Journal articles: 'Metric geometry of singularities'

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Sabourau, Stéphane, and Zeina Yassine. "A systolic-like extremal genus two surface." Journal of Topology and Analysis 11, no. 03 (September 2019): 721–38. http://dx.doi.org/10.1142/s1793525319500298.

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It is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics. We prove that this piecewise flat metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes.

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Stoica, Ovidiu-Cristinel. "Spacetimes with singularities." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 2 (June 1, 2012): 213–38. http://dx.doi.org/10.2478/v10309-012-0050-3.

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Abstract We report on some advances made in the problem of singularities in general relativity.First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein's equation, which works for degenerate metric too.Once we make the singularities manageable from mathematical viewpoint, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. Then we find space-like foliations for these regions, with the implication that the initial data can be preserved in reasonable situations. We propose qualitative models of non-primordial and/or evaporating black holes.We supplement the material with a brief note reporting on progress made since this talk was given, which shows that we can analytically extend the Schwarzschild and Reissner-Nordström metrics at and beyond the singularities, and the singularities can be made degenerate and handled with the mathematical apparatus we developed.

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Li, Chi. "On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold." Compositio Mathematica 148, no. 6 (October 12, 2012): 1985–2003. http://dx.doi.org/10.1112/s0010437x12000334.

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AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

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García Ariza, M. Á. "Degenerate Hessian structures on radiant manifolds." International Journal of Geometric Methods in Modern Physics 15, no. 06 (May 8, 2018): 1850087. http://dx.doi.org/10.1142/s0219887818500871.

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We present a rigorous mathematical treatment of Ruppeiner geometry, by considering degenerate Hessian metrics defined on radiant manifolds. A manifold [Formula: see text] is said to be radiant if it is endowed with a symmetric, flat connection and a global vector field [Formula: see text] whose covariant derivative is the identity mapping. A degenerate Hessian metric on [Formula: see text] is a degenerate metric tensor that can locally be written as the covariant Hessian of a function, called potential. A function on [Formula: see text] is said to be extensive if its Lie derivative with respect to [Formula: see text] is the function itself. We show that the Hessian metrics appearing in equilibrium thermodynamics are necessarily degenerate, owing to the fact that their potentials are extensive (up to an additive constant). Manifolds having degenerate Hessian metrics always contain embedded Hessian submanifolds, which generalize the manifolds defined by constant volume in which Ruppeiner geometry is usually studied. By means of examples, we illustrate that linking scalar curvature to microscopic interactions within a thermodynamic system is inaccurate under this approach. In contrast, thermodynamic critical points seem to arise as geometric singularities.

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Birbrair, Lev, and Alexandre Fernandes. "Inner metric geometry of complex algebraic surfaces with isolated singularities." Communications on Pure and Applied Mathematics 61, no. 11 (November 2008): 1483–94. http://dx.doi.org/10.1002/cpa.20244.

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LUO, FENG. "COMBINATORIAL YAMABE FLOW ON SURFACES." Communications in Contemporary Mathematics 06, no. 05 (October 2004): 765–80. http://dx.doi.org/10.1142/s0219199704001501.

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In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface. We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric. If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation. We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity.

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Botvinnik, Boris. "Manifolds with singularities accepting a metric of positive scalar curvature." Geometry & Topology 5, no. 2 (September 26, 2001): 683–718. http://dx.doi.org/10.2140/gt.2001.5.683.

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Larrañaga, Alexis, Natalia Herrera, and Juliana Garcia. "Geometric Description of the Thermodynamics of the Noncommutative Schwarzschild Black Hole." Advances in High Energy Physics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/641273.

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The thermodynamics of the noncommutative Schwarzschild black hole is reformulated within the context of the recently developed formalism of geometrothermodynamics (GTD). Using a thermodynamic metric which is invariant with respect to Legendre transformations, we determine the geometry of the space of equilibrium states and show that phase transitions, which correspond to divergencies of the heat capacity, are represented geometrically as singularities of the curvature scalar. This further indicates that the curvature of the thermodynamic metric is a measure of thermodynamic interaction.

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Ashtekar, Abhay, and Javier Olmedo. "Properties of a recent quantum extension of the Kruskal geometry." International Journal of Modern Physics D 29, no. 10 (July 2020): 2050076. http://dx.doi.org/10.1142/s0218271820500765.

