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December  2015, 35(12): 5787-5798. doi: 10.3934/dcds.2015.35.5787

Short-time existence of the second order renormalization group flow in dimension three

Laura Cremaschi 1, and  Carlo Mantegazza 1,

1. 

Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, 56126, Italy, Italy

Received  January 2014 Published  May 2015

Given a compact three--manifold together with a Riemannian metric, we prove the short--time existence of a solution to the renormalization group flow, truncated at the second order term, under a suitable hypothesis on the sectional curvature of the initial metric.
Keywords: evolution equation, Renormalization group, Ricci flow, Riemann tensor, quasilinear PDE..
Mathematics Subject Classification: Primary: 53C44; Secondary: 35K4.
Citation: Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787
References:
[1]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[2]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 2008.  Google Scholar

[3]

V. Bour, Fourth order curvature flows and geometric applications, preprint, 2010. Google Scholar

[4]

J. A. Buckland, Short-time existence of solutions to the cross curvature flow on 3-manifolds, Proc. Amer. Math. Soc., 134 (2006), 1803-1807 (electronic). doi: 10.1090/S0002-9939-05-08204-3.  Google Scholar

[5]

M. Carfora, Renormalization group and the Ricci flow, Milan J. Math., 78 (2010), 319-353. doi: 10.1007/s00032-010-0110-y.  Google Scholar

[6]

M. Carfora and A. Marzuoli, Model geometries in the space of Riemannian structures and Hamilton's flow, Classical Quantum Gravity, 5 (1988), 659-693. doi: 10.1088/0264-9381/5/5/005.  Google Scholar

[7]

B. Chow and R. S. Hamilton, The cross curvature flow of 3-manifolds with negative sectional curvature, Turkish J. Math., 28 (2004), 1-10.  Google Scholar

[8]

B. Chow and D. Knopf, The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, 110, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/surv/110.  Google Scholar

[9]

D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom., 18 (1983), 157-162.  Google Scholar

[10]

D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors (improved version), in Collected Papers on Ricci Flow (eds. H.-D. Cao, B. Chow, S.-C. Chu and S.-T. Yau), Series in Geometry and Topology, 37, Int. Press, 2003, 163-165. Google Scholar

[11]

J. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. doi: 10.2307/2373037.  Google Scholar

[12]

D. H. Friedan, Nonlinear models in $2+\varepsilon $ dimensions, Phys. Rev. Lett., 45 (1980), 1057-1060. doi: 10.1103/PhysRevLett.45.1057.  Google Scholar

[13]

D. H. Friedan, Nonlinear models in $2+\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419. doi: 10.1016/0003-4916(85)90384-7.  Google Scholar

[14]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, NJ, 1964.  Google Scholar

[15]

S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer-Verlag, 1990. doi: 10.1007/978-3-642-97242-3.  Google Scholar

[16]

K. Gimre, C. Guenther and J. Isenberg, A geometric introduction to the 2-loop renormalization group flow, J. Fixed Point Theory Appl., 14 (2013), 3-20. doi: 10.1007/s11784-014-0162-7.  Google Scholar

[17]

K. Gimre, C. Guenther and J. Isenberg, Second-order renormalization group flow of three-dimensional homogeneous geometries, Comm. Anal. Geom., 21 (2013), 435-467. doi: 10.4310/CAG.2013.v21.n2.a7.  Google Scholar

[18]

K. Gimre, C. Guenther and J. Isenberg, Short-time existence for the second order renormalization group flow in general dimensions, preprint, 2014. Google Scholar

[19]

C. Guenther and T. A. Oliynyk, Stability of the (two-loop) renormalization group flow for nonlinear sigma models, Lett. Math. Phys., 84 (2008), 149-157. doi: 10.1007/s11005-008-0245-8.  Google Scholar

[20]

R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255-306.  Google Scholar

[21]

I. Jack, D. R. T. Jones and N. Mohammedi, A four-loop calculation of the metric $\beta$-function for the bosonic $\sigma$-model and the string effective action, Nuclear Phys. B, 322 (1989), 431-470. doi: 10.1016/0550-3213(89)90422-7.  Google Scholar

[22]

J. Lott, Renormalization group flow for general $\sigma$-models, Comm. Math. Phys., 107 (1986), 165-176. doi: 10.1007/BF01206956.  Google Scholar

[23]

C. Mantegazza and L. Martinazzi, A note on quasilinear parabolic equations on manifolds, Ann. Sc. Norm. Sup. Pisa, 11 (2012), 857-874.  Google Scholar

[24]

T. A. Oliynyk, The second-order renormalization group flow for nonlinear sigma models in two dimensions, Classical Quantum Gravity, 26 (2009), 105020, 8pp. doi: 10.1088/0264-9381/26/10/105020.  Google Scholar

[25]

T. A. Oliynyk, V. Suneeta and E. Woolgar, Metric for gradient renormalization group flow of the worldsheet sigma model beyond first order, Phys. Rev. D, 76 (2007), 045001, 7pp. doi: 10.1103/PhysRevD.76.045001.  Google Scholar

[26]

