Metric Of Constant Curvature Research Articles

Holomorphic curvature of Finsler metrics and complex geodesics

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kähler, has constant holomorphic sectional curvature −4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature −4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.

On the geometry and topology of flows and foliations on surfaces and the Anosov problem

A study is made of flows with finitely many equilibrium states and of foliations with finitely many singularities of saddle type with integer and half-integer index on closed surfaces, and for a metric of constant curvature the role of geodesics is established in the asymptotic behaviour of semitrajectories of flows and semileaves of foliations upon lifting to the unbranched or branched universal covering.

A rigidity theorem for Haken manifolds

A compact, orientable 3-manifoldMis calledhyperbolicif intMadmits a complete hyperbolic structure (Riemannian metric of constant curvature − 1) of finite volume. Any hyperbolic 3-manifoldMis irreducible, and each component of∂Mis an incompressible torus. Letf:M→Nbe a proper, continuous map between hyperbolic 3-manifolds. By Mostow's Rigidity Theorem [8], iffis π1-isomorphic thenfis properly homotopic to a diffeomorphismg:M→Nsuch thatg| intM: intM→ intNis isometric. In particular, the topological type of intMdetermines uniquely the hyperbolic structure on intMup to isometry, so the volume vol (intM) of intMis well-defined. This Rigidity Theorem is generalized by Thurston[11, theorem 6·4] so that any proper, continuous mapf:M→Nbetween hyperbolic 3-manifolds with vol(intM) = deg(f) vol(intN) is properly homotopic to a deg(f)-fold coveringg:M→Nsuch thatg| intM: intM→ intNis locally isometric.

On metrics of constant sectional curvature

On metrics of constant sectional curvature

Toeplitz and Hankel type operators on an annulus

Let Ω be a regular domain in the extended complex plane, i.e., it is a bounded domain and its boundary consists of a finite number of disjoint analytic simple closed curves. Let dm(z) be the Lebesgue area measure on Ω and let ds = dm(z)/ω(z) be the Poincare metric on Ω, a Riemannian metric of negative constant curvature. It may be proved that Ω(z) ≈ Euclidean distance from z to the boundary of Ω (see [8]).

On the Functional logdet and Related Flows on the Space of Closed Embedded Curves on S2

For any two-dimensional Riemannian manifold ( M, g) we introduce a new functional, h g , on the space of closed simple nonparametrized curves on M. This functional associates to any simple curve Γ the regularize determinant of the Laplace operator on the manifold obtained by cutting M along Γ and imposing Dirichlet boundary conditions. When M is of genus zero we derive a formula for the variation of h g , we prove that the critical points are conformal circles (i.e., the curves which, with respect to the unique metric of constant curvature 1 in the conformal class { e 2α g:α ∈ C ∞( S 2, R )} of g, have constant geodesic curvature), and that the hessian of the functional at a critical point is nondegenerate in directions normal the critical submanifold (Theorem 1.1). We also construct smooth flows on the space of nonparametrized curves retracting the space onto the critical sub-manifold and show that they are gradient-like for our function. These flows deform a given closed embedded curve on S 2 to a conformal circle keeping the area of the domain bounded by each curve of the deformation constant (Theorem 1.3).

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

Correspondence Principle for the Quantized Annulus, Romanovski Polynomials, and Morse Potential

A general theory of quantization has been proposed by Berezin (see, e.g., his survey in Comm. Math. Phys. 40 (1975), 153-174). In this paper we establish a weak form of the correspondence principle for the annulus, quantized according to Berezin. More precisely, we show that B ħ → I as ħ → 0, where I is the identity operator and B ħ the Berezin transform. We consider also spectral analysis on the annulus. In particular, we express the eigenfunctions of the relevant Laplacian in terms of certain Romanovski polynomials. Finally, we write down the expression for the analogue of the Morse potential in this case. We remark that similar considerations can be made, in principle, on any circular domain in the presence of a radial Hermitian metric of constant curvature.