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Recently, it was shown that, in an effective description motivated by loop quantum gravity, singularities of the Kruskal spacetime are naturally resolved [A. Ashtekar, J. Olmedo and P. Singh, Phys. Rev. Lett. 121 (2018) 241301; A. Ashtekar, J. Olmedo and P. Singh, Phys. Rev. D 98 (2018) 126003]. In this paper, we explore a few properties of this quantum corrected effective metric. In particular, we (i) calculate the Hawking temperature associated with the horizon of the effective geometry and show that the quantum correction to the temperature is completely negligible for macroscopic black holes, just as one would hope; (ii) discuss the subtleties associated with the asymptotic properties of the spacetime metric, and show that the metric is asymptotically flat in a precise sense; (iii) analyze the asymptotic fall-off of curvature; and, (iv) show that the ADM energy is well defined (and agrees with that determined by the horizon area), even though the curvature falls off less rapidly than in the standard asymptotically flat context.

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Han, Yiwen, and XiaoXiong Zeng. "Legendre Invariance and Geometrothermodynamics Description of the 3D Charged-Dilaton Black Hole." Advances in High Energy Physics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/865354.

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We first review Weinhold information geometry and Ruppeiner information geometry of 3D charged-dilaton black hole. Then, we use the Legendre invariant to introduce a 2-dimensional thermodynamic metric in the space of equilibrium states, which becomes singular at those points. According to the analysis of the heat capacities, these points are the places where phase transitions occur. This result is valid for the black hole, therefore, provides a geometrothermodynamics description of black hole phase transitions in terms of curvature singularities.

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Stoica, O. C. "On singular semi-Riemannian manifolds." International Journal of Geometric Methods in Modern Physics 11, no. 05 (May 2014): 1450041. http://dx.doi.org/10.1142/s0219887814500418.

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On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.

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SPINDEL, PH, and M. ZINQUE. "ASYMPTOTIC BEHAVIOR OF THE BIANCHI IX COSMOLOGICAL MODELS IN THE R2 THEORY OF GRAVITY." International Journal of Modern Physics D 02, no. 03 (September 1993): 279–94. http://dx.doi.org/10.1142/s0218271893000210.

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The asymptotic method of Belinskii, Khalatnikov and Lifshitz is applied to the study of the behavior near singularities of generic Bianchi IX cosmological models in the framework of the R2 theory of gravity. Three main kinds of asymptotic forms for the metric are obtained: a de Sitter geometry, a monotonic fall on a curvature singularity after a finite number of oscillations, an infinite sequence of regular oscillations. No chaos appears.

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Wang, Yong. "Affine connections on singular warped products." International Journal of Geometric Methods in Modern Physics 18, no. 05 (February 22, 2021): 2150076. http://dx.doi.org/10.1142/s0219887821500766.

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In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated with the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated with the almost product structure and their curvature of semi-regular almost product warped products are expressed in terms of those of the factor manifolds. Furthermore, we generalize the results in [O. Stoica, The geometry of warped product singularities, Int. J. Geom. Methods Mod. Phys. 14(2) (2017) 1750024, arXiv:1105.3404.] to singular multiply warped products.

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St-Pierre, Luc, and Paul Zsombor-Murray. "SINGULARITIES OF REDUNDANT 4R POSITIONING MANIPULATORS." Transactions of the Canadian Society for Mechanical Engineering 31, no. 4 (December 2007): 373–90. http://dx.doi.org/10.1139/tcsme-2007-0027.

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Singularities can be elusive but geometric considerations can reveal the singularities of a redundant 4R manipulator for positioning tasks. Points on a line, called transversal, that intersects all R-joint axes, cannot move in the direction of this line. Conditions governing the existence of transversals to four lines will be discussed. A way to find transversals is developed and tested with a numerical example. A possible metric, or singularity proximity measure for this type of singularity is investigated. This metric is based on the shortest distance between a line and a quadric and various methods will be proposed to solve this geometric problem.

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Molnár, Emil, Jenő Szirmai, and Andrei Vesnin. "Projective metric realizations of cone-manifolds with singularities along 2-bridge knots and links." Journal of Geometry 95, no. 1-2 (November 2009): 91–133. http://dx.doi.org/10.1007/s00022-009-0013-7.

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Bu, Wanghui. "Closeness to singularities of manipulators based on geometric average normalized volume spanned by weighted screws." Robotica 35, no. 7 (June 9, 2016): 1616–26. http://dx.doi.org/10.1017/s0263574716000345.

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SUMMARYIn order to prevent robot manipulators from reaching singularities, the “distance” from the current configuration to a singular configuration should be measured. This paper presents a novel metric based on geometric average normalized volume spanned by weighted screws to measure closeness to singularities for both serial and parallel manipulators. The weighted screws are proposed to reinterpret the physical meaning of twists and wrenches, so the problem of inconsistent dimensions in the dot product of screws has been eliminated. Compared with other existing methods, the proposed metric can obtain an identical result for similar manipulators with different sizes. Furthermore, the metric is independent of the selection of base screws, which is very suitable for the overconstrained or lower mobility parallel manipulator whose base screws are not uniquely definite. Besides, the geometrical meaning of the metric is related to the dimensionless volume of a high dimensional polyhedron, and hence the metric is insensitive to screw magnitude.