P. Topping, Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511721465.  Google Scholar

[27]

A. A. Tseytlin, Sigma model renormalization group flow, "central charge'' action and Perelman's entropy, Phys. Rev. D, 75 (2007), 064024, 6pp. doi: 10.1103/PhysRevD.75.064024.  Google Scholar

show all references

References:
[1]

T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer-Verlag, 1998. doi: 10.1007/978-3-662-13006-3.  Google Scholar

[2]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 2008.  Google Scholar

[3]

V. Bour, Fourth order curvature flows and geometric applications, preprint, 2010. Google Scholar

[4]

J. A. Buckland, Short-time existence of solutions to the cross curvature flow on 3-manifolds, Proc. Amer. Math. Soc., 134 (2006), 1803-1807 (electronic). doi: 10.1090/S0002-9939-05-08204-3.  Google Scholar

[5]

M. Carfora, Renormalization group and the Ricci flow, Milan J. Math., 78 (2010), 319-353. doi: 10.1007/s00032-010-0110-y.  Google Scholar

[6]

M. Carfora and A. Marzuoli, Model geometries in the space of Riemannian structures and Hamilton's flow, Classical Quantum Gravity, 5 (1988), 659-693. doi: 10.1088/0264-9381/5/5/005.  Google Scholar

[7]

B. Chow and R. S. Hamilton, The cross curvature flow of 3-manifolds with negative sectional curvature, Turkish J. Math., 28 (2004), 1-10.  Google Scholar

[8]

B. Chow and D. Knopf, The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, 110, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/surv/110.  Google Scholar

[9]

D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom., 18 (1983), 157-162.  Google Scholar

[10]

D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors (improved version), in Collected Papers on Ricci Flow (eds. H.-D. Cao, B. Chow, S.-C. Chu and S.-T. Yau), Series in Geometry and Topology, 37, Int. Press, 2003, 163-165. Google Scholar

[11]

J. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. doi: 10.2307/2373037.  Google Scholar

[12]

D. H. Friedan, Nonlinear models in $2+\varepsilon $ dimensions, Phys. Rev. Lett., 45 (1980), 1057-1060. doi: 10.1103/PhysRevLett.45.1057.  Google Scholar

[13]

D. H. Friedan, Nonlinear models in $2+\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419. doi: 10.1016/0003-4916(85)90384-7.  Google Scholar

[14]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, NJ, 1964.  Google Scholar

[15]

S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer-Verlag, 1990. doi: 10.1007/978-3-642-97242-3.  Google Scholar

[16]

K. Gimre, C. Guenther and J. Isenberg, A geometric introduction to the 2-loop renormalization group flow, J. Fixed Point Theory Appl., 14 (2013), 3-20. doi: 10.1007/s11784-014-0162-7.  Google Scholar

[17]

K. Gimre, C. Guenther and J. Isenberg, Second-order renormalization group flow of three-dimensional homogeneous geometries, Comm. Anal. Geom., 21 (2013), 435-467. doi: 10.4310/CAG.2013.v21.n2.a7.  Google Scholar

[18]

K. Gimre, C. Guenther and J. Isenberg, Short-time existence for the second order renormalization group flow in general dimensions, preprint, 2014. Google Scholar

[19]

C. Guenther and T. A. Oliynyk, Stability of the (two-loop) renormalization group flow for nonlinear sigma models, Lett. Math. Phys., 84 (2008), 149-157. doi: 10.1007/s11005-008-0245-8.  Google Scholar

[20]

R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255-306.  Google Scholar

[21]

I. Jack, D. R. T. Jones and N. Mohammedi, A four-loop calculation of the metric $\beta$-function for the bosonic $\sigma$-model and the string effective action, Nuclear Phys. B, 322 (1989), 431-470. doi: 10.1016/0550-3213(89)90422-7.  Google Scholar

[22]

J. Lott, Renormalization group flow for general $\sigma$-models, Comm. Math. Phys., 107 (1986), 165-176. doi: 10.1007/BF01206956.  Google Scholar

[23]

C. Mantegazza and L. Martinazzi, A note on quasilinear parabolic equations on manifolds, Ann. Sc. Norm. Sup. Pisa, 11 (2012), 857-874.  Google Scholar

[24]

T. A. Oliynyk, The second-order renormalization group flow for nonlinear sigma models in two dimensions, Classical Quantum Gravity, 26 (2009), 105020, 8pp. doi: 10.1088/0264-9381/26/10/105020.  Google Scholar

[25]

T. A. Oliynyk, V. Suneeta and E. Woolgar, Metric for gradient renormalization group flow of the worldsheet sigma model beyond first order, Phys. Rev. D, 76 (2007), 045001, 7pp. doi: 10.1103/PhysRevD.76.045001.  Google Scholar

[26]

P. Topping, Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511721465.  Google Scholar

[27]

A. A. Tseytlin, Sigma model renormalization group flow, "central charge'' action and Perelman's entropy, Phys. Rev. D, 75 (2007), 064024, 6pp. doi: 10.1103/PhysRevD.75.064024.  Google Scholar

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