Spherical Space Forms Research Articles

Evolving hypersurfaces by their mean curvature in the background manifold evolving by Ricci flow

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial background manifold is an approximation of a spherical space form and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point in finite time or converge to a totally geodesic sphere as the time tends to infinity.

Entanglement entropy and mutual information of circular entangling surfaces in the 2 + 1-dimensional quantum Lifshitz model

We investigate the entanglement entropy (EE) of circular entangling cuts in the 2 + 1-dimensional quantum Lifshitz model. The ground state in this model is a spatially conformal invariant state of the Rokhsar–Kivelson type, whose amplitude is the Gibbs weight of 2D Euclidean free boson. We show that the finite subleading corrections of EE to the area-law term, as well as the mutual information, are conformal invariants and calculate them for cylinder, disk-like and spherical manifolds with various spatial cuts. The subtlety due to the boson compactification in the replica trick is carefully addressed. We find that in the geometry of a punctured plane with many small holes, the constant piece of EE is proportional to the number of holes, indicating the ability of entanglement to detect topological information of the configuration. Finally, we compare the mutual information of two small distant disks with Cardy’s relativistic CFT scaling proposal. We find that in the quantum Lifshitz model, the mutual information also scales at long distance with a power determined by the lowest scaling dimension local operator in the theory.

A note on a three-dimensional sphere theorem with integral curvature condition

In this paper, we obtain a three-dimensional sphere theorem with integral curvature condition. On a closed three manifold [Formula: see text] with constant positive scalar curvature, if a certain combination of [Formula: see text] norm of the Ricci curvature and [Formula: see text] norm of the scalar curvature is positive, then [Formula: see text] is diffeomorphic to a spherical space form.

Alternating Heegaard diagrams and Williams solenoid attractors in $3$-manifolds

We find all Heegaard diagrams with the property ``alternating'' or ``weakly alternating'' on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two $3$-manifolds, each admits an automorphism whose non-wandering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincaré's homology $3$-sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of $3$-manifolds admit a kind of ``translation'' with certain stability.

On conformal qc geometry, spherical qc manifolds and convex cocompact subgroups of $$\mathrm{Sp}{(n+1,1)}$$ Sp ( n + 1 , 1 )

Conformal qc geometry of spherical qc manifolds is investigated. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal qc transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. On a scalar positive spherical qc manifold, we can construct the Green function of the qc Yamabe operator, which can be applied to construct a conformally invariant tensor. It becomes a spherical qc metric if the qc positive mass conjecture is true. Conformal qc geometry of spherical qc manifolds can be applied to study convex cocompact subgroups of $$\mathrm{Sp}(n+1,1).$$ On a spherical qc manifold constructed from such a discrete subgroup, we construct a spherical qc metric of Nayatani type. As a corollary, we prove that such a spherical qc manifold is scalar positive, negative or vanishing if and only if the Poincare critical exponent of the discrete subgroup is less than, greater than or equal to $$2n+2$$ , respectively.

Fundamental Domain and Cellular Decomposition of Tetrahedral Spherical Space Forms

Given a free isometric action of the binary tetrahedral group on a (4n − 1)-dimensional sphere, we obtain an explicit finite cellular decomposition of the sphere, equivariant with respect to the group action. A cell decomposition of the correspondent spherical space form and an explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group of the space form follows. In particular, the construction provides a simple explicit 4-periodic free resolution for the binary tetrahedral group.

Sequential Bayesian Algorithms for Identification and Blind Equalization of Unit-Norm Channels

In many estimation problems of interest, the unknown parameters reside on spherical manifolds. As most common filtering algorithms assume that parameters have Gaussian prior distributions, their application to such problems leads to suboptimal performance. In this letter, we propose a model in which the unknown unit-norm parameter vectors have Fisher–Bingham (F-B) prior distributions. We show that if the observations relate to the parameters via Gaussian likelihoods, the F-B priors form a conjugate model that yields closed-form, recursive estimators that naturally take into account the restrictions on the unknowns. We apply this model to a communication setup with multiple gain-controlled FIR frequency-selective channels, deriving a novel maximum a posteriori (MAP) channel parameter estimator and a blind equalizer based on Rao–Blackwellized particle filters. As we verify via Monte Carlo numerical simulations, the F-B model leads to superior performance compared to previous algorithms that adopt mismatched Gaussian prior models.

Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background.

THE METRIC DIMENSION OF METRIC MANIFOLDS

In this paper we determine the metric dimension of $n$-dimensional metric $(X,G)$-manifolds. This category includes all Euclidean, hyperbolic and spherical manifolds as special cases.

Half-integral finite surgeries on knots in S 3

Supposing that a hyperbolic knot in S 3 admits a finite surgery, Boyer and Zhang proved that the surgery slope must be either integral or half-integral, and they conjectured that the latter case does not happen. Using the correction terms in Heegaard Floer homology, we prove that if a hyperbolic knot in S 3 admits a half-integral finite surgery, then the knot must have the same knot Floer homology as one of the eight non-hyperbolic knots which are known to admit such surgeries, and the resulting manifold must be one of ten spherical space forms. As knot Floer homology carries a lot of information about the knot, this gives a strong evidence to Boyer–Zhang’s conjecture.

Fixed point theory of spherical 3-manifolds

Let f:M→M be a self-map on a three-dimensional manifold M with S3-geometry. In this paper, we determine the possible values of N(f) where N(f) is the Nielsen number of f.