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ESCHER, JOACHIM, and MARCUS WUNSCH. "RESTRICTIONS ON THE GEOMETRY OF THE PERIODIC VORTICITY EQUATION." Communications in Contemporary Mathematics 14, no. 03 (June 2012): 1250016. http://dx.doi.org/10.1142/s0219199712500162.

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We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF∞(𝕊1) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 1377–1389], the axisymmetric Euler flow in ℝd (see [H. Okamoto and J. Zhu, Some similarity solutions of the Navier–Stokes equations and related topics, Taiwanese J. Math. 4 (2000) 65–103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 1251–1263].

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CASTRO, C., J. A. NIETO, L. RUIZ, and J. SILVAS. "ON TIME-DEPENDENT BLACK HOLES AND COSMOLOGICAL MODELS FROM A KALUZA–KLEIN MECHANISM." International Journal of Modern Physics A 24, no. 07 (March 20, 2009): 1383–415. http://dx.doi.org/10.1142/s0217751x09042931.

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Novel static, time-dependent and spatial–temporal solutions to Einstein field equations, displaying singularities, with and without horizons, and in several dimensions, are found based on a dimensional reduction procedure widely used in Kaluza–Klein-type theories. The Kerr–Newman black hole entropy as well as the Reissner–Nordstrom, Kerr and Schwarzschild black hole entropy are derived from the corresponding Euclideanized actions. A very special cosmological model based on the dynamical interior geometry of a black hole is found that has no singularities at t = 0 due to the smoothing of the mass distribution. We conclude with another cosmological model equipped also with a dynamical horizon and which is related to Vaidya's metric (associated with the Hawking radiation of black holes) by interchanging t ↔ r, which might render our universe a dynamical black hole.

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HUSAIN, VIQAR. "NAKED SINGULARITIES AND THE WILSON LOOP." Modern Physics Letters A 17, no. 15n17 (June 7, 2002): 955–65. http://dx.doi.org/10.1142/s021773230200693x.

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We give observations about dualities where one of the dual theories is geometric. These are illustrated with a duality between the simple harmonic oscillator and a topological field theory. We then discuss the Wilson loop in the context of the AdS/CFT duality. We show that the Wilson loop calculation for certain asymptotically AdS scalar field space–times with naked singularities gives results qualitatively similar to that for the AdS black hole. In particular, it is apparent that (dimensional) metric parameters in the singular space–times permit a "thermal screening" interpretation for the uark potential in the boundary theory, just like black hole mass. This suggests that the Wilson loop calculation merely captures metric parameter information rather than true horizon information.

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Dey, Subhadip. "Spherical metrics with conical singularities on 2-spheres." Geometriae Dedicata 196, no. 1 (December 9, 2017): 53–61. http://dx.doi.org/10.1007/s10711-017-0306-1.

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Riedi, Rudolf H., and Istvan Scheuring. "Conditional and Relative Multifractal Spectra." Fractals 05, no. 01 (March 1997): 153–68. http://dx.doi.org/10.1142/s0218348x97000152.

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In the study of the involved geometry of singular distributions, the use of fractal and multifractal analysis has shown results of outstanding significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to the ordinary metric or volume. Most prominent examples include self-similar measures and attractors of dynamical systems. In certain cases, the multifractal spectrum is known explicitly, providing a characterization in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel methods, the conditional and the relativemultifractal spectrum, which allow for a direct comparison of two distributions. These notions measure the extent to which the singularities of two distributions 'correlate'. Being based on multifractal concepts, however, they go beyond calculating correlations. As a particularly useful tool, we develop the multifractal formalism and establish some basic properties of the new notions. With the simple example of Binomial multifractals, we demonstrate how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and sparse parts of the distributions.

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Lakzian, Sajjad, and Michael Munn. "On Weak Super Ricci Flow through Neckpinch." Analysis and Geometry in Metric Spaces 9, no. 1 (January 1, 2021): 120–59. http://dx.doi.org/10.1515/agms-2020-0123.

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Abstract In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion subspaces. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x} × 𝕊 n . We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.

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Ghosal, Ashitava, and Bahram Ravani. "A Differential-Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators." Journal of Mechanical Design 123, no. 1 (July 1, 1998): 80–89. http://dx.doi.org/10.1115/1.1325008.

