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

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.
Citation: Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete & Continuous Dynamical Systems - A, 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.

[2]

A. L. Besse, Einstein Manifolds,, Springer-Verlag, (2008).

[3]

V. Bour, Fourth order curvature flows and geometric applications,, preprint, (2010).

[4]

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

[5]

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

[6]

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

[7]

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

[8]

B. Chow and D. Knopf, The Ricci Flow: An Introduction,, Mathematical Surveys and Monographs, (2004). doi: 10.1090/surv/110.

[9]

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

[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, (2003), 163.

[11]

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

[12]

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

[13]

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

[14]

A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall Inc., (1964).

[15]

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

[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. doi: 10.1007/s11784-014-0162-7.

[17]

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

[18]

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

[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. doi: 10.1007/s11005-008-0245-8.

[20]

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

[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. doi: 10.1016/0550-3213(89)90422-7.

[22]

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

[23]

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

[24]

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

[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). doi: 10.1103/PhysRevD.76.045001.

[26]

P. Topping, Lectures on the Ricci Flow,, London Mathematical Society Lecture Note Series, (2006). doi: 10.1017/CBO9780511721465.

[27]

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

show all references

References:
[1]

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

[2]

A. L. Besse, Einstein Manifolds,, Springer-Verlag, (2008).

[3]

V. Bour, Fourth order curvature flows and geometric applications,, preprint, (2010).

[4]

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

[5]

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

[6]

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

[7]

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

[8]

B. Chow and D. Knopf, The Ricci Flow: An Introduction,, Mathematical Surveys and Monographs, (2004). doi: 10.1090/surv/110.

[9]

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

[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, (2003), 163.

[11]

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

[12]

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

[13]

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

[14]

A. Friedman, Partial Differential Equations of Parabolic Type,, Prentice-Hall Inc., (1964).

[15]

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

[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. doi: 10.1007/s11784-014-0162-7.

[17]

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

[18]

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

[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. doi: 10.1007/s11005-008-0245-8.

[20]

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

[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. doi: 10.1016/0550-3213(89)90422-7.

[22]

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

[23]

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

[24]

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

[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). doi: 10.1103/PhysRevD.76.045001.

[26]

P. Topping, Lectures on the Ricci Flow,, London Mathematical Society Lecture Note Series, (2006). doi: 10.1017/CBO9780511721465.

[27]

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

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