Metric Of Constant Curvature Research Articles

Every noncompact surface is a leaf of a minimal foliation

We show that any noncompact oriented surface is homeomorphic to the leaf of a minimal foliation of a closed 3 -manifold. These foliations are (or are covered by) suspensions of continuous minimal actions of surface groups on the circle. Moreover, the above result is also true for any prescription of a countable family of topologies of noncompact surfaces: they can coexist in the same minimal foliation. All the given examples are hyperbolic foliations, meaning that they admit a leafwise Riemannian metric of constant negative curvature. Many oriented Seifert manifolds with a fibered incompressible torus and whose associated orbifold is hyperbolic admit minimal foliations as above. The given examples are not transversely C^{2} -smoothable.

The Holonomy of Spherically Symmetric Projective Finsler Metrics of Constant Curvature

In this paper, we investigate the holonomy group of n-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to Diffo(Sn-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}i\\!f \\hspace{-3pt} f_o({\\mathbb {S}}^{n-1})$$\\end{document}, the connected component of the identity of the group of smooth diffeomorphism on the n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${n-1}$$\\end{document}-dimensional sphere. In particular, the holonomy group of the n-dimensional standard Funk metric and the Bryant–Shen metrics are maximal and isomorphic to Diffo(Sn-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}i\\!f \\hspace{-3pt} f_o({\\mathbb {S}}^{n-1})$$\\end{document}. These results are the firsts describing explicitly the holonomy group of n-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case.

Reversible Finsler metrics of constant flag curvature

Without the restriction of quadratic form as a Riemannian metric, a Finsler metric F=F(x,y) on a smooth manifold M can be reversible (symmetric in y) or not. Reversible Finsler metrics have different properties from Riemannian metrics though it seems they are very close to Riemannian metrics. Hilbert metric is the famous reversible Finsler metric of negative constant flag curvature in the history, and it is projectively flat. Then it is natural to ask the question how to classify reversible projectively flat Finsler metrics of constant flag curvature and give more new examples? In this paper, we answer the above question by giving the classification when the flag curvature K=−1,0,1 respectively. Especially, for the case when K=−1, we show that the only reversible projectively flat Finsler metrics are just Hilbert metrics. For the case when K=1, we give an algebraic way to construct explicit metric function by solving algebraic equations, such as by solving a quartic equation. When the flag curvature is zero, it is much easier to construct reversible projectively flat Finsler metrics than before.

Elliptic KdV potentials and conical metrics of positive constant curvature, I

Elliptic KdV potentials and conical metrics of positive constant curvature, I

Weighted combinatorial Ricci flow and metrics defined by degenerate circle packings

Chow and Luo [1] in 2003 had shown that the combinatorial analogue of the Hamilton Ricci flow on surfaces under certain conditions converges to Thruston?s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. Crucial assumption in the paper [1] was that the weights are nonnegative. Recently we have shown that same statement on convergence can be proved under weaker condition: some weights can be negative and should satisfy certain inequalities, [3]. Moreover, in [6] notions of degenerate circle packing and corresponding metric were introduced. In [6] theory of combinatorial Ricci flow for such metrics was developed, which includes Chow-Luo theory as a partial case for nondegenerate circle packing and nonnegative weights on edges. On the other hand, in [2] the combinatorial Yamabe flow was introduced and investigated. In [7, 8] we developed weighted modification of Yamabe flow. In this paper we merge ideas from these two theories and introduce weighted combinatorial Ricci flow on metrics defined by degenerate circle packings. We prove that under certain conditions for any initial metric the flow converges to a unique metric of constant curvature.

Determinants of Laplacians for constant curvature metrics with three conical singularities on the 2-sphere

We deduce an explicit closed formula for the zeta-regularized spectral determinant of the Friedrichs Laplacian on the Riemann sphere equipped with arbitrary constant curvature (flat, spherical, or hyperbolic) metric having three conical singularities of order \(\beta _j\in (-1,0)\) (or, equivalently, of angle \(2\pi (\beta _j+1)\)). We show that among the metrics with a fixed value of the sum \(\beta _1+\beta _2+\beta _3\) and a fixed surface area, those with \(\beta _1=\beta _2=\beta _3\) correspond to a stationary point of the determinant. If, in addition, the surface area is sufficiently small, then the stationary point is a minimum. As a crucial step towards obtaining these results, we find a new anomaly formula for the determinant of Laplacian that includes (as one of its terms) the Liouville action, introduced by A. Zamolodchikov and Al. Zamolodchikov in connection with the celebrated DOZZ formula for the three-point structure constants of the Liouville field theory. The Liouville action satisfies a system of differential equations that can be easily integrated.

Classification of Left Invariant Riemannian Metrics on Complex Hyperbolic Space

It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that each of those metrics is of constant negative scalar curvature, only one of them being Einstein (up to isometry and scaling).

Ricci flow and diffeomorphism groups of 3-manifolds

We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature.

