scholarly journals Equivariant Schrödinger map flow on two dimensional hyperbolic space

2020 ◽  
Vol 40 (7) ◽  
pp. 4379-4425
Author(s):  
Jiaxi Huang ◽  
◽  
Youde Wang ◽  
Lifeng Zhao ◽  
◽  
...  
Keyword(s):  
Hyperbolic Space ◽  
Two Dimensional ◽  
Schrödinger Map

scholarly journals Bra-ket wormholes in gravitationally prepared states

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Yiming Chen ◽  
Victor Gorbenko ◽  
Juan Maldacena
Keyword(s):  
Hyperbolic Space ◽  
Path Integral ◽  
Flat Space ◽  
Low Energy ◽  
Two Dimensional ◽  
De Sitter ◽  
Euclidean Case ◽  
Fine Grained ◽  
Entropy Formula ◽  
First Case

Abstract We consider two dimensional CFT states that are produced by a gravitational path integral.As a first case, we consider a state produced by Euclidean AdS2 evolution followed by flat space evolution. We use the fine grained entropy formula to explore the nature of the state. We find that the naive hyperbolic space geometry leads to a paradox. This is solved if we include a geometry that connects the bra with the ket, a bra-ket wormhole. The semiclassical Lorentzian interpretation leads to CFT state entangled with an expanding and collapsing Friedmann cosmology.As a second case, we consider a state produced by Lorentzian dS2 evolution, again followed by flat space evolution. The most naive geometry also leads to a similar paradox. We explore several possible bra-ket wormholes. The most obvious one leads to a badly divergent temperature. The most promising one also leads to a divergent temperature but by making a projection onto low energy states we find that it has features that look similar to the previous Euclidean case. In particular, the maximum entropy of an interval in the future is set by the de Sitter entropy.

Schwarzite nets: a wealth of 3-valent examples sharing similar topologies and symmetries

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200372
Author(s):  
Stephen T. Hyde ◽  
Martin Cramer Pedersen
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Two Dimensional ◽  
Hyperbolic Surfaces ◽  
Euclidean Three Space ◽  
Three Space ◽  
Planar Graphene

We enumerate trivalent reticulations of two- and three-periodic hyperbolic surfaces by equal-sided n -gonal faces, ( n , 3), where n  = 7, 8, 9, 10, 12. These are the simplest hyperbolic generalizations of the planar graphene net, (6, 3) and dodecahedrane, (5, 3). The enumeration proceeds by deleting isometries of regular reticulations of two-dimensional hyperbolic space until the ( n , 3) nets can be embedded in euclidean three-space via periodic hyperbolic surfaces. Those nets are then symmetrized in euclidean space retaining their net topology, leading to explicit crystallographic net embeddings whose edges are as equal as possible, affording candidtae patterns for graphitic schwarzites. The resulting schwarzites are the most symmetric examples. More than one hundred topologically distinct nets are described, most of which are novel.

scholarly journals KINEMATICS OF FLOWS ON CURVED, DEFORMABLE MEDIA

2009 ◽  
Vol 06 (04) ◽  
pp. 645-666 ◽  
Author(s):  
ANIRVAN DASGUPTA ◽  
HEMWATI NANDAN ◽  
SAYAN KAR
Keyword(s):  
Hyperbolic Space ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Flat Space ◽  
Singular Solutions ◽  
Geodesic Flows ◽  
Two Dimensional ◽  
Singular Behavior ◽  
Deformable Media ◽  
Raychaudhuri Equations

Kinematics of geodesic flows on specific, two-dimensional, curved surfaces (the sphere, hyperbolic space and the torus) are investigated by explicitly solving the evolution (Raychaudhuri) equations for the expansion, shear and rotation, for a variety of initial conditions. For flows on the sphere and on hyperbolic space, we show the existence of singular (within a finite value of the time parameter) as well as non-singular solutions. We illustrate our results through a phase diagram which demonstrates under which initial conditions (or combinations thereof) we end up with a singularity in the congruence and when, if at all, we can obtain non-singular solutions for the kinematic variables. Our analysis portrays the differences which arise due to positive or negative curvature and also explores the role of rotation in controlling singular behavior. Subsequently, we move on to geodesic flows on two-dimensional spaces with varying curvature. As an example, we discuss flows on a torus. Characteristic oscillatory features, dependent on the ratio of the two radii of the torus, emerge in the solutions for the expansion, shear and rotation. Singular (within a finite time) and non-singular behavior of the solutions are also discussed. Finally, we conclude with a generalization to three-dimensional spaces of constant curvature, a summary of some of the generic features obtained and a comparison of our results with those for flows in flat space.

