Metric Of Constant Negative Curvature Research Articles

Flexibility of entropies for surfaces of negative curvature

We consider a smooth closed surface M of fixed genus ⩾ 2 with a Rie-mannian metric g of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for g is greater than or equal to the topological entropy for the metric of constant negative curvature on M with the same total area which is greater than or equal to the metric entropy with respect to the Liouville measure of geodesic flow for g. Equality holds only in the case of constant negative curvature. We prove that those are the only restrictions on the values of topological and metric entropies for metrics of negative curvature.

Rigid Levi degenerate hypersurfaces with vanishing CR-curvature

We continue the study, initiated in our earlier article, of a class of rigid hypersurfaces in C 3 that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of the earlier paper and give a complete description of the class. Surprisingly, the answer is expressed in terms of solutions of several well-known differential equations, in particular, the equation characterizing conformal metrics of constant negative curvature and a nonlinear ∂ ¯ -equation.

Generalized two-field α-attractor models from geometrically finite hyperbolic surfaces

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface Σ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field α-attractor models, being parameterized by a positive constant α, by the choice of a finitely-generated surface group Γ⊂PSL(2,R) (which is isomorphic with the fundamental group of Σ) and by the choice of a scalar potential defined on Σ. The traditional two-field α-attractor models arise when Γ is the trivial group, in which case Σ is the Poincaré disk. We give a general prescription for the study of such models through uniformization in the so-called “non-elementary” case and discuss some of their qualitative features in the gradient flow approximation, which we relate to Morse theory. We also discuss some aspects of the SRST approximation in these models, showing that it is generally not well-suited for studying dynamics near cusp ends. When Σ is non-compact and the scalar potential is “well-behaved” at the ends, we show that, in the naive local one-field truncation, our generalized models have the same universal behavior as ordinary one-field α-attractors if inflation happens near any of the ends of Σ where the extended potential has a local maximum, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends; we also discuss spiral trajectories near the ends. Generalized two field α-attractors illustrate interesting consequences of nonlinear sigma models whose scalar manifold is not simply connected. They provide a large class of tractable cosmological models with non-trivial topology of the scalar field space.

Geodesics and nodal sets of Laplace eigenfunctions on hyperbolic manifolds

Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be bounded just in terms of the area of the surface. We also provide constructions of geodesics in hyperbolic surfaces that lie in a nodal set but that do not lie in the fixed point set of a reflection symmetry.

Bounding hyperbolic and spherical coefficients of Maass forms

On développe une nouvelle méthode pour majorer les coefficients de Fourier hyperboliques et sphériques des formes de Maass définies par rapport à des réseaux uniformes généraux.

Hermitian-holomorphic Deligne cohomology, Deligne pairing for singular metrics, and hyperbolic metrics

We introduce a model for Hermitian-holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular Hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with “good” Hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the singularity divisor of a hyperbolic metric: its cup square (corrected by the total area) is shown to be a functional whose extrema are the metrics of constant negative curvature.

Recent progress on the Poincaré conjecture and the classification of 3-manifolds

If M is a closed 3-manifold with trivial fundamental group, then is M diffeomorphic to S? The Poincare Conjecture is that the answer to this question is “Yes.” Developing tools to attack this problem formed the basis for much of the work in 3-dimensional topology over the last century, including for example, the proof of Dehn’s lemma and the loop theorem and the study of surgery on knots and links. In the 1980’s Thurston developed another approach to 3-manifolds, see [24] and [4]. He considered 3-manifolds with riemannian metrics of constant negative curvature −1. These manifolds, which are locally isometric to hyperbolic 3-space, are called hyperbolic manifolds. There are fairly obvious obstructions showing that not every 3-manifold can admit such a metric. Thurston formulated a general conjecture that roughly says that the obvious obstructions are the only ones; should they vanish for a particular 3-manifold then that manifold admits such a metric. His proof of various important special cases of this conjecture led him to formulate a more general conjecture about the existence of locally homogeneous metrics, hyperbolic or otherwise, for all manifolds; this is called Thurston’s Geometrization Conjecture for 3-manifolds. The statement of this conjecture is somewhat complicated, so it is deferred until Section 2. An important point is that Thurston’s Geometrization Conjecture includes the Poincare Conjecture as a very special case. In addition, Thurston’s conjecture has two advantages over the Poincare Conjecture: • It applies to all closed oriented 3-manifolds. • It posits a close relationship between topology and geometry in dimension three.

