Metric Of Constant Curvature Research Articles

Combinatorial Ricci Flow for Degenerate Circle Packing Metrics

Chow and Luo [3] showed in 2003 that the combinatorial analogue of the Hamilton Ricci flow on surfaces converges under certain conditions to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [3] was that the weights are nonnegative. Recently we have shown that the same statement on convergence can be proved under a weaker condition: some weights can be negative and should satisfy certain inequalities [4]. On the other hand, for weights not satisfying conditions of Chow — Luo’s theorem we observed in numerical simulation a degeneration of the metric with certain regular behaviour patterns [5]. In this note we introduce degenerate circle packing metrics, and under weakened conditions on weights we prove that under certain assumptions for any initial metric an analogue of the combinatorial Ricci flow has a unique limit metric with a constant curvature outside of singularities.

A Characterisation for Finsler Metrics of Constant Curvature and a Finslerian Version of Beltrami Theorem

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function. This way we provide a Finslerian version of Beltrami Theorem.

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.

6D attractors and black hole microstates

We find a family of AdS2×ℳ4 supersymmetric solutions of the six-dimensional F(4) gauged supergravity coupled to one vector multiplet that arises as a low energy description of massive type IIA supergravity on (warped) AdS6 × S4. ℳ4 is either a Kähler-Einstein manifold or a product of two Riemann surfaces with a constant curvature metric. These solutions correspond to the near-horizon region of a family of static magnetically charged black holes. In the case where ℳ4 is a product of Riemann surfaces, we successfully compare their entropy to a microscopic counting based on the recently computed topologically twisted index of the five-dimensional mathcal{N} = 1 USp(2N) theory with Nf fundamental flavors and an antisymmetric matter field. Furthermore, our results suggest that the near-horizon regions exhibit an attractor mechanism for the scalars in the matter coupled F(4) gauged supergravity, and we give a proposal for it.

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.

Centroaffine Surfaces of Cohomogeneity One

We characterize centroaffine surfaces of cohomogeneity one which have vanishing Tchebychev vector fields. Moreover, we classify centroaffine minimal surfaces of cohomogeneity one which have centroaffine metrics of constant curvature.

A discrete uniformization theorem for polyhedral surfaces

A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a discrete Yamabe flow with surgery.

On a combinatorial curvature for surfaces with inversive distance circle packing metrics

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton [20]. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to α-curvature and prove that it is also globally rigid, which is in fact a generalized Bowers–Stephenson conjecture [6]. We also use the combinatorial Ricci flow to study the corresponding α-Yamabe problem.

On flops and canonical metrics

This article is concerned with an observation for proving non-existence of canonical Kahler metrics. The idea is to use a rather explicit type of degeneration that applies in many situations. Namely, in a variation on a theme introduced by Ross-Thomas, we consider flops of the deformation to the normal cone. This yields a rather widely applicable notion of stability that is still completely explicit and readily computable, but with wider scope. We describe some applications, among them, a proof of one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1. Motivation and results A variety is slope stable in the sense of Ross-Thomas if, roughly, it is K-stable with respect to degenerations to the normal cone of its subvarieties. This notion has been studied extensively by a number of authors and has yielded many non-existence results for canonical metrics on projective Kahler manifolds. Our main purpose in this article is to introduce a slight variation on this theme by considering a somewhat more involved notion of stability that involves additional flops on the degeneration to the normal cone but that is still geometric and computable, and is partly inspired by the work of Li-Xu. This gives many new non-existence results, and most notably allows us to resolve one direction of the Calabi conjecture for asymptotically logarithmic del Pezzo surfaces. 1.1. Existence theorem for KEE metrics. Kahler-Einstein edge (KEE) metrics are a nat- ural generalization of Kahler-Einstein metrics: they are smooth metrics on the complement of a divisor, and have a conical singularity of angle 2πβ transverse to that 'complex edge'. When β = 1, of course, this is just an ordinary Kahler-Einstein metric, that extends smoothly across the divisor. One can think of the metric as being 'bent' at an angle 2πβ along the divisor. In the case of Riemann surfaces, KEE metrics are just the familiar constant curvature metrics with isolated cone singularities, that have been studied since the late 19th century, e.g., by Picard (25). A basic question, whose origins trace back to Tian's 1994 lectures in the setting of nonpositive curvature (32), extended in Donaldson's 2009 lectures to the setting of anticanonical divisors on Fano manifolds (10), and further extended in our previous work (5) (see also the survey (27, §8)), is the following: Problem 1.1. Under what analytic conditions on the triple (X,D,β) does a KEE metric exist on the Kahler manifold X bent at an angle 2πβ along the divisor D ⊂ X? This is partly motivated by Troyanov's solution in the Riemann surface case (36), and was settled by Jeffres-Mazzeo-Rubinstein (sufficient condition)and Darvas-Rubinstein (sufficient and necessary conditions) in higher dimensions for smooth D (17, 7). These results give an analytic criterion characterizing existence, once the cohomological condition

Proper Curvature Collineations of Plane Symmetric Static Spacetime in F(R) Theory of Gravity

The purpose of this paper is to study the proper curvature collineations of plane symmetric static spacetime in f(R) theory of gravity. We have taken the metrics of both constant and non-constant curvature and formulated the curvature vector fields by using vanishing Lie derivative and direct integration technique. Here, the solutions of the Einstein’s field equations (EFEs) for both cases involve inverse curvature terms which may cause an increase in gravity that justifies the accelerated expansion of the universe. It turns out that curvature collineations become Killing vector fields and therefore, proper curvature collineations do not exists in both the cases.