Weierstrass points and multiple intersection points of closed geodesics

Every Riemann surfaceM g of genusg ⩾ 2 carries a canonical metric of constant curvature −1. We show:

The upper Perron method for labelled complexes with applications to circle packings

The construction of geometric surfaces via labelled complexes was introduced by Thurston[16, chapter 13], and subsequent applications and developments have appeared in [1, 3, 4, 5, 14, 15]. The basic idea of using labelled complexes to produce geometric structures is that the vertices of a simplicial triangulation of a surface can be labelled with positive real numbers that collectively determine a metric of constant curvature ±1 or 0, with possible singularities at vertices, by using the label values to identify 2-simplices of the triangulation with geometric triangles. Beardon and Stephenson[1] developed a particularly simple method for producing non-singular surfaces via labelled complexes that is modelled after the classical Perron method for producing harmonic functions, and they applied their method in [2] to construct a fairly comprehensive theory of circle packings in general Riemann surfaces. This Perron method was developed more fully by Stephenson and the author in [3, 4] and applied to the study of circle packing points in moduli space. At about the same time and independently of Beardon, Stephenson, and Bowers, Carter and Rodin [5] and Doyle [8] developed the method for flat surfaces and Minda and Rodin [14] developed the method for finite type surfaces. Minda and Rodin [14] applied their development to give partial solutions to the labelled complex version of the classical Schwarz-Picard problem that concerns the construction of singular hyperbolic metrics on surfaces with prescribed singularities. In this paper, we modify the aforementioned approaches and examine the upper Perron method for producing non-singular geometric surfaces. This upper method has several advantages over the Perron method as developed previously and provides a complete solution to the labelled complex version of the Schwarz-Picard problem.

Curvature Pinching Based on Integral Norms of the Curvature

AbstractA compact Riemannian manifold (M, g) of dimension 3 or higher admits a metric of constant (positive or negative) sectional curvature if the following conditions hold: the diameter is bounded from above, the part of the Ricci curvature which lies below some fixed negative number is bounded in LP norm for p > n/2, and the metric is almost spherical or almost hyperbolic in the LP sense. The idea of the proof is to obtain stronger (i.e. L∞) pinching by deforming the initial metric using the Ricci flow, thus reducing the problem to the theorems of Gromov in the case rg < 0 and of Grove, Karcher and Ruh in the case rg > 0. The reduced curvature tensor changes along the flow according to the heat equation, which implies a weak nonlinear parabolic inequality for its norm. The iteration method of De Giorgi, Nash and Moser is applied to obtain the estimate for the maximum norm of the reduced curvature tensor. The crucial step in the iteration consists of controlling the Sobolev constant of the appropriate imbedding (which also changes along the flow, but behaves well) by the isoperimetric constant, which, in turn, can be bounded in terms independent of the particular manifold.

All universal coverings of two-dimensional gravity with torsion

Global solutions of two-dimensional gravity with torsion are found and elucidated. The model admits complete analysis of possible maximally continued space-times by means of Penrose’s diagrams. There are 13 types of orientable universal coverings. For zero cosmological constant there is flat two-dimensional Minkowskian space-time. For nonzero cosmological constant the universal covering of zero torsion covers one-sheet hyperboloid with Lorenzian metric of constant curvature. All other universal coverings are the surfaces of nonzero torsion and nonconstant curvature. The global solutions include surfaces of a black hole configuration.

The Weil-Petersson symplectic structure at Thurston’s boundary

The Weil-Petersson Kähler structure on the Teichmüller space $\mathcal {T}$ of a punctured surface is shown to extend, in an appropriate sense, to Thurston’s symplectic structure on the space $\mathcal {M}{\mathcal {F}_0}$ of measured foliations of compact support on the surface. We introduce a space ${\widetilde {\mathcal {M}\mathcal {F}}_0}$ of decorated measured foliations whose relationship to $\mathcal {M}{\mathcal {F}_0}$ is analogous to the relationship between the decorated Teichmüller space $\tilde {\mathcal {T}}$ and $\mathcal {T}$. $\widetilde {\mathcal {M}{\mathcal {F}_0}}$ is parametrized by a vector space, and there is a natural piecewise-linear embedding of $\mathcal {M}{\mathcal {F}_0}$ in $\widetilde {\mathcal {M}{\mathcal {F}_0}}$ which pulls back a global differential form to Thurston’s symplectic form. We exhibit a homeomorphism between $\tilde {\mathcal {T}}$ and ${\widetilde {\mathcal {M}\mathcal {F}}_0}$ which preserves the natural two-forms on these spaces. Following Thurston, we finally consider the space $\mathcal {Y}$ of all suitable classes of metrics of constant Gaussian curvature on the surface, form a natural completion $\overline {\mathcal {Y}}$ of $\mathcal {Y}$, and identify $\overline {\mathcal {Y}} - \mathcal {Y}$ with $\mathcal {M}{\mathcal {F}_0}$. An extension of the Weil-Petersson Kähler form to $\mathcal {Y}$ is found to extend continuously by Thurston’s symplectic pairing on $\mathcal {M}{\mathcal {F}_0}$ to a two-form on $\overline {\mathcal {Y}}$ itself.