On Hermitian-Holomorphic Classes Related to Uniformization, the Dilogarithm, and the Liouville Action

Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Cech-de Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bundles. We reprove the relation with the determinant of cohomology construction. Furthermore, if we specialize the covering to the one provided by a Kleinian uniformization (thereby allowing possibly disconnected surfaces) the same class can be reinterpreted as the transgression of the regulator class expressed by the Bloch-Wigner dilogarithm.

Peaks in the Hartle–Hawking wavefunction from sums over topologies

Recent developments in ‘Einstein Dehn filling’ allow the construction of infinitely many Einstein manifolds that have different topologies but are geometrically close to each other. Using these results, we show that for many spatial topologies, the Hartle–Hawking wavefunction for a spacetime with a negative cosmological constant develops sharp peaks at certain calculable geometries. The peaks we find are all centred on spatial metrics of constant negative curvature, suggesting a new mechanism for obtaining local homogeneity in quantum cosmology.

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e 2u=Ko(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore 2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem.

Entropy rigidity of Anosov flows in dimension three

We show that for a smooth contact Anosov flow on a closed three manifold the measure of maximal entropy is in the Lebesgue class if and only if the flow is up to finite covers conjugate to the geodesic flow of a metric of constant negative curvature on a closed surface.This shows that the ratio between the measure theoretic entropy and the topological entropy of a contact Anosov flow is strictly smaller than one on any closed three manifold which is not a Seifert bundle.

Quantum effective action from the AdS/CFT correspondence

We obtain an Einstein metric of constant negative curvature given an arbitrary boundary metric in three dimensions, and a conformally flat one given an arbitrary conformally flat boundary metric in other dimensions. In order to compute the on-shell value of the gravitational action for these solutions, we propose to integrate the radial coordinate from the boundary till a critical value where the bulk volume element vanishes. The result, which is a functional of the boundary metric, provides a sector of the quantum effective action common to all conformal field theories that have a gravitational description. We verify that the so-defined boundary effective action is conformally invariant in odd (boundary) dimensions and has the correct conformal anomaly in even (boundary) dimensions. In three dimensions and for arbitrary static boundary metric the bulk metric takes a rather simple form. We explicitly carry out the computation of the corresponding effective action and find that it equals the non-local Polyakov action.

The Witten Laplacian on Negatively Curved Simply Connected Manifolds

The Hopf conjecture states that the Euler characteristic of a compact Riemannian $2n$-manifold $\overline{M}$ of negative sectional curvature satisfies $(-1)^{n}\chi(\overline{M})>0[6]$ . Applying the Chern-Gauss-Bonnet theorem gives the conjecture for $n=1,2$ , for spaces of constant curvature, and for spaces of sufficiently pinched curvature [5]. Singer’s idea of instead using the $L^{2}$ index theorem to establish the Hopf conjecture has been successfully carried out for K\ahler manifolds by Gromov [18] (cf. [11]). It is worth noting that the first examples of negatively curved manifolds not admitting metrics of constant negative curvature are rather recent [20], [19]. Singer’s method depends on the vanishing of $L^{2}$ harmonic forms (except in the middle dimension) on the universal cover of a compact negatively curved manifold, as explained in \S 4. This raised the question of such vanishing for arbitrary simply connected negatively curved manifolds. Anderson’s paper [1] shows that such vanishing results are not possible without a pinching condition; however, his examples admit no compact quotient, so Singer’s approach is not ruled out. One of our main results (Corollary 4.4) is that for one-forms vanishing occurs except in the pinching region ruled out by Anderson’s examples. In general, we obtain vanishing results and hence $(-1)^{n}\chi(\overline{M})\geq 0$ (Theorem 4.5) for manifolds of pinched negative curvature, where the pinching constant is more relaxed than in previous work, e.g. [5]. The vanishing theorems depend upon Witten’s deformation $\coprod_{\tau}$ of the Laplacian $\cdot$ on forms on $M[21]$ . In contrast to Witten’s work, in which the Morse inequalities are recovered by letting the deformation parameter $\tau$ go to infinity, the vanishing theorems arise through the study of small deformations. Moreover, instead of deforming the Laplacian by a Morse function as in [21], we use the distance function to a point. The

