Hyperbolic Manifolds Research Articles

Asymptotics for minimizers of a Donaldson functional and mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds

It has been shown in [17] that, for given |c|<1, the moduli space of constant mean curvature (CMC) c-immersions of a closed orientable surface of genus g≥2 into a hyperbolic 3-manifold can be parametrized by elements of the tangent bundle of the corresponding Teichmüller space. This is attained by showing the unique solvability of the Gauss-Codazzi equations governing (CMC) c-immersions. The corresponding unique solution is identified as the global minimum (and only critical point) of the Donaldson functional Dt (introduced in [11]) given in (1.11) with t=1−c2.When |c|≥1 (i.e. t≤0), so far nothing is known about the existence of analogous (CMC) c-immersions. Indeed, for t≤0 the functional Dt may no longer be bounded from below and evident non-existence situations do occur.Already the case |c|=1 (i.e. t=0) appears rather involved and actually (CMC) 1-immersions can be attained only as “limits” of (CMC) c-immersions for |c|⟶1− (see Theorem 2). To handle this situation, here we analyze the asymptotic behavior of minimizers of Dt as t⟶0+.We use an accurate asymptotic analysis (see [35]) to describe possible blow-up phenomena. In this way, we can relate the existence of (CMC) 1-immersions to the Kodaira map defined in (1.9). As a consequence, we obtain the first existence and uniqueness result about (CMC) 1-immersions of surfaces of genus g=2 into hyperbolic 3-manifolds.

Stable isoperimetric ratios and the Hodge Laplacian of hyperbolic manifolds

AbstractWe show that for a closed hyperbolic 3‐manifold, the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1‐forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3‐manifolds with injectivity radius bounded below and volume going to infinity for which the 1‐form Laplacian has spectral gap vanishing exponentially fast in the volume.

Automorphisms of the k-Curve Graph

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k satisfying k≤|χ(S)|−512. We prove the same result for the so-called systolic complex, a variant of the curve graph with many complete subgraphs coming from interesting collections of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.

Static potentials on asymptotically hyperbolic manifolds

Static potentials on asymptotically hyperbolic manifolds

Well-Posedness of the Ambient Metric Equations and Stability of Even Dimensional Asymptotically de Sitter Spacetimes

The vanishing of the Fefferman–Graham obstruction tensor was used by Anderson and Chruściel to show stability of the asymptotically de Sitter spaces in even dimensions. However, the existing proofs of hyperbolicity of this equation contain gaps. We show in this paper that it is indeed a well-posed hyperbolic system with unique up to diffeomorphism and conformal transformations smooth development for smooth Cauchy data. Our method applies also to equations defined by various versions of the Graham–Jenne–Mason–Sparling operators. In particular, we use one of these operators to propagate Gover’s condition of being almost Einstein (basically conformal to Einsteinian metric). This allows us to study initial data also for Cauchy surfaces which cross the conformal boundary. As a by-product we show that on globally hyperbolic manifolds one can always choose a conformal factor such that Branson Q-curvature vanishes.

Constraining Weil–Petersson volumes by universal random matrix correlations in low-dimensional quantum gravity

Based on the discovery of the duality between Jackiw–Teitelboim quantum gravity and a double-scaled matrix ensemble by Saad, Shenker and Stanford in 2019, we show how consistency between the two theories in the universal random matrix theory (RMT) limit imposes a set of constraints on the volumes of moduli spaces of Riemannian manifolds. These volumes are given in terms of polynomial functions, the Weil–Petersson (WP) volumes, solving a celebrated nonlinear recursion formula that is notoriously difficult to analyse. Since our results imply linear relations between the coefficients of the WP volumes, they therefore provide both a stringent test for their symbolic calculation and a possible way of simplifying their construction. In this way, we propose a long-term program to improve the understanding of mathematically hard aspects concerning moduli spaces of hyperbolic manifolds by using universal RMT results as input.

Volume comparison with respect to scalar curvature

In this article, we investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison holds for small geodesic balls of metrics near a V-static metric. For closed manifold, we prove the volume comparison for metrics near a strictly stable Einstein metric. As applications, we give a partial answer to a conjecture of Bray and recover a result of Besson, Courtois and Gallot, which partially confirms a conjecture of Schoen about closed hyperbolic manifold. Applying analogous techniques, we obtain a different proof of a local rigidity result due to Dai, Wang and Wei, which shows it admits no metric with positive scalar curvature near strictly stable Ricci-flat metrics.