scholarly journals JØRGENSEN’S INEQUALITY FOR QUATERNIONIC HYPERBOLIC SPACE WITH ELLIPTIC ELEMENTS

2009 ◽  
Vol 81 (1) ◽  
pp. 121-131 ◽  
Author(s):  
WENSHENG CAO ◽  
HAIOU TAN
Keyword(s):  
Hyperbolic Space ◽  
Two Dimensional ◽  
Möbius Group ◽  
Quaternionic Hyperbolic Space ◽  
Groups Of Isometries ◽  
Jørgensen’S Inequality

AbstractIn this paper, we give an analogue of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jørgensen’s inequality in the two-dimensional Möbius group of the above case.

Diophantine approximation and Hausdorff dimension in Fuchsian groups

1993 ◽  
Vol 113 (2) ◽  
pp. 343-354 ◽  
Author(s):  
S. L. Velani
Keyword(s):  
Hausdorff Dimension ◽  
Unit Circle ◽  
Hyperbolic Space ◽  
Diophantine Approximation ◽  
Fuchsian Groups ◽  
Two Dimensional ◽  
Disc Model ◽  
Image Position ◽  
Straight Lines

The Poincaré disc modelof two-dimensional hyperbolic space supports a metric ρ derived from the differentialGeodesics for the metric ρ are arcs of circles orthogonal to the unit circle S, and straight lines through the origin.

scholarly journals Existence results for semilinear problems in the two-dimensional hyperbolic space involving critical growth

2017 ◽  
Vol 147 (1) ◽  
pp. 141-176 ◽  
Author(s):  
Debdip Ganguly ◽  
Debabrata Karmakar
Keyword(s):  
Hyperbolic Space ◽  
Critical Growth ◽  
Two Dimensional ◽  
Disc Model ◽  
Semilinear Problems ◽  
Semilinear Elliptic Problems ◽  
Dimension 2 ◽  
Image Position ◽  
Ps Condition ◽  
Sign Changing Solutions

We consider semilinear elliptic problems on two-dimensional hyperbolic space. A model problem of our study iswhere H1(𝔹2) denotes the Sobolev space on the disc model of the hyperbolic space and f(x, t) denotes the function of critical growth in dimension 2. We first establish the Palais–Smale (PS) condition for the functional corresponding to the above equation, and using the PS condition we obtain existence of solutions. In addition, using a concentration argument, we also explore existence of infinitely many sign-changing solutions.

The Keller--Segel System on the Two-Dimensional-Hyperbolic Space

2020 ◽  
Vol 52 (5) ◽  
pp. 5036-5065
Author(s):  
Patrick Maheux ◽  
Vittoria Pierfelice
Keyword(s):  
Hyperbolic Space ◽  
Two Dimensional

Non-Euclidean Extensions

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Positive Curvature ◽  
Spherical Geometry ◽  
Two Dimensional ◽  
Actual Infinity ◽  
Constant Positive Curvature ◽  
Negative Constant ◽  
The Given ◽  
Line P

This chapter adapts the foregoing results to present two non-Euclidean theories, both in line with the (semi-)Aristotelian theme of rejecting points, as parts of regions (but working with actual infinity). The first theory is a two-dimensional hyperbolic space, that is, one that has a negative constant curvature. The second theory captures a space of constant positive curvature, a two-dimensional spherical geometry. The task here is to formulate axioms on regions which allow us to prove that (i) there are no infinitesimal regions and (ii) that there are no parallels to any given “line” through any “point” not on the given “line”.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

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