# On Perelman’s functional with curvature corrections

Annales UMCS, Mathematica (2012)

- Volume: 66, Issue: 2, page 47-55
- ISSN: 2083-7402

## Access Full Article

top## Abstract

top## How to cite

topRami Ahmad El-Nabulsi. "On Perelman’s functional with curvature corrections." Annales UMCS, Mathematica 66.2 (2012): 47-55. <http://eudml.org/doc/267916>.

@article{RamiAhmadEl2012,

abstract = {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.},

author = {Rami Ahmad El-Nabulsi},

journal = {Annales UMCS, Mathematica},

keywords = {Perelman’s functional; logarithmic curvature correction.; Perelman's functional; Ricci flow; Riemannian metric; logarithmic curvature correction},

language = {eng},

number = {2},

pages = {47-55},

title = {On Perelman’s functional with curvature corrections},

url = {http://eudml.org/doc/267916},

volume = {66},

year = {2012},

}

TY - JOUR

AU - Rami Ahmad El-Nabulsi

TI - On Perelman’s functional with curvature corrections

JO - Annales UMCS, Mathematica

PY - 2012

VL - 66

IS - 2

SP - 47

EP - 55

AB - 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.

LA - eng

KW - Perelman’s functional; logarithmic curvature correction.; Perelman's functional; Ricci flow; Riemannian metric; logarithmic curvature correction

UR - http://eudml.org/doc/267916

ER -

## References

top- [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] Zbl0703.53075
- [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. Zbl1200.53057
- [4] Calcagni, G., Geometry and field theory in multi-fractional spacetime, JHEP01 (2012) 065, 61 pp.[Crossref] Zbl1306.83018
- [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. Zbl0961.83056
- [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] Zbl1179.53070
- [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. Zbl1172.26305
- [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] Zbl1196.28014
- [14] El-Nabulsi, R. A., Glaeske-Kilbas-Saigo fractional integration and fractional Dixmier trace, Acta Math. Viet. 37, no. 2 (2012), 149-160. Zbl1254.26013
- [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. Zbl1267.49037
- [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] Zbl1221.81175
- [19] Hamilton, R., Three manifolds with positive Ricci curvature, Jour. Diff. Geom. 17 (1982), 255-306. Zbl0504.53034
- [20] Hamilton, R., Four-manifolds with positive curvature operator, J. Diff. Geom. 24 (2) (1986), 153-179. Zbl0628.53042
- [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] Zbl1114.83007
- [23] Herrmann, R., Gauge invariance in fractional field theories, Phys. Lett. A 372 (2008), 5515-5522. Zbl1223.70062
- [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. Zbl1130.53003
- [27] Perelman, G., Ricci flow with surgery on three-manifolds, math.DG/0303109v1. Zbl1130.53002
- [28] Perelman, G., The entropy formula for the Ricci flow and its geometric applications, http://arXiv.org/abs/math.DG/0211159. Zbl1130.53001
- [29] Thurston, William P., Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, American Mathematical Society. Bulletin. New Series 6 (3) (1982), 357-381.[Crossref] Zbl0496.57005
- [30] Vacaru, S., Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows, J. Math. Phys. 50 (2009), 073503, 24 pp.[WoS][Crossref] Zbl1256.58008
- [31] Vacaru, S., Fractional dynamics from Einstein gravity, general solutions, and black holes, Int. J. Theor. Phys. 51 (2012), 1338-1359.[Crossref] Zbl1252.83087
- [32] Vacaru, S., Fractional nonholonomic Ricci flows, Chaos, Solitons and Fractals 45 (2012), 1266-1276.[WoS] Zbl1258.35202
- [33] Vacaru, S., Nonholonomic Clifford and Finsler structures, non-commutative Ricci flows, and mathematical relativity, arXiv: 1205.5387.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.