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In this paper, we present a differential-geometric approach to analyze the singularities of task space point trajectories of two and three-degree-of-freedom serial and parallel manipulators. At non-singular configurations, the first-order, local properties are characterized by metric coefficients, and, geometrically, by the shape and size of a velocity ellipse or an ellipsoid. At singular configurations, the determinant of the matrix of metric coefficients is zero and the velocity ellipsoid degenerates to an ellipse, a line or a point, and the area or the volume of the velocity ellipse or ellipsoid becomes zero. The degeneracies of the velocity ellipsoid or ellipse gives a simple geometric picture of the possible task space velocities at a singular configuration. To study the second-order properties at a singularity, we use the derivatives of the metric coefficients and the rate of change of area or volume. The derivatives are shown to be related to the possible task space accelerations at a singular configuration. In the case of parallel manipulators, singularities may lead to either loss or gain of one or more degrees-of-freedom. For loss of one or more degrees-of-freedom, the possible velocities and accelerations are again obtained from a modified metric and derivatives of the metric coefficients. In the case of a gain of one or more degrees-of-freedom, the possible task space velocities can be pictured as growth to lines, ellipses, and ellipsoids. The theoretical results are illustrated with the help of a general spatial 2R manipulator and a three-degree-of-freedom RPSSPR-SPR parallel manipulator.

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BOJOWALD, MARTIN. "QUANTUM GEOMETRY AND ITS IMPLICATIONS FOR BLACK HOLES." International Journal of Modern Physics D 15, no. 10 (October 2006): 1545–59. http://dx.doi.org/10.1142/s0218271806008942.

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General relativity successfully describes space–times at scales that we can observe and probe today, but it cannot be complete as a consequence of singularity theorems. For a long time, there have been indications that quantum gravity will provide a more complete, non-singular extension which, however, was difficult to verify in the absence of a quantum theory of gravity. By now there are several candidates which show essential hints as to what a quantum theory of gravity may look like. In particular, loop quantum gravity is a non-perturbative formulation which is background independent, two properties which are essentially close to a classical singularity with strong fields and a degenerate metric. In cosmological and black hole settings, one can indeed see explicitly how classical singularities are removed by quantum geometry: there is a well-defined evolution all the way down to, and across, the smallest scales. As for black holes, their horizon dynamics can be studied showing characteristic modifications to the classical behavior. Conceptual and physical issues can also be addressed in this context, providing lessons for quantum gravity in general. Here, we conclude with some comments on the uniqueness issue often linked to quantum gravity in some form or another.

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Singh, Parampreet. "Is classical flat Kasner spacetime flat in quantum gravity?" International Journal of Modern Physics D 25, no. 08 (July 2016): 1642001. http://dx.doi.org/10.1142/s0218271816420013.

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Quantum nature of classical flat Kasner spacetime is studied using effective spacetime description in loop quantum cosmology (LQC). We find that even though the spacetime curvature vanishes at the classical level, nontrivial quantum gravitational effects can arise. For the standard loop quantization of Bianchi-I spacetime, which uniquely yields universal bounds on expansion and shear scalars and results in a generic resolution of strong singularities, we find that a flat Kasner metric is not a physical solution of the effective spacetime description, except in a limit. The lack of a flat Kasner metric at the quantum level results from a novel feature of the loop quantum Bianchi-I spacetime: quantum geometry induces nonvanishing spacetime curvature components, making it not Ricci flat even when no matter is present. The noncurvature singularity of the classical flat Kasner spacetime is avoided, and the effective spacetime transits from a flat Kasner spacetime in asymptotic future, to a Minkowski spacetime in asymptotic past. Interestingly, for an alternate loop quantization which does not share some of the fine features of the standard quantization, flat Kasner spacetime with expected classical features exists. In this case, even with nontrivial quantum geometric effects, the spacetime curvature vanishes. These examples show that the character of even a flat classical vacuum spacetime can alter in a fundamental way in quantum gravity and is sensitive to the quantization procedure.

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Khatsymovsky, Vladimir. "On the Discrete Version of the Schwarzschild Problem." Universe 6, no. 10 (October 17, 2020): 185. http://dx.doi.org/10.3390/universe6100185.

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We consider a Schwarzschild type solution in the discrete Regge calculus formulation of general relativity quantized within the path integral approach. Earlier, we found a mechanism of a loose fixation of the background scale of Regge lengths. This elementary length scale is defined by the Planck scale and some free parameter of such a quantum extension of the theory. Besides, Regge action was reduced to an expansion over metric variations between the tetrahedra and, in the main approximation, is a finite-difference form of the Hilbert–Einstein action. Using for the Schwarzschild problem a priori general non-spherically symmetrical ansatz, we get finite-difference equations for its discrete version. This defines a solution which at large distances is close to the continuum Schwarzschild geometry, and the metric and effective curvature at the center are cut off at the elementary length scale. Slow rotation can also be taken into account (Lense–Thirring-like metric). Thus, we get a general approach to the classical background in the quantum framework in zero order: it is an optimal starting point for the perturbative expansion of the theory, finite-difference equations are classical, and the elementary length scale has quantum origin. Singularities, if any, are resolved.

