In recent ten years, there has been much concentration and increased research activities on Hamilton’s Ricci flow evolving on a Riemannian metric and Perelman’s functional. In this paper, we extend Perelman’s functional approach to include logarithmic curvature corrections induced by quantum effects. Many interesting consequences are revealed.
@article{bwmeta1.element.doi-10_2478_v10062-012-0012-2,
author = {Rami Ahmad El-Nabulsi},
title = {On Perelman's functional with curvature corrections},
journal = {Annales UMCS, Mathematica},
volume = {66},
year = {2012},
pages = {47-55},
zbl = {1297.49083},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0012-2}
}
Rami Ahmad El-Nabulsi. On Perelman’s functional with curvature corrections. Annales UMCS, Mathematica, Tome 66 (2012) pp. 47-55. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0012-2/
[1] Caianiello, E. R., Feoli, A., Gasperini, M., Scarpetta G., Quantum corrections to the spacetime metric from geometric phase space quantization, Int. J. Theor. Phys. 29 (2) (1990), 131-139.[Crossref] | Zbl 0703.53075
[2] Cao, H.-D, Existence of gradient Kahler-Ricci solitons, Elliptic and Parabolic Methods in Geometry (Minneapolis, 1994), 1-16, A. K. Peters, Wellesley MA, 1996.
[3] Cao, H.-D., Zhu, X.-P., A complete proof of the Poincare and Geometrization conjectures - application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math., 10, no. 2 (2006), 165-492. | Zbl 1200.53057
[4] Calcagni, G., Geometry and field theory in multi-fractional spacetime, JHEP01 (2012) 065, 61 pp.[Crossref] | Zbl 1306.83018
[5] D’Hoker, E., String Theory, in Quantum Fields and Strings: A Course for Mathematicians, vol. 2, American Mathematical Society, Providence, 1999.
[6] Davis, S., Luckock, H., The effect of higher-order curvature terms on string quantum cosmology, Phys. Lett. B 485 (2000), 408-421. | Zbl 0961.83056
[7] Dowker, H. F., Topology change in quantum gravity, The Future of Theoretical Physics and Cosmology, eds. G. W. Gibbons, S. J. Rankin, E. P. S. Shellard, Cambridge Univ. Press, (2003), p. 879.
[8] Dzhunushaliev, V., Quantum wormhole as a Ricci flow, Int. J. Geom. Meth. Mod. Phys. 6 (2009), 1033-1046.[Crossref] | Zbl 1179.53070
[9] Dzhunushaliev, V., Serikbayev, N., Myrzakulov, R., Topology change in quantum gravity and Ricci flows, arXiv:0912.5326v2.
[10] El-Nabulsi, R. A., Fractional field theories from multidimensional fractional variational problems, Int. J. Mod. Geom. Meth. Mod. Phys. 5, no. 6 (2008), 863-892. | Zbl 1172.26305
[11] El-Nabulsi, R. A., Complexified quantum field theory and mass without mass from multidimensional fractional actionlike variational approach with time-dependent fractional exponent, Chaos, Solitons Fractals 42, no. 4 (2009), 2384-2398.[WoS]
[12] El-Nabulsi, R. A., Fractional quantum field theory on multifractal sets, American. J. Eng. Appl. Sci. 4 (1) (2010), 133-141.
[13] El-Nabulsi, R. A., Modifications at large distance from fractional and fractal arguments, Fractals 18, no. 2 (2010), 185-190.[WoS] | Zbl 1196.28014
[14] El-Nabulsi, R. A., Glaeske-Kilbas-Saigo fractional integration and fractional Dixmier trace, Acta Math. Viet. 37, no. 2 (2012), 149-160. | Zbl 1254.26013
[15] El-Nabulsi, R. A, Wu, G.-C., Fractional complexified field theory from Saxena-Kumbhat fractional integral, fractional derivative of order and dynamical fractional integral exponent, Afric. Disp. J. Math. 13, no. 2 (2012), 45-61. | Zbl 1267.49037
[16] Gibbons, G. W., Topology change in classical and quantum gravity, Published in Mt. Sorak Symposium 1991: 159-185 (QCD161:S939:1991) Developments in Field Theory, ed. Jihn E Kim (Min Eum Sa, Seoul) (1992); arXiv:1110.0611v1.
[17] Gibbons, G. W., Hawking, S. W., Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977), 2752-2756.
[18] Goldfain, E., Fractional dynamics and the standard model for particle physics, Comm. Nonlinear Sci. Numer. Simul. 13 (2008), 1397-1404.[Crossref] | Zbl 1221.81175
[19] Hamilton, R., Three manifolds with positive Ricci curvature, Jour. Diff. Geom. 17 (1982), 255-306. | Zbl 0504.53034
[20] Hamilton, R., Four-manifolds with positive curvature operator, J. Diff. Geom. 24 (2) (1986), 153-179. | Zbl 0628.53042
[21] Hamilton, R., Formation of singularities in the Ricci flow, Surveys in Diff. Geom. 2 (1997), 7-136.
[22] Headrick, M., Wiseman, T., Ricci flow and black holes, Class. Quant. Grav. 23 (2006), 6683-6708.[Crossref] | Zbl 1114.83007
[23] Herrmann, R., Gauge invariance in fractional field theories, Phys. Lett. A 372 (2008), 5515-5522. | Zbl 1223.70062
[24] Jolany, H., Perelman’s functional and reduced volume, arXiv:1004.1785.
[25] Mohaupt, T., Strings, higher curvature corrections, and black holes, talk given at the 2nd Workshop on Mathematical and Physical Aspects of Quantum Gravity, Blaubeuren, 28 July-1 August, (2005); hep-th/0512048.
[26] Perelman, G., Finite extinction time for the solutions to the Ricci flow on certain three manifolds, math.DG/0307245v1. | Zbl 1130.53003
[27] Perelman, G., Ricci flow with surgery on three-manifolds, math.DG/0303109v1. | Zbl 1130.53002
[28] Perelman, G., The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math.DG/0211159. | Zbl 1130.53001
[29] Thurston, William P., Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, American Mathematical Society. Bulletin. New Series 6 (3) (1982), 357-381.[Crossref] | Zbl 0496.57005
[30] Vacaru, S., Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows, J. Math. Phys. 50 (2009), 073503, 24 pp.[WoS][Crossref] | Zbl 1256.58008
[31] Vacaru, S., Fractional dynamics from Einstein gravity, general solutions, and black holes, Int. J. Theor. Phys. 51 (2012), 1338-1359.[Crossref] | Zbl 1252.83087
[32] Vacaru, S., Fractional nonholonomic Ricci flows, Chaos, Solitons and Fractals 45 (2012), 1266-1276.[WoS] | Zbl 1258.35202
[33] Vacaru, S., Nonholonomic Clifford and Finsler structures, non-commutative Ricci flows, and mathematical relativity, arXiv: 1205.5387.