Rigidity Theorems for Nonpositive Einstein Metrics

In this paper we study the following problem: When must a complete Einstein metric $g$ on an $n$-manifold with $\operatorname {Ric} = (n - 1)\lambda {\text {g,}}\lambda \leq 0$, be a metric of constant curvature $\lambda ?$

Boost symmetries in spatially compact spacetimes with a cosmological constant

The author characterizes those boost symmetric spacetimes which, in addition to de Sitter space, admit spatially compact, (locally) conformally flat, time symmetric Cauchy hypersurfaces and satisfy Einstein's vacuum field equations with a cosmological constant. The Cauchy surfaces all turn out to be (finite quotients of) Riemannian products and warped products of (S1,dy2) with (S2, gamma k) where gamma k is the canonical metric of constant positive sectional curvature k.

Hamiltonian systems of hydrodynamic type and constant curvature metrics

We consider the one-dimensional systems of hydrodynamic type. The general relations for the nonlocal Poisson brackets of hydrodynamic type are presented. For a nondegenerate nonlocal Poisson bracket of hydrodynamic type the relations mean exactly that the bracket is generated by a metric of constant Riemannian curvature.

A remark on foliations on a complex projective space with complex leaves

Let EF be a foliation on a Riemannian manifold M. The distribution on M which is defined to be orthogonal to SF is said to be normal to £Fand denoted by 3L. Nakagawa and Takagi [8] showed that any harmonic foliation on a compact Riemannian manifold of non-negative constant sectional curvature is totally geodesic if the normal distribution is minnimal. And succesively the present author [2] proved a complex version of their result, that is, the above result holds also on a complex projective space with a Fubini-Study metric. However, recently, Li [4] pointed out a serious mistake in the proof of the result of Nakagawa and Takagi, and so of the author's. Therefore those results are now open yet. On the other hand, Li [4] have studied a harmonic foliationon the sphere along the method of Nakagawa and Takagi, and obtained some interesting results. The purpose of this paper is to give a complex analogue of the Li's results. Let Pn+p(C) be the complex projective space with the Fubini-Study metric of constant holomorphic sectional curvature c. Let £Fbe a complex foliation on Pn+p(C) with ^-complex codimention and h the second fundamental tensor of <7. Then we shall Drove the following:

Minimal immersions of S2 and ℝP2 into ℂPn with few higher order singularities

In this paper we extend ideas developed in 2, 4 to study certain minimal immersions of S2 and ℝP2 into ℂPn. Here S2 denotes the unit sphere in ℝ3 with its standard conformal structure and ℝP2 is S2 factored out by the antipodal map, while ℂPn denotes complex projective n-space equipped with the FubiniStudy metric of constant holomorphic sectional curvature 4. Since ℝPn with its standard metric of constant curvature 1 is included in ℂPn as a totally geodesic submanifold, this includes the case of minimal immersions into the unit sphere Sn(1) with its standard metric.

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

Integrability conditions for a certain class of nonlinear evolution equations and Kähler geometry

Multicomponent evolution equations associated with linear connections on complex manifolds are considered. It is proved that under some general assumptions an equation from this class is integrable by inverse scattering method if the corresponding linear connection is the Levi-Civita connection of an indefinite Kählerian metric of constant holomorphic sectional curvature. This result is based on a certain characterization of the above-mentioned Levi-Civita connections. It is shown that the obtained integrable equations are generalized ferromagnetics, and recurrent formulas for their local conservation laws are given.