A random cover of a compact hyperbolic surface has relative spectral gap frac{3}{16}-varepsilon

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each nin {mathbf {N}}, let X_{n} be a random degree-n cover of X sampled uniformly from all degree-n Riemannian covering spaces of X. An eigenvalue of X or X_{n} is an eigenvalue of the associated Laplacian operator Delta _{X} or Delta _{X_{n}}. We say that an eigenvalue of X_{n} is new if it occurs with greater multiplicity than in X. We prove that for any varepsilon >0, with probability tending to 1 as nrightarrow infty , there are no new eigenvalues of X_{n} below frac{3}{16}-varepsilon . We conjecture that the same result holds with frac{3}{16} replaced by frac{1}{4}.

Area-minimizing properties of Pansu spheres in the sub-Riemannian 3-sphere

Abstract We consider the sub-Riemannian 3-sphere ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} obtained by restriction of the Riemannian metric of constant curvature 1 to the planar distribution orthogonal to the vertical Hopf vector field. It was shown in [A. Hurtado and C. Rosales, Area-stationary surfaces inside the sub-Riemannian three-sphere, Math. Ann. 340 2008, 3, 675–708] that ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} contains a family of spherical surfaces { 𝒮 λ } λ ⩾ 0 {\{\mathcal{S}_{\lambda}\}_{\lambda\geqslant 0}} with constant mean curvature λ. In this work, we first prove that the two closed half-spheres of 𝒮 0 {\mathcal{S}_{0}} with boundary C 0 = { 0 } × 𝕊 1 {C_{0}=\{0\}\times\mathbb{S}^{1}} minimize the sub-Riemannian area among compact C 1 {C^{1}} surfaces with the same boundary. We also see that the only C 2 {C^{2}} solutions to this Plateau problem are vertical translations of such half-spheres. Second, we establish that the closed 3-ball enclosed by a sphere 𝒮 λ {\mathcal{S}_{\lambda}} with λ > 0 {\lambda>0} uniquely solves the isoperimetric problem in ( 𝕊 3 , g h ) {(\mathbb{S}^{3},g_{h})} for C 1 {C^{1}} sets inside a vertical solid tube and containing a horizontal section of the tube. The proofs mainly rely on calibration arguments.

On the deformation of ball packings

In this paper, we study the geometric aspects of ball packings on (M,T), where T is a triangulation on a 3-manifold M. First, we introduce a combinatorial Yamabe invariant YT, depending on the topology of M and the combinatoric of T. Then we prove that YT is attainable if and only if there is a constant curvature packing, and the combinatorial Yamabe problem can be solved by minimizing the Cooper-Rivin-Glickenstein functional. We also study the combinatorial Yamabe flow introduced by Glickenstein [23–25]. First, we prove a small energy convergence theorem that the flow would converge to a constant curvature metric if the initial energy is close in a quantitative way to the energy of a constant curvature metric. We also show even in case that the flow develops singularities in finite time, there is a natural way to extend the flow through the singularities such that the flow exists for all time. Finally, if the triangulation T is regular (i.e., all the numbers of tetrahedrons surrounding each vertex are equal), the combinatorial Yamabe flow converges exponentially fast to a constant curvature packing.

Bergman–Calabi diastasis and Kähler metric of constant holomorphic sectional curvature

We prove that for a bounded domain in $\mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation and characterize pseudoconvex domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set.

Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

For a one-parameter family of simple metrics of constant curvature ( 4\kappa for \kappa\in (-1,1) ) on the unit disk M , we first make explicit the Pestov–Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions à la Helgason–Ludwig or Gel'fand–Graev. We then derive an explicit Singular Value Decomposition for the geodesic X-ray transform. Computations dictate a specific choice of weighted L^2-L^2 setting which is equivalent to the L^2(M, \operatorname{dVol}_\kappa)\to L^2(\partial_+SM,d\Sigma^2) one for any \kappa\in (-1,1) .

Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ⩽n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

On Projectively Flat Finsler Warped Product Metrics of Constant Flag Curvature

In this paper, we study locally projectively flat Finsler metrics of constant flag curvature. We find equations that characterize these metrics by warped product. Using the obtained equations, we manufacture new locally projectively flat Finsler warped product metrics of vanishing flag curvature. These metrics contain the metric introduced by Berwald and the spherically symmetric metric given by Mo-Zhu.

Curvatures of Spherically Symmetric Metrics on Vector Bundles

This paper is devoted to study the geometry of vector bundle manifolds equipped with spherically symmetric metrics. We will compute the curvature tensor, the sectional curvature, the Ricci curvature tensor and the scalar curvature. We will then characterize spherically symmetric metrics of constant sectional curvature. Moreover, we will establish some rigidity results regarding the scalar curvature and the Ricci curvature. Finally, we investigate geodesics using the fundamental tensors of a submersion.

A class of Finsler metrics admitting first integrals

We use two non-Riemannian curvature tensors, the χ-curvature and the mean Berwald curvature to characterise a class of Finsler metrics admitting first integrals. This class includes Finsler metrics of constant flag curvature.

Moduli spaces for Lamé functions and Abelian differentials of the second kind

The topology of the moduli space for Lamé functions of degree [Formula: see text] is determined: this is a Riemann surface which consists of two connected components when [Formula: see text]; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle [Formula: see text] on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves.

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.

Isometry groups of infinite-genus hyperbolic surfaces

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannian metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.