Closed hyperbolic manifolds without $$\text {spin}^c$$ structures

Combinatorial Calabi flow on 3-manifolds with toroidal boundary

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic.

In order to solve the problem of the traditional image matching algorithm based on the spectral feature, the spectral matching algorithm based on hyperbolic mahalanobis metric is proposed in this paper. The algorithm first introduces a hyperbolic metric that has better adaptability to the sample data. And the hyperbolic mahalanobis metric is defined according to the statistical properties of the data. For a point in a given pointset, the sub point-set is selected according to the hyperbolic mahalanobis metric and the weighted graph of the sub point-set is constructed. The eigenvalue vector and the spectral gap vector are obtained by the singular value decomposition (SVD) of the adjacency matrix of the weighted graph, which construct the hyperbolic mahalanobis metric spectral feature. Finally, the matching matrix is constructed based on the similarity between the hyperbolic mahalanobis metric spectral feature and geometric relations between feature points. Thereby establish the matching mathematical model and introduce the greedy algorithm to obtain the matching results. A large number of experimental results show that the proposed algorithm improves the matching accuracy and the robustness. And the algorithm extends the application range of the spectral matching algorithm.

The persistence of a (compact) normally hyperbolic manifold and the existence of its stable and unstable manifolds, corresponding to the action of C r maps or semiflows, r ≥ 1, are well known facts (see, for instance, [102] and [176]). The more general case of (compact) hyperbolic sets that are invariant under (not necessarily injective) maps was also carefully studied (see Ruelle [176], Shub [189] and Palis and Takens [163]). The persistence and smoothness of (compact) hyperbolic invariant manifolds for RFDE were considered in detail in Magalhães [132] based on skew-product semiflows defined for RFDE, locally around hyperbolic invariant manifolds, and their spectral properties, associated with exponential dichotomies, following the lines of work developed by Sacker and Sell for flows([186], [187]).KeywordsBanach SpaceInvariant ManifoldCompact ManifoldUnstable ManifoldExponential DichotomyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we adopt combinatorial Ricci flow to study the existence of hyperbolic structure on cusped 3-manifolds. The long-time existence and the uniqueness for the extended combinatorial Ricci flow are proven for general pseudo 3-manifolds. We prove that the extended combinatorial Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature, and the flow converges exponentially fast in this case. For an ideally triangulated cusped 3-manifold admitting a complete hyperbolic metric, the flow provides an effective algorithm for finding the hyperbolic metric.

The natural setup for our work here is a metric space (G, mG) where G is a subdomain of \(\mathbb {R}^n\,, n\ge 2\). For our studies, the distance mG(x, y), x, y ∈ G is required to take into account both how close the points x, y are to each other and the position of the points relative to the boundary ∂G. Metrics of this type are called hyperbolic type metrics and they are substitutes for the hyperbolic metric in dimensions n ≥ 3. The quasihyperbolic metric and the distance ratio metric are both examples of hyperbolic type metrics. A key problem is to study a quasiconformal mapping between metric spaces $$\displaystyle f: (G, m_G) \to (f(G), m_{f(G)}) $$ and to estimate its modulus of continuity. We expect Holder continuity, but a concrete form of these results may differ from metric to metric. Another question is the comparison of the metrics to each other.

A notion of discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying Gauss–Bonnet formula. Furthermore, the hyperbolic polyhedral metric with given curvature can be obtained using a discrete Yamabe flow with surgery. In particular, each hyperbolic polyhedral metric on a closed surface with negative Euler characteristic is discrete conformal to a unique hyperbolic metric.

We introduce a new notion called the extended hyperbolic metrics, as a hyperbolic metric (i.e. metric of constant curvature − 1) with certain kinds of singularities defined on a Riemann surface, and we give several fundamental properties of such metrics. Extended hyperbolic metrics are closely related to space-like surfaces of constant mean curvature one (i.e. CMC-1 surfaces) in de Sitter 3-space S 1 3. For example, the singular set of a given CMC-1 surface in S 1 3 is contained in the singular set of the associated extended hyperbolic metric. We then classify all catenoids in S 1 3 (i.e. weakly complete constant mean curvature 1 surfaces in S 1 3 of genus zero with two regular ends whose hyperbolic Gauss map is of degree one). Such surfaces are called S 1 3-catenoids. Since there is a bijection between the moduli space of S 1 3-catenoids and the moduli space of co-orientable extended hyperbolic metrics with two regular singularities, a classification of such hyperbolic metrics is also given. (Co-orientability of extended hyperbolic metrics is defined in this paper.)

We consider the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold (as some or as the only boundary component). We discuss some explicit examples of hyperbolic surfaces, in particular the surface associated to the small stellated dodecahedron (one of the four Kepler-Poinsot polyhedra) which is the boundary of a hyperbolic icosahedral 3-manifold.

In connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa, Molnár (eds) Non-Euclidean geometries, János Bolyai memorial volume mathematics and its applications, Springer, Berlin, 2006), Molnár et al. (Symmetry Cult Sci 22(3–4):435–459, 2011) our computer program (Prok in Period Polytech Ser Mech Eng 36(3–4):299–316, 1992) found 5079 equivariance classes for combinatorial face pairings of the double-simplex. From this list we have chosen those 7 classes which can form charts for hyperbolic manifolds by double-simplices with ideal vertices. In such a way we have obtained the orientable manifold of Thurston (The geometry and topology of 3-manifolds (Lecture notes), Princeton University, Princeton, 1978), that of Fomenko–Matveev–Weeks (Fomenko and Matveev in Uspehi Mat Nauk 43:5–22, 1988; Weeks in Hyperbolic structures on three-manifolds. Ph.D. dissertation, Princeton, 1985) and a nonorientable manifold M_{c^2} with double simplex {widetilde{{mathcal {D}}}}_1, seemingly known by Adams (J Lond Math Soc (2) 38:555–565, 1988), Adams and Sherman (Discret Comput Geom 6:135–153, 1991), Francis (Three-manifolds obtainable from two and three tetrahedra. Master Thesis, William College, 1987) as a 2-cusped one. This last one is represented for us in 5 non-equivariant double-simplex pairings. In this paper we are going to determine the possible Dehn type surgeries of M_{c^2}={widetilde{{mathcal {D}}}}_1, leading to compact hyperbolic cone manifolds and multiple tilings, especially orbifolds (simple tilings) with new fundamental domain to {widetilde{{mathcal {D}}}}_1. Except the starting regular ideal double simplex, we do not get further surgery manifold. We compute volumes for starting examples and limit cases by Lobachevsky method. Our procedure will be illustrated by surgeries of the simpler analogue, the Gieseking manifold (1912) on the base of our previous work (Molnár et al. in Publ Math Debr, 2020), leading to new compact cone manifolds and orbifolds as well. Our new graphic analysis and tables inform you about more details. This paper is partly a survey discussing as new results on Gieseking manifold and on M_{c^2} as well, their cone manifolds and orbifolds which were partly published in Molnár et al. (Novi Sad J Math 29(3):187–197, 1999) and Molnár et al. (in: Karáné, Sachs, Schipp (eds) Proceedings of “Internationale Tagung über geometrie, algebra und analysis”, Strommer Gyula Nemzeti Emlékkonferencia, Balatonfüred-Budapest, Hungary, 1999), updated now to Memory of Professor Gyula Strommer. Our intention is to illustrate interactions of Algebra, Analysis and Geometry via algorithmic and computational methods in a classical field of Geometry and of Mathematics, in general.

A projective manifold [Formula: see text] is algebraically hyperbolic if there exists a positive constant [Formula: see text] such that the degree of any curve of genus [Formula: see text] on [Formula: see text] is bounded from above by [Formula: see text]. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by -1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, closed hyperbolic manifolds minimize volume over all hyperbolic cone manifolds in the same bilipschitz class with cone angles not greater than 2pi. The proof uses techniques developed by Besson-Courtois-Gallot. In 3 dimensions, this result provides a partial solution to a conjecture in Kleinian groups concerning acylindrical manifolds.

Let $M$ be a Carathéodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete Kähler, we show that $M$ admits the Bergman metric. When $M$ is strongly Carathéodory hyperbolic and is the universal covering of a quasi-projective manifold $X$, the Bergman metric can be estimated in terms of a Poincaré-type metric on $X$. It is also proved that any quasi-projective (resp. projective) subvariety of $X$ is of log-general type (resp. general type), a result consistent with a conjecture of Lang.

We show in this article that Kähler hyperbolic manifolds satisfy a family of optimal Chern number inequalities and that the equality cases can be attained by some compact ball quotients. These present restrictions to complex structures on negatively curved compact Kähler manifolds, thus providing evidence for the rigidity conjecture of S.-T. Yau. The main ingredients in our proof are Gromov’s results on the L 2 L^2 -Hodge numbers, the − 1 -1 -phenomenon of the χ y \chi _y -genus and Hirzebruch’s proportionality principle. Similar methods can be applied to obtain parallel results on Kähler nonelliptic manifolds. In addition to these, we term a condition called “Kähler exactness”, which includes Kähler hyperbolic and nonelliptic manifolds and has been used by B.-L. Chen and X. Yang in their work, and we show that the canonical bundle of a Kähler exact manifold of the general type is ample. Some of its consequences and remarks are discussed as well.

Geometric proof for normally hyperbolic invariant manifolds