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Д Брюно, А., and А. А Азимов. "LOCAL PARAMETRIZATION OF AN ALGEBRAIC VARIETY NEAR SOME ITS SINGULARITIES." 2022-yil, 3-son (133/1) ANIQ FANLAR SERIYASI 5, no. 135/1 (June 25, 2022): 1–15. http://dx.doi.org/10.59251/2181-1296.v5.1351.1113.

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In theoretical physics, when studying invariant Einstein metrics, there was a need to study an algebraic variety, which is described by an equation of the twelfth degree in three variables. 4 singular points and 2 curves of singular points of the variety were found early. Using algorithms of power geometry and computer algebra programs, local parametrizations of a manifold near 2 its singular points of the third order and near one curve of singular points are obtained.

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CHANG, ZHE, and CHENG-BO GUAN. "DYNAMICS OF MASSIVE SCALAR FIELDS IN dS SPACE AND THE dS/CFT CORRESPONDENCE." International Journal of Modern Physics A 17, no. 30 (December 10, 2002): 4591–600. http://dx.doi.org/10.1142/s0217751x02012302.

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Global geometric properties of dS space are presented explicitly in various coordinates. A Robertson–Walker-like metric is deduced, which is convenient to be used in the study of dynamics in dS space. Singularities of wave functions of massive scalar fields at boundary are demonstrated. A bulk-boundary propagator is constructed by making use of the solutions of equations of motion. The dS/CFT correspondence and the Strominger mass bound is shown.

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Bruno, Alexander D., and Alijon A. Azimov. "Parametric Expansions of an Algebraic Variety near Its Singularities." Axioms 12, no. 5 (May 13, 2023): 469. http://dx.doi.org/10.3390/axioms12050469.

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Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, (2) separating truncated equations, and (3) power transformations of coordinates. Here, we describe the method for the simplest case, a single algebraic equation, and apply it to an algebraic variety, as described by an algebraic equation of order 12 in three variables. The variety was considered in study of Einstein’s metrics and has several singular points and singular curves. Near some of them, we compute a local parametric expansion of the variety.

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GORODSKI, CLAUDIO. "CLOSED MINIMAL HYPERSURFACES IN COMPACT SYMMETRIC SPACES." International Journal of Mathematics 03, no. 05 (October 1992): 629–51. http://dx.doi.org/10.1142/s0129167x92000291.

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W.Y. Hsiang, W.T. Hsiang and P. Tomter conjectured that every simply-connected, compact symmetric space of dimension ≥4 must contain some minimal hypersurfaces of sphere type. With the aid of equivariant differential geometry, they showed that this is in fact the case for many symmetric spaces of rank one and two. Let M be one of the symmetric spaces: Sn(1)×Sn(1)(n≥4), SU(6)/Sp(3), E6/F4, ℍP2 (quaternionic proj. plane) or CaP2 (Cayley proj. plane). We prove the existence of infmitely many immersed, minimal hypersurfaces of sphere type in M which are invariant under a certain group G of isometries of M. Following Hsiang and the others, the equivariant method is also used here to reduce the problem to an investigation of geodesics in M/G equipped with a metric (with singularities) depending only on the orbital geometry of the transformation group (G, M). However, our constructions are based on area minimizing homogeneous cones, corresponding to a corner singularity of M/G with the local geometry of nodal type; this can be viewed as a variation of some of their constructions which depended on some unstable minimal cones of focal type. We further apply the equivariant method to construct a minimal embedding of S1×Sn−1×Sn−1 into Sn(1)×Sn(1)(n≥2) and a minimal, embedded hypersurface of sphere type in [Formula: see text], ℍPn×ℍPn (n≥2) and CaP2×CaP2.

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Bowers, Philip L. "The upper Perron method for labelled complexes with applications to circle packings." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 2 (September 1993): 321–45. http://dx.doi.org/10.1017/s0305004100071619.

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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.

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32

Bielawski, Roger. "On the hyperk�hler metrics associated to singularities of nilpotent varieties." Annals of Global Analysis and Geometry 14, no. 2 (May 1996): 177–91. http://dx.doi.org/10.1007/bf00127972.

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Bartolucci, Daniele. "On the Best Pinching Constant of Conformal Metrics on $\mathbb {S}^{2}$ with One and Two Conical Singularities." Journal of Geometric Analysis 23, no. 2 (September 23, 2011): 855–77. http://dx.doi.org/10.1007/s12220-011-9266-0.

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Takayama, Shigeharu. "Singularities of Narasimhan-Simha type metrics on direct images of relative pluricanonical bundles." Annales de l’institut Fourier 66, no. 2 (2016): 753–83. http://dx.doi.org/10.5802/aif.3025.