On the curvature ODE associated to the Ricci flow

In the vector space of algebraic curvature operators we study the reaction ODE $$\begin{aligned} \frac{dR}{dt} = R^2+R^{\#}= Q(R) \end{aligned}$$ which is associated to the evolution equation of the Riemann curvature operator along the Ricci flow. More precisely, we give a partial classification of the zeros of this ODE up to suitable normalization and analyze the stability of a special class of zeros of the same. In particular, we show that the ODE is unstable near the curvature operators of the Riemannian product spaces \(M \times \mathbb {R}^k, k \ge 0\) where \(M\) is an Einstein (locally) symmetric space of compact type and not a spherical space form when \(k=0\).

Kaluza–Klein mass spectra on extended dimensional branes

We study the hierarchy problem on the four- and five-branes which are constructed by attaching a circle and a sphere to the standard three-brane, respectively. The effective masses in their excited spectra on these extended dimensional branes imply intriguing characteristics associated with the quantization of the compact circular and spherical manifolds. In particular, their lightest effective masses are shown to suppress exponentially with respect to the Planck mass, similar to the three-brane case. We also investigate the effective couplings in these extended dimensional branes.

Classifying non-splitting fiber preserving actions on prism manifolds

In this paper, we classify the finite groups of fiber preserving isometries G which act on a prism manifold M(b,d) and do not leave a Heegaard Klein bottle invariant. We construct the following groups of isometries Z3, Z6, Dih(Z3) and Dih(Z6), which act on M(b,2) preserving the longitudinal fibering and not leaving any fibered Heegaard Klein bottle invariant. For the prism manifold M(1,d) we construct the following groups of isometries S4, A5, and A4, which act on M(1,d) preserving the meridian fibering and not leaving any Heegaard Klein bottle invariant. Let G0 be the normal subgroup of G consisting of the isometries which leave every fiber invariant. We show that if G acts on M(b,d) preserving the longitudinal fibering and not leaving any fibered Klein bottle invariant, then M(b,d)=M(b,2) and G/G0 is isomorphic to one of the groups Z3, Z6, Dih(Z3) and Dih(Z6); and if G preserves the meridian fibering not leaving any fibered Klein bottle invariant, then M(b,d)=M(1,d) and G/G0 is isomorphic to one of the groups S4, A5, and A4. We give complete descriptions of the group G in each of these cases.

Differentiable rigidity of hypersurface in space forms

Let M be a compact n-dimensional hypersurface in the space form \({F^{n+1}(c)}\) where n = 3 or 4. We prove that if the scalar curvature of M is positive and the integral of the length of the second fundamental form of M satisfies certain inequality, then M must be diffeomorphic to the spherical space form. On the other hand, we prove that if M is homeomorphic to 2-dimensional sphere \({\mathbb{S}^2}\) and f1, f2, f3 are first eigenfunctions of M such that \({\sum\nolimits_{i=1}^3|\nabla f_{i}|^2}\) is a constant, then M is isometric to a sphere with constant curvature.

Coupling of gravity to matter, spectral action and cosmic topology

We consider a model of modified gravity based on the spectral action functional, for a cosmic topology given by a spherical space form, and the associated slow-roll inflation scenario. We consider then the coupling of gravity to matter determined by an almost-commutative geometry over the spherical space form. We show that this produces a multiplicative shift of the amplitude of the power spectra for the density fluctuations and the gravitational waves, by a multiplicative factor equal to the total number of fermions in the matter sector of the model. We obtain the result by an explicit nonperturbative computation, based on the Poisson summation formula and the spectra of twisted Dirac operators on spherical space forms, as well as, for more general spacetime manifolds, using a heat kernel computation.

Non-fiber preserving actions on prism manifolds

In this paper we classify the finite groups of isometries which act on a prism manifolds M(b,d) and do not preserve any fibering. We construct nine distinct finite groups of isometries which act on M(1,2), and do not preserve any fibering. We then show that if a finite group of isometries G acts on M(b,d) and does not preserve any fibering, then M(b,d) = M(1,2) and G is conjugate to one of these nine groups which are: Z3 × T, T, O, S3 × O, Z3 ∘ O, S3 × T, Z3 × O, Z3 × I, and S3 × I , where T, O, I and S3 are the tetrahedral, octahedral, icosahedral, and symmetric groups respectively.

Spherically symmetric Riemannian manifolds of constant scalar curvature and their conformally flat representations

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in three, four and six dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in three-dimensional spaces because of the link to General Relativity. In particular, all three-dimensional constant negative scalar curvature spherical manifolds can be embedded as constant mean curvature surfaces in appropriate Schwarzschild solutions. Our approach, although not the simplest one, is linked to the Lichnerowicz–York method of finding initial data for Einstein equations.

Nielsen coincidence numbers, Hopf invariants and spherical space forms

Given two maps between smooth manifolds, the obstruction to removing their coincidences (via homotopies) is measured by minimum numbers. In order to determine them we introduce and study an infinite hierarchy of Nielsen numbers N_i, i = 0, 1, ..., \infty. They approximate the minimum numbers from below with decreasing accuracy, but they are (in principle) more easily computable as i grows. If the domain and the target manifold have the same dimension (e.g. in the fixed point setting) all these Nielsen numbers agree with the classical definition. However, in general they can be quite distinct. While our approach is very geometric the computations use the techniques of homotopy theory and, in particular, all versions of Hopf invariants (a la Ganea, Hilton, James..). As an illustration we determine all Nielsen numbers and minimum numbers for pairs of maps from spheres to spherical space forms. Maps into even dimensional real projective spaces turn out to produce particularly interesting coincidence phenomena.

Uniformization of spherical CR manifolds

Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n+1. Let M˜ be the universal covering of M. Let Φ denote a CR developing mapΦ:M˜→S2n+1 where S2n+1 is the standard unit sphere in complex n+1-space Cn+1. Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n⩾3. In the case n=2, we also show that Φ is injective under the condition: s(M)<1 where s(M) means the minimum exponent of the integrability of the Green's function for the CR invariant sublaplacian on M˜. It then follows that M is uniformizable.