Orthogonal Approximation of Invariant Manifolds in the Circular Restricted Three-Body Problem

Methods to parameterize and approximate the hyperbolic invariant manifolds of particular solutions in the circular restricted three-body problem (CR3BP) are presented in this paper. Analytical representations obtained from these manifold approximations are instrumental in the synthesis of optimal trajectories for cislunar transport. A multivariate Chebyshev series is used to approximate the surfaces, thereby serving as tractable parametric representations of the complex properties of motion. It is demonstrated that the continuum of ballistic capture trajectories and their associated sensitivities on the manifold can be realized in functional form using simple algebraic operations. Two applications making use of the Chebyshev manifold approximations as a terminal constraint surface are presented. The first is a low-thrust trajectory optimization problem formulated such that the optimal free final state lying on the manifold is determined as an additional set of design parameters. The second is a guidance law designed to target the manifold in the vicinity of the nominal patch point. Each of these methods takes advantage of the Chebyshev approximations to provide additional flexibility for mission design in multibody dynamic environments. These applications offer tremendous optimism about the utility of function approximation methods in arriving at a formal representation for the invariant manifolds in the three-body problem for efficient generation of optimal trajectories.

Harmonic Forms, Minimal Surfaces, and Norms on Cohomology of Hyperbolic 3-Manifolds

Abstract We bound the $L^2$-norm of an $L^2$ harmonic $1$-form in an orientable cusped hyperbolic $3$-manifold $M$ by its topological complexity, measured by the Thurston norm, up to a constant depending on $M$. This generalizes two inequalities of Brock and Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction between minimal surfaces and harmonic forms. We unify various results by defining two functionals on closed and cusped hyperbolic $3$-manifolds and formulate several questions and conjectures.

Selberg Zeta function and hyperbolic Eisenstein series

Let [Formula: see text] be a Fuchsian group acting on the hyperbolic upper half-plane [Formula: see text], such that [Formula: see text] is a geometrically finite Riemann surface with respect to the natural hyperbolic metric induced from [Formula: see text]. If [Formula: see text] is hyperbolic then following [J. Kudla and S. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979) 193–211; J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293 (1977) 143–203], there is a corresponding hyperbolic Eisenstein series. In this paper, we study the limiting behavior of hyperbolic Eisenstein series on a degenerating family of geometrically finite hyperbolic surfaces. In particular, we give a partial lightening to a question of Ji, concerning the approximation of Eisenstein series during degeneration (see Proposition 5.2 and Theorem 5.2).

Equivariant hyperbolization of 3-manifolds via homology cobordisms

The main result of this paper is that any 3-dimensional manifold with a finite group action is equivariantly invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolic homology 3-sphere. The theorem has other corollaries, including the existence of infinitely many hyperbolic homology spheres that support free Zp-actions that do not extend over any contractible manifolds, and (from the non-equivariant version of the theorem) infinitely many that bound homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric, and thus a partial order.

Combinatorial Curvature Flows for Generalized Circle Packings on Surfaces with Boundary

Abstract In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the $(-1,-1,-1)$-type generalized circle packing metric introduced by Guo and Luo [ 16]. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide effective algorithms for finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

For $$n \ge 2$$ , we prove that a finite volume complex hyperbolic n-manifold containing infinitely many maximal properly immersed totally geodesic submanifolds of real dimension at least two is arithmetic, paralleling our previous work for real hyperbolic manifolds. As in the real hyperbolic case, our primary result is a superrigidity theorem for certain representations of complex hyperbolic lattices. The proof requires developing new general tools not needed in the real hyperbolic case. Our main results also have a number of other applications. For example, we prove nonexistence of certain maps between complex hyperbolic manifolds, which is related to a question of Siu, that certain hyperbolic 3-manifolds cannot be totally geodesic submanifolds of complex hyperbolic manifolds, and that arithmeticity of complex hyperbolic manifolds is detected purely by the topology of the underlying complex variety, which is related to a question of Margulis. Our results also provide some evidence for a conjecture of Klingler that is a broad generalization of the Zilber–Pink conjecture.

On Schwarz–Pick-Type Inequality and Lipschitz Continuity for Solutions to Nonhomogeneous Biharmonic Equations

The purpose of this paper is to study the Schwarz–Pick type inequality and the Lipschitz continuity for the solutions to the nonhomogeneous biharmonic equation: $$\Delta (\Delta f)=g$$ , where g : $$\overline{{\mathbb D}}\rightarrow {\mathbb {C}}$$ is a continuous function and $$\overline{{\mathbb D}}$$ denotes the closure of the unit disk $${\mathbb D}$$ in the complex plane $${\mathbb {C}}$$ . In fact, we establish the following properties for these solutions: First, we show that the solutions f do not always satisfy the Schwarz–Pick-type inequality $$\begin{aligned} \frac{1-|z|^2}{1-|f(z)|^2} \le C, \end{aligned}$$ where C is a constant. Second, we establish a general Schwarz–Pick-type inequality of f under certain conditions. Third, we discuss the Lipschitz continuity of f, and as applications, we get the Lipschitz continuity with respect to the distance ratio metric and the Lipschitz continuity with respect to the hyperbolic metric.