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Shabani, Hamid, and Amir Hadi Ziaie. "Static vacuum solutions on curved space–times with torsion." International Journal of Modern Physics A 33, no. 16 (June 7, 2018): 1850095. http://dx.doi.org/10.1142/s0217751x18500951.

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The Einstein–Cartan–Kibble–Sciama ( ECKS ) theory of gravity naturally extends Einstein’s general relativity ( GR ) to include intrinsic angular momentum (spin) of matter. The main feature of this theory consists of an algebraic relation between space–time torsion and spin of matter, which indeed deprives the torsion of its dynamical content. The Lagrangian of ECKS gravity is proportional to the Ricci curvature scalar constructed out of a general affine connection so that owing to the influence of matter energy–momentum and spin, curvature and torsion are produced and interact only through the space–time metric. In the absence of spin, the space–time torsion vanishes and the theory reduces to GR . It is however possible to have torsion propagation in vacuum by resorting to a model endowed with a nonminimal coupling between curvature and torsion. In the present work we try to investigate possible effects of the higher order terms that can be constructed from space–time curvature and torsion, as the two basic constituents of Riemann–Cartan geometry. We consider Lagrangians that include fourth-order scalar invariants from curvature and torsion and then investigate the resulting field equations. The solutions that we find show that there could exist, even in vacuum, nontrivial static space–times that admit both black holes and naked singularities.

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HERRERA-AGUILAR, ALFREDO, DAGOBERTO MALAGÓN-MOREJÓN, REFUGIO RIGEL MORA-LUNA, and ULISES NUCAMENDI. "ASPECTS OF THICK BRANE WORLDS: 4D GRAVITY LOCALIZATION, SMOOTHNESS, AND MASS GAP." Modern Physics Letters A 25, no. 24 (August 10, 2010): 2089–97. http://dx.doi.org/10.1142/s0217732310033244.

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We review some interrelated aspects of thick braneworlds constructed within the framework of 5D gravity coupled to a scalar field depending on the extra dimension. It turns out that when analyzing localization of 4D gravity in this smooth version of the Randall–Sundrum model, a kind of dichotomy emerges. In the first case the geometry is completely smooth and the spectrum of the transverse traceless modes of the metric fluctuations shows a single massless bound state, corresponding to the 4D graviton, and a tower of massive states described by a continuous spectrum of Kaluza–Klein excitations starting from zero mass, indicating the lack of a mass gap. In the second case, there are two bound states, a massless 4D graviton and a massive excitation, separated by a mass gap from a continuous spectrum of massive modes; nevertheless, the presence of a mass gap in the graviton spectrum of the theory is responsible for a naked singularity at the boundaries (or spatial infinity) of the Riemannian manifold. However, the imposition of unitary boundary conditions, which is equivalent to eliminating the continuous spectrum of gravitational massive modes, renders these singularities harmless from the physical point of view, providing the viability of the model.

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Kothawala, Dawood. "Relics of the quantum spacetime: from Synge’s world function as the fundamental probe of spacetime architecture to the emergent description of gravity." Journal of Physics: Conference Series 2533, no. 1 (June 1, 2023): 012012. http://dx.doi.org/10.1088/1742-6596/2533/1/012012.

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Abstract All our observations that characterise space and time are expressed in terms of non-local, bi-tensorial objects such as geodesic intervals between events and two-point (Green) functions. In this contribution, I highlight the importance of characterising spacetime geome-try in terms of such non-local objects, focusing particularly on two important bi-tensors that play a particular fundamental role – Synge’s World function and the van Vleck determinant. I will first discuss how these bi-tensors help capture information about spacetime geometry, and then describe their role in characterising quantum spacetime endowed with a lower bound, say ℓ 0, on spacetime intervals. Incorporating such a length scale in a Lorentz covariant manner necessitates a description of spacetime geometry in terms of above bi-tensors, and naturally replaces the conventional description based on the metric tensor gab (x) with a description in terms of a non-local bi-tensor qab (x, y). The non-analytic structure of qab (x, y) which renders a perturbative expansion in ℓ 0 meaningless, also generically leaves a non-trivial “relic” in the limit ℓ 0 → 0. I present some results where such a relic term is manifest; specifically, I will discuss how this: (i) suggests a description of gravitational dynamics different from the one based on Einstein-Hilbert lagrangian, (ii) implies dimensional reduction to 2 at small scales, (iii) connects with the notion of cosmological constant itself being a non-local vestige of the small scale structure of spacetime, (iv) helps address the issues of spacetime singularities. I will conclude by discussing the ramifications of these ideas for quantum gravity.

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Lin, Chang Shou, and Xiaohua Zhu. "Explicit construction of extremal Hermitian metrics with finite conical singularities on $S^2$." Communications in Analysis and Geometry 10, no. 1 (2002): 177–216. http://dx.doi.org/10.4310/cag.2002.v10.n1.a8.