Riemann surfaces and the geometrization of 3-manifolds

About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of hyperbolic geometry, as natural as Euclid’s regular polyhedra. For a closed manifold, Mostow rigidity assures that a hyperbolic structure is unique when it exists [Mos], so topology and geometry mesh harmoniously in dimension 3. This remarkable theorem applies to all 3-manifolds which can be built up in an inductive way from 3-balls, i.e. Haken manifolds. Thurston’s construction of a hyperbolic structure is also inductive. At the inductive step one must find the right geometry on an open 3-manifold so that its ends may be glued together. Using quasiconformal deformations, the gluing problem can be formulated as a fixed-point problem for a map of Teichmuller space to itself. Thurston proposes to find the fixed point by iterating this map. Here we outline Thurston’s construction, and sketch a new proof that the iteration converges. Our argument rests on a result entirely in the theory of Riemann surfaces: an extremal quasiconformal mapping can be relaxed (isotoped to a map of lesser dilatation) when lifted to a sufficiently

Finite parts of the spectrum of a Riemann surface

Let S and $S'$ be compact Riemann surfaces of the same genus g (g$ge 2)$ endowed with the Poincare metric of constant negative curvature -1. par The authors show that for every $epsilon >0$, there exists an integer $m=m(g,epsilon)$ with the property: Assume that (1) injectivity radius of S and $S'ge epsilon$, and (2) the first $m=m(g,epsilon)$ eigenvalues of the Laplacian of S and $S'$ coincide, then S and $S'$ are isospectral. par The authors conjecture that the integer m will not depend on $epsilon$ that is, the theorem holds with an integer which depends only on the genus. par For the proof, a model of Teichmuller space is described and the analyticity of the resolvent of Laplacian on this space is proved. Also the authors note that the injectivity radius is estimated by a finite part of the spectrum $(=$ the number of eigenvalues of the Laplacian in the interval [1/4,1]).

Characteristic classes of Lagrangian foliations

Of course a symplectic manifold is a special case of quasisymplectic manifold. An example of a quasicontact manifold with a Lagrangian foliation is a contact manifold (M =n+~, ~) and a foliation ~"+* on it such that the characteristic field Ker dA is tangent to~ r and~cuts out a Lagrangian subbundle in the symplectic distribution Ker I. In particular, the horocyclic foliation on the manifold of contact elements of a flat torus or of a surface of genus ~2 with metric of constant negative curvature (a leaf is formed by parallel contact elements) is such.

Four applications of conformal equivalence to geometry and dynamics

AbstractConformal equivalence theorem from complex analysis says that every Riemannian metric on a compact surface with negative Euler characteristics can be obtained by multiplying a metric of constant negative curvature by a scalar function. This fact is used to produce information about the topological and metric entropies of the geodesic flow associated with a Riemannian metric, geodesic length spectrum, geodesic and harmonic measures of infinity and Cheeger asymptotic isoperimetric constant. The method is rather uniform and is based on a comparison of extremals for variational problems for conformally equivalent metrics.

Entropy and closed geodesies

AbstractWe study asymptotic growth of closed geodesies for various Riemannian metrics on a compact manifold which carries a metric of negative sectional curvature. Our approach makes use of both variational and dynamical description of geodesies and can be described as an asymptotic version of length-area method. We also obtain various inequalities between topological and measure-theoretic entropies of the geodesic flows for different metrics on the same manifold. Our method works especially well for any metric conformally equivalent to a metric of constant negative curvature. For a surface with negative Euler characteristics every Riemannian metric has this property due to a classical regularization theorem. This allows us to prove that every metric of non-constant curvature has strictly more close geodesies of length at most T for sufficiently large T then any metric of constant curvature of the same total area. In addition the common value of topological and measure-theoretic entropies for metrics of constant negative curvature with the fixed area separates the values of two entropies for other metrics with the same area.

Closed geodesics on Riemann surfaces

This note deals with an elementary property of closed geodesics on surfaces with a complete metric of constant negative curvature.

Mappings of nonpositively curved manifolds

In recent papers with S. S. Chern [3] and T.Ishihara [4], the author studied both the volume—and distance—decreasing properties of harmonic mappings thereby obtaining real analogues and generalizations of the classical Schwarz-Ahlfors lemma, as well as Liouville’s theorem and the little Picard theorem. The domain M in the first case was the open ball with the hyperbolic metric of constant negative curvature, and the target was a negatively curved Riemannian manifold with sectional curvature bounded away from zero. In this paper, it is shown that M may be taken to be any complete Riemannian manifold of non-positive curvature.