . [ABSTRACT FROM AUTHOR]

We examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics do not lie on any totally geodesic surfaces?, and Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics come from distinct totally geodesic surfaces? [ABSTRACT FROM AUTHOR]

We study M-theory and D-brane quantum partition functions for microscopic black hole ensembles within the context of the AdS/CFT correspondence in terms of highest weight representations of infinite-dimensional Lie algebras, elliptic genera, and Hilbert schemes, and describe their relations to elliptic modular forms. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras, and in the role of spectral functions of hyperbolic three-geometry associated with q-series in the calculation of elliptic genera. We present new calculations of supergravity elliptic genera on local Calabi-Yau threefolds in terms of BPS invariants and spectral functions, and also of equivariant D-brane elliptic genera on generic toric singularities. We use these examples to conjecture a link between the black hole partition functions and elliptic cohomology. [ABSTRACT FROM AUTHOR]

An idealized scenario of extratropical transition ( ET) is investigated, in which a tropical cyclone interacts with a high-amplitude, upper-level wave pattern and well-developed surface cyclones. Early during the interaction, the external forcing of the upper-level wave by the ET system is quantified based on a metric for the waviness of the midlatitude flow. Local amplification of the wave pattern is diagnosed, associated prominently with the trough downstream of ET. This amplified trough, however, exhibits pronounced anticyclonic breaking and thus, in contrast to many previous ET studies, it is not clear that the amplification of the upper-level wave propagates into the farther downstream region. Subsequently, the ET system merges with the upstream cyclone. The upstream trough undergoes strong deformation and cyclonic breaking associated with straining due to the cyclonic circulation of the ET system. With the decay of this trough, the ET system weakens considerably and the upper-level wave pattern changes locally to a zonal flow orientation. This zonal flow pattern then extends into the downstream region and promotes the decay of the downstream baroclinic systems. As in previous studies, the evolution of ET exhibits large sensitivity to the initial location of the tropical cyclone. Examining the steering flow's topology, i.e. identifying the stagnation points and the streamlines emanating from these points, helps to identify three different regimes: a no- ET regime and two ET regimes reminiscent of the northwest and northeast patterns, respectively, introduced previously by Harr et al.. A stagnation point located on the axis of the upstream trough governs the bifurcation into no- ET and ET regimes. A stagnation point located on the axis of the downstream ridge governs the bifurcation into northwest and northeast patterns. [ABSTRACT FROM AUTHOR]

We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert and Stein, the proof of the full result relies on the interplay of the latter ideas and a finiteness theorem for Riemann surfaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume. It depends on a representation of the fundamental group. For manifolds of odd dimension, we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group. [Copyright &y& Elsevier]

The nonintersecting classes ℋ are defined, with p, q ∈ ℕ and p ≥ q ≥ 1, of orientable hyperbolic 3-manifolds with geodesic boundary. If M ∈ ℋ, then the complexity c( M) and the Euler characteristic χ( M) of M are related by the formula c( M) = p− χ( M). The classes ℋ, q ≥ 1, and ℋ are known to contain infinite series of manifolds for each of which the exact values of complexity were found. There is given an infinite series of manifolds from ℋ and obtained exact values of complexity for these manifolds. The method of proof is based on calculating the ɛ-invariants of manifolds. [ABSTRACT FROM AUTHOR]

We show that complementary series representations of SO( n, 1), which are sufficiently close to a cohomological representation contain discretely, complementary series of SO( m, 1) also sufficiently close to cohomological representations, provided that the degree of the cohomological representation does not exceed m/2. We prove, as a consequence, that the cohomological representation of degree i of the group SO( n, 1) contains discretely, the cohomological representation of degree i of the subgroup SO( m, 1) if i ≤≤ m/2. As a global application, we show that if G/ℚℚ is a semisimple algebraic group such that G(ℝℝ) = SO( n, 1) up to compact factors, and if we assume that for all n, the tempered cohomological representations are not limits of complementary series in the automorphic dual of SO( n, 1), then for all n, non-tempered cohomological representations are isolated in the automorphic dual of G. This reduces conjectures of Bergeron to the case of tempered cohomological representations. [ABSTRACT FROM AUTHOR]

The present paper examines the vortex breakdown and large-scale stirring during the final warming of the Southern Hemisphere stratosphere during the spring of 2005. A unique set of in situ observations collected by 27 superpressure balloons (SPBs) is used. The balloons, which were launched from McMurdo, Antarctica, by the Stratéole/VORCORE project, drifted for several weeks on two different isopycnic levels in the lower stratosphere. We describe balloon trajectories and compare them with simulations obtained on the basis of the velocity field from the GEOS-5 and NCEP/NCAR reanalyses performed with and without VORCORE data. To gain insight on the mechanisms responsible for the horizontal transport of air inside and outside the well-isolated vortex we examine the balloon trajectories in the framework of the Lagrangian properties of the stratospheric flow. Coherent structures of the flow are visualized by computing finite-time Lyapunov exponents (FTLE). A combination of isentropic analysis and FTLE distributions reveals that air is stripped away from the vortex's interior as stable manifolds eventually cross the vortex's edge. It is shown that two SPBs escaped from the vortex within high potential vorticity tongues that developed in association with wave breaking at locations along the vortex's edge where forward and backward FTLE maxima approximately intersect. The trajectories of three SPBs flying as a group at the same isopycnic level are examined and their behavior is interpreted in reference to the FTLE field. These results support the concept of stable and unstable manifolds governing transport of air masses across the periphery of the stratospheric polar vortex. [ABSTRACT FROM AUTHOR]

Using a three degrees of freedom quasi-integrable Hamiltonian as a model problem, we numerically compute the unstable manifolds of the hyperbolic manifolds of the phase space related to single resonances. We measure an exponential dependence of the splitting of these manifolds through many orders of magnitude of the perturbing parameter. This is an indirect numerical verification of the exponential decay of the normal form, as predicted by the Nekhoroshev theorem. We also detect different transitions in the topology of these manifolds related to the local rational approximations of the frequencies. The variation of the size of the homoclinic tangle as well as the topological transitions turn out to be correlated to the speed of Arnold diffusion. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the hyperbolicity of arborescent tangles and arborescent links. We will explicitly determine all essential surfaces in arborescent tangle complements with non-negative Euler characteristic, and show that given an arborescent tangle T, the complement is non-hyperbolic if and only if T is a rational tangle, for some , or T contains for some . We use these results to prove a theorem of Bonahon and Siebenmann which says that a large arborescent link L is non-hyperbolic if and only if it contains . [Copyright &y& Elsevier]

Abstract: We show that the Dirac operator on a spin manifold does not admit eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing. [Copyright &y& Elsevier]

Abstract: Results of H. Cartan about holomorphic automorphisms on bounded domains are generalized to the case of hyperbolic manifolds in the sense of Kobayashi. In this setting, we give an identity theorem together with its topological version. We show also that a sequence of automorphisms which converges uniformly on some nonempty open set, has a limit on the whole space which is an automorphism. At the end of the paper, conditions are given for the sequence of iterates of a self holomorphic map in order to be an automorphism. [Copyright &y& Elsevier]

We first prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the quotients G\H n +1 with rational nonmaximal rank cusps previously studied by Froese-Hislop-Perry. [ABSTRACT FROM AUTHOR]