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Wei, Zhiqiang, and Yingyi Wu. "Multi-valued holomorphic functions and non-CSC extremal Kähler metrics with singularities on compact Riemann surfaces." Differential Geometry and its Applications 60 (October 2018): 66–79. http://dx.doi.org/10.1016/j.difgeo.2018.05.008.

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Yan, Jian-Ming, Qiang Wu, Cheng Liu, Tao Zhu, and Anzhong Wang. "Constraints on self-dual black hole in loop quantum gravity with S0-2 star in the galactic center." Journal of Cosmology and Astroparticle Physics 2022, no. 09 (September 1, 2022): 008. http://dx.doi.org/10.1088/1475-7516/2022/09/008.

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Abstract One of remarkable features of loop quantum gravity (LQG) is that it can provide resolutions to both the black hole and big bang singularities. In the mini-superspace approach based on the polymerization procedure in LQG, a quantum corrected black hole metric is constructed. This metric is also known as self-dual spacetime since the form of the metric is invariant under the exchange r ⟶ a 0/r with a 0 being proportional to the minimum area in LQG and r is the standard radial coordinate at asymptotic infinity. It modifies the Schwarzschild spacetime by the polymeric function P, purely due to the geometric quantum effects from LQG. Here P is related to the polymeric parameter δ which is introduced to define the paths one integrates the connection along to define the holonomies in the quantum corrected Hamiltonian constraint in the polymerization procedure in LQG. In this paper, we consider its effects on the orbital signatures of S0-2 star orbiting Sgr A* in the central region of our Milky Way, and compare it with the publicly available astrometric and spectroscopic data, including the astrometric positions, the radial velocities, and the orbital precession for the S0-2 star. We perform Monte Carlo Markov Chain (MCMC) simulations to probe the possible LQG effects on the orbit of S0-2 star. No significant evidence of the self-dual spacetime arisIng from LQG is found. We thus place an upper bounds at 95% confidence level on the polymeric function P < 0.043 and P < 0.056, for Gaussian and uniform priors on orbital parameters, respectively.

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Dymnikova, Irina, and Evgeny Galaktionov. "Dynamics of Electromagnetic Fields and Structure of Regular Rotating Electrically Charged Black Holes and Solitons in Nonlinear Electrodynamics Minimally Coupled to Gravity." Universe 5, no. 10 (September 27, 2019): 205. http://dx.doi.org/10.3390/universe5100205.

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We study the dynamics of electromagnetic fields of regular rotating electrically charged black holes and solitons replacing naked singularities in nonlinear electrodynamics minimally coupled to gravity (NED-GR). They are related by electromagnetic and gravitational interactions and described by the axially symmetric NED-GR solutions asymptotically Kerr-Newman for a distant observer. Geometry is described by the metrics of the Kerr-Schild class specified by T t t = T r r ( p r = − ρ ) in the co-rotating frame. All regular axially symmetric solutions obtained from spherical solutions with the Newman-Janis algorithm belong to this class. The basic generic feature of all regular objects of this class, both electrically charged and electrically neutral, is the existence of two kinds of de Sitter vacuum interiors. We analyze the regular solutions to dynamical equations for electromagnetic fields and show which kind of a regular interior is favored by electromagnetic dynamics for NED-GR objects.

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Antoniadis, I., and B. Pioline. "Higgs Branch, Hyper-Kähler Quotient and Duality in SUSY N = 2 Yang–Mills Theories." International Journal of Modern Physics A 12, no. 27 (October 30, 1997): 4907–31. http://dx.doi.org/10.1142/s0217751x97002620.

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Low-energy limits of N = 2 supersymmetric field theories in the Higgs branch are described in terms of a nonlinear four-dimensional σ-model on a hyper-Kähler target space, classically obtained as a hyper-Kähler quotient of the original flat hypermultiplet space by the gauge group. We review in a pedagogical way this construction, and illustrate it in various examples, with special attention given to the singularities emerging in the low-energy theory. In particular, we thoroughly study the Higgs branch singularity of Seiberg–Witten SU(2) theory with Nf flavors, interpreted by Witten as a small instanton singularity in the moduli space of one instanton on ℝ4. By explicitly evaluating the metric, we show that this Higgs branch coincides with the Higgs branch of a U(1) N = 2 SUSY theory with the number of flavors predicted by the singularity structure of Seiberg–Witten's theory in the Coulomb phase. We find another example of Higgs phase duality, namely between the Higgs phases of U(Nc)Nf flavors and U(Nf-Nc)Nf flavors theories, by using a geometric interpretation due to Biquard et al. This duality may be relevant for understanding Seiberg's conjectured duality Nc ↔ Nf-Nc in N = 1 SUSY SU(Nc) gauge theories.

