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

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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

Abstract
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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 and 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.
[2]	A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 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), 1803-1807 (electronic). doi: 10.1090/S0002-9939-05-08204-3.
[5]	M. Carfora, Renormalization group and the Ricci flow, Milan J. Math., 78 (2010), 319-353. 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), 659-693. 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), 1-10.
[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.
[9]	D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom., 18 (1983), 157-162.
[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.
[11]	J. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. doi: 10.2307/2373037.
[12]	D. H. Friedan, Nonlinear models in $2+\varepsilon $ dimensions, Phys. Rev. Lett., 45 (1980), 1057-1060. doi: 10.1103/PhysRevLett.45.1057.
[13]	D. H. Friedan, Nonlinear models in $2+\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419. doi: 10.1016/0003-4916(85)90384-7.
[14]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, NJ, 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), 3-20. 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), 435-467. 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), 149-157. doi: 10.1007/s11005-008-0245-8.
[20]	R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255-306.
[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.
[22]	J. Lott, Renormalization group flow for general $\sigma$-models, Comm. Math. Phys., 107 (1986), 165-176. 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), 857-874.
[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.
[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.
[26]	P. Topping, Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 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), 064024, 6pp. doi: 10.1103/PhysRevD.75.064024.

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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, Berlin, 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), 1803-1807 (electronic). doi: 10.1090/S0002-9939-05-08204-3.
[5]	M. Carfora, Renormalization group and the Ricci flow, Milan J. Math., 78 (2010), 319-353. 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), 659-693. 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), 1-10.
[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.
[9]	D. M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom., 18 (1983), 157-162.
[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.
[11]	J. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. doi: 10.2307/2373037.
[12]	D. H. Friedan, Nonlinear models in $2+\varepsilon $ dimensions, Phys. Rev. Lett., 45 (1980), 1057-1060. doi: 10.1103/PhysRevLett.45.1057.
[13]	D. H. Friedan, Nonlinear models in $2+\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419. doi: 10.1016/0003-4916(85)90384-7.
[14]	A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc., Englewood Cliffs, NJ, 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), 3-20. 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), 435-467. 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), 149-157. doi: 10.1007/s11005-008-0245-8.
[20]	R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255-306.
[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.
[22]	J. Lott, Renormalization group flow for general $\sigma$-models, Comm. Math. Phys., 107 (1986), 165-176. 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), 857-874.
[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.
[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.
[26]	P. Topping, Lectures on the Ricci Flow, London Mathematical Society Lecture Note Series, 325, Cambridge University Press, Cambridge, 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), 064024, 6pp. doi: 10.1103/PhysRevD.75.064024.

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