On Stretch curvature of Finsler manifolds

In this paper, Finsler metrics with relatively non-negative (resp. non-positive), isotropic and constant stretch curvature are studied. In particular, it is showed that every compact Finsler manifold with relatively non-positive (resp. non-negative) stretch curvature is a Landsberg metric. Also, it is proved that every (α,β)-metric of non-zero constant flag curvature and non-zero relatively isotropic stretch curvature on a manifold of dimension n>2 has a constant characteristic scalar along the geodesics. Two dimensional Finsler manifolds of relatively stretch curvature are studied, too../files/site1/files/0Abstract4.pdf

Homogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitons

The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.

Wind Riemannian spaceforms and Randers–Kropina metrics of constant flag curvature

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to obtain the local classification. For the global one, a suitable result on completeness for WRS yields the complete simply connected models. In particular, any of the local models in the Randers classification does admit an extension to a unique model of wind Riemannian structure, even if it cannot be extended as a complete Finslerian manifold. Thus, WRS's emerge as the natural framework for the analysis of Randers spaceforms and, prospectively, wind Finslerian structures would become important for other global problems too. For the sake of completeness, a brief overview about WRS (including a useful link with the conformal geometry of a class of relativistic spacetimes) is also provided.

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We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of all such transforms in a neighborhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equations are injective in the absence of connection, then for any smooth enough connection multiplied by a complex parameter, the corresponding transform is injective for all values of that parameter outside a discrete set. Range characterizations are also provided.

Metrics of Constant Positive Curvature with Conical Singularities, Hurwitz Spaces, and Determinants of Laplacians

Let $f: X\to {\Bbb C}P^1$ be a meromorphic function of degree $N$ with simple poles and simple critical points on a compact Riemann surface $X$ of genus $g$ and let $\mathsf m$ be the standard round metric of curvature $1$ on the Riemann sphere ${\Bbb C}P^1$. Then the pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ is a metric of curvature $1$ with conical singularities of conical angles $4\pi$ at the critical points of $f$. We study the $\zeta$-regularized determinant of the Laplace operator on $X$ corresponding to the metric $f^*\mathsf m$ as a functional on the moduli space of the pairs $(X, f)$ (i.e. on the Hurwitz space $H_{g, N}(1, \dots, 1)$) and derive an explicit formula for the functional.

Weak and smooth solutions for a fractional Yamabe flow: The case of general compact and locally conformally flat manifolds

ABSTRACTAs a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the manifold is locally conformally flat with positive Yamabe invariant, we also prove that the flow is smooth and converges to a constant fractional curvature metric. We provide different proofs using extension properties introduced by Chang and González (2011) for the conformally covariant fractional order operators.

Equilibrium for a combinatorial Ricci flow with generalized weights on a tetrahedron

Chow and Lou [2] showed in 2003 that under certain conditions the combinatorial analogue of the Hamilton Ricci flow on surfaces converges to Thruston’s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. A crucial assumption in [2] was that the weights are nonnegative. We have recently shown that the same statement on convergence can be proved under weaker conditions: some weights can be negative and should satisfy certain inequalities. In this note we show that there are some restrictions for weakening the conditions. Namely, we show that in some situations the combinatorial Ricci flow has no equilibrium or has several points of equilibrium and, in particular, the convergence theorem is no longer valid.

Boundary behaviors for Liouville's equation in planar singular domains

We study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary. The conformal structure plays an essential role.

Strain tensor selection and the elastic theory of incompatible thin sheets.

The existing theory of incompatible elastic sheets uses the deviation of the surface metric from a reference metric to define the strain tensor [Efrati et al., J. Mech. Phys. Solids 57, 762 (2009)JMPSA80022-509610.1016/j.jmps.2008.12.004]. For a class of simple axisymmetric problems we examine an alternative formulation, defining the strain based on deviations of distances (rather than distances squared) from their rest values. While the two formulations converge in the limit of small slopes and in the limit of an incompressible sheet, for other cases they are found not to be equivalent. The alternative formulation offers several features which are absent in the existing theory. (a) In the case of planar deformations of flat incompatible sheets, it yields linear, exactly solvable, equations of equilibrium. (b) When reduced to uniaxial (one-dimensional) deformations, it coincides with the theory of extensible elastica; in particular, for a uniaxially bent sheet it yields an unstrained cylindrical configuration. (c) It gives a simple criterion determining whether an isometric immersion of an incompatible sheet is at mechanical equilibrium with respect to normal forces. For a reference metric of constant positive Gaussian curvature, a spherical cap is found to satisfy this criterion except in an arbitrarily narrow boundary layer.

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.