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Stoica, O. C. "Spacetime Causal Structure and Dimension from Horismotic Relation." Journal of Gravity 2016 (May 25, 2016): 1–6. http://dx.doi.org/10.1155/2016/6151726.

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A reflexive relation on a set can be a starting point in defining the causal structure of a spacetime in General Relativity and other relativistic theories of gravity. If we identify this relation as the relation between lightlike separated events (the horismos relation), we can construct in a natural way the entire causal structure: causal and chronological relations, causal curves, and a topology. By imposing a simple additional condition, the structure gains a definite number of dimensions. This construction works with both continuous and discrete spacetimes. The dimensionality is obtained also in the discrete case, so this approach can be suited to prove the fundamental conjecture of causal sets. Other simple conditions lead to a differentiable manifold with a conformal structure (the metric up to a scaling factor) as in Lorentzian manifolds. This structure provides a simple and general reconstruction of the spacetime in relativistic theories of gravity, which normally requires topological structure, differential structure, and geometric structure (which decomposes in the conformal structure, giving the causal relations and the volume element). Motivations for such a reconstruction come from relativistic theories of gravity, where the conformal structure is important, from the problem of singularities, and from Quantum Gravity, where various discretization methods are pursued, particularly in the causal sets approach.

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Dymnikova, Irina, and Anna Dobosz. "Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons." Symmetry 15, no. 2 (January 18, 2023): 273. http://dx.doi.org/10.3390/sym15020273.

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We briefly overview the basic properties and generic behavior of circular equatorial particle orbits and light rings around regular rotating compact objects with dark energy interiors, which are described by regular metrics of the Kerr–Schild class and include rotating black holes and self-gravitating spinning solitons replacing naked singularities. These objects have an internal de Sitter vacuum disk and can have two types of dark interiors, depending on the energy conditions. The first type reduces to the de Sitter disk, the second contains a closed de Sitter surface and an S surface with the de Sitter disk as the bridge and an anisotropic phantom fluid in the regions between the S surface and the disk. In regular geometry, the potentials decrease from V(r)→∞ to their minima, which ensures the existence of the innermost stable photon and particle orbits that are essential for processes of energy extraction occurring within the ergoregions, which for the second type of interiors contain the phantom energy. The innermost orbits provide a diagnostic tool for investigation of dark interiors of de Sitter–Kerr objects. They include light rings which confine these objects and ensure the most informative observational signature for rotating black holes presented by their shadows.

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Larsen, Jens Chr. "Metric Singularities." Rocky Mountain Journal of Mathematics 29, no. 3 (September 1999): 909–56. http://dx.doi.org/10.1216/rmjm/1181071616.

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Struve, Rolf. "Ordered metric geometry." Journal of Geometry 106, no. 3 (February 19, 2015): 551–70. http://dx.doi.org/10.1007/s00022-015-0265-3.

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Gaussier, Hervé, and Harish Seshadri. "Metric geometry of the Kobayashi metric." European Journal of Mathematics 3, no. 4 (October 11, 2017): 1030–44. http://dx.doi.org/10.1007/s40879-017-0177-x.

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Birbrair, Lev, and Jean-Paul Brasselet. "Metric homology for isolated conical singularities." Bulletin des Sciences Mathématiques 126, no. 2 (February 2002): 87–95. http://dx.doi.org/10.1016/s0007-4497(01)01085-5.

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Zeng, Yu. "Removable Singularities of the cscK Metric." International Mathematics Research Notices 2016, no. 19 (November 20, 2015): 5860–74. http://dx.doi.org/10.1093/imrn/rnv335.

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Stoica, Ovidiu Cristinel. "The Geometry of Black Hole Singularities." Advances in High Energy Physics 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/907518.

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Recent results show that important singularities in General Relativity can be naturally described in terms of finite and invariant canonical geometric objects. Consequently, one can write field equations which are equivalent to Einstein's at nonsingular points but, in addition remain well-defined and smooth at singularities. The black hole singularities appear to be less undesirable than it was thought, especially after we remove the part of the singularity due to the coordinate system. Black hole singularities are then compatible with global hyperbolicity and do not make the evolution equations break down, when these are expressed in terms of the appropriate variables. The charged black holes turn out to have smooth potential and electromagnetic fields in the new atlas. Classical charged particles can be modeled, in General Relativity, as charged black hole solutions. Since black hole singularities are accompanied by dimensional reduction, this should affect Feynman's path integrals. Therefore, it is expected that singularities induce dimensional reduction effects in Quantum Gravity. These dimensional reduction effects are very similar to those postulated in some approaches to make Quantum Gravity perturbatively renormalizable. This may provide a way to test indirectly the effects of singularities, otherwise inaccessible.

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Journal articles on the topic 'Metric geometry of singularities'

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