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Hyperbolic Plane (hyperbolic + plane)
Selected AbstractsExistence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in T3 -GowdyCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2006Hans Ringström This is the first of two papers that together prove strong cosmic censorship in T3 -Gowdy space-times. In the end, we prove that there is a set of initial data, open with respect to the C2 × C1 topology and dense with respect to the C, topology, such that the corresponding space-times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, i.e., the Riemann tensor contracted with itself, blows up in the incomplete direction. In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions. In this paper, we shall, however, focus on the concept of asymptotic velocity. Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint. The target of the wave map is the hyperbolic plane. There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy. We define the asymptotic velocity v, to be the nonnegative square root of the limit of the kinetic energy density. The asymptotic velocity has some very important properties. In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v,. It also has properties such that if 0 < v,(,0) < 1, then v, is smooth in a neighborhood of ,0. Furthermore, if v,(,0) > 1 and v, is continuous at ,0, then v, is smooth in a neighborhood of ,0. Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to C2 × C1 topology on initial data. © 2005 Wiley Periodicals, Inc. [source] On a wave map equation arising in general relativityCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2004Hans Ringström We consider a class of space-times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t , ,. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t,1/2 as t , ,. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half-plane (after applying an isometry of hyperbolic space if necessary): 1The solution converges to a point. 2The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary). 3The solution goes to infinity along a curve y = const. 4The solution oscillates around a circle inside the upper half-plane. Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space-times. For instance, one obtains the leading-order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness. © 2004 Wiley Periodicals, Inc. [source] Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planesMATHEMATISCHE NACHRICHTEN, Issue 11 2005Bang-Yen Chen Abstract A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of AH . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP2 as well as in complex hyperbolic plane CH2. We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP2 and 41 families in CH2. All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP2 or in CH2 is a surface obtained from these 55 families. As an immediate by-product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] | ||||||||||