Strictification theorems for the homotopy time-slice axiom

It is proven that the homotopy time-slice axiom for many types of algebraic quantum field theories (AQFTs) taking values in chain complexes can be strictified. This includes the cases of Haag–Kastler-type AQFTs on a fixed globally hyperbolic Lorentzian manifold (with or without time-like boundary), locally covariant conformal AQFTs in two spacetime dimensions, locally covariant AQFTs in one spacetime dimension, and the relative Cauchy evolution. The strictification theorems established in this paper prove that, under suitable hypotheses that hold true for the examples listed above, there exists a Quillen equivalence between the model category of AQFTs satisfying the homotopy time-slice axiom and the model category of AQFTs satisfying the usual strict time-slice axiom.

Complex hierarchical structures in single-cell genomics data unveiled by deep hyperbolic manifold learning.

With the advances in single-cell sequencing techniques, numerous analytical methods have been developed for delineating cell development. However, most are based on Euclidean space, which would distort the complex hierarchical structure of cell differentiation. Recently, methods acting on hyperbolic space have been proposed to visualize hierarchical structures in single-cell RNA-seq (scRNA-seq) data and have been proven to be superior to methods acting on Euclidean space. However, these methods have fundamental limitations and are not optimized for the highly sparse single-cell count data. To address these limitations, we propose scDHMap, a model-based deep learning approach to visualize the complex hierarchical structures of scRNA-seq data in low-dimensional hyperbolic space. The evaluations on extensive simulation and real experiments show that scDHMap outperforms existing dimensionality-reduction methods in various common analytical tasks as needed for scRNA-seq data, including revealing trajectory branches, batch correction, and denoising the count matrix with high dropout rates. In addition, we extend scDHMap to visualize single-cell ATAC-seq data.

Computing Natural Transitions Between Tori Near Resonances in the Earth–Moon System

Natural transitions between bounded motions near mean-motion resonances occur throughout our solar system and are valuable in trajectory design. Such phenomena have been examined for natural transitions between periodic orbits near resonances within multi-body systems. However, families of quasi-periodic trajectories, tracing the surface of invariant 2-tori, significantly expand the solution space of bounded motions near resonances. Yet, identifying natural transitions between spatial 2-tori has previously been cumbersome due to the high dimensionality of the associated solution space. This paper approaches the challenge in constructing these natural transfers by using a combination of Poincaré mapping, a well-known technique from dynamical systems theory, and manifold learning, a technique for dimension reduction. The presented approach involves projecting a higher-dimensional dataset of intersections recorded from the hyperbolic invariant manifolds of two 2-tori onto a lower-dimensional embedding, enabling rapid identification of initial guesses for natural transfers. These initial guesses are then corrected and input to a continuation scheme to recover families of geometrically similar transfers connecting families of invariant 2-tori. This approach is demonstrated by constructing families of natural transitions between tori near distinct resonances in the Earth–Moon circular restricted three-body problem.

Complete topological descriptions of certain Morse boundaries

We study direct limits of embedded Cantor sets and embedded Sierpiński curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called \omega -Cantor spaces, and, similarly, all limits of Sierpiński curves give homeomorphic spaces, called \omega -Sierpiński curves. We then show that the former occur naturally as Morse boundaries of right-angled Artin groups and fundamental groups of non-geometric graph manifolds, while the latter occur as Morse boundaries of fundamental groups of finite-volume, cusped hyperbolic 3-manifolds.

On Asymptotically Locally Hyperbolic Metrics with Negative Mass

We construct families of asymptotically locally hyperbolic Riemannian metrics with constant scalar curvature (i.e., time symmetric vacuum general relativistic initial data sets with negative cosmological constant), with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular we obtain new classes of solutions with negative mass.

Infinitesimal rigidity for cubulated manifolds

We prove the infinitesimal rigidity of some geometrically infinite hyperbolic 4- and 5-manifolds. These examples arise as infinite cyclic coverings of finite-volume hyperbolic manifolds obtained by colouring right-angled polytopes, already described in the papers (Battista in Trans Am Math Soc 375(04):2597–2625, 2022; Italiano et al. in Invent Math, 2022. https://doi.org/10.1007/s00222-022-01141-w). The 5-dimensional example is diffeomorphic to Ntimes {{mathbb {R}}} for some aspherical 4-manifold N which does not admit any hyperbolic structure. To this purpose, we develop a general strategy to study the infinitesimal rigidity of cyclic coverings of manifolds obtained by colouring right-angled polytopes.