The conjugate linearized Ricci flow on closed 3-manifolds

Carfora, Mauro

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 681-724.

We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three-manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. We also provide an integral representation of the Ricci flow metric itself and of its Ricci tensor in terms of the heat kernel of the conjugate linearized Ricci flow. These results, which readily extend to closed $n$-dimensional manifolds, yield various conservation laws, monotonicity and asymptotic formulas for the Ricci flow and its linearization.

[1] B. Amman and C. Bär, The Einstein-Hilbert action as a spectral action, In: “Noncommutative Geometry and the Standard Model of Elementary Particle Physics”, F. Scheck and H. Upmeier (eds.), Springer Lecture Notes in Physics, Vol. 596, 2002. | MR | Zbl

[2] G. Anderson and B. Chow, A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds, Calc. Var. Partial Differential Equations 23 (2005), 1–12. | MR | Zbl

[3] T. Aubin, “Some Nonlinear Problems in Riemannian Geometry”, Springer Verlag, 1998. | MR | Zbl

[4] A. Avez, Le laplacien de Lichnerowicz sur les tenseurs, C. R. Acad. Sci. Paris. Sér. A-B Math. 284 (1977), 1219–1220. | MR | Zbl

[5] I. Bakas, Geometric flows and (some of) their physical applications, In: “AvH conference Advances in Physics and Astrophysics of the 21st Century”, 6-11 September 2005, Varna, Bulgaria, hep-th/0511057. | Zbl

[6] I. Bakas, Renormalization group flows and continual Lie algebras, J. High Energy Phys. 0308, 013 (2003). | MR

[7] N. Berline, E. Getzler and M. Vergne, “Heat kernels and Dirac Operators”, Grundlehren Math. Wiss., Vol. 298, Springer-Verlag, New York, 1992. | MR | Zbl

[8] T. Buchert and M. Carfora, Regional averaging and scaling in relativistic cosmology, Classical Quantum Gravity 19 (2002), 6109–6145. | MR | Zbl

[9] T. Buchert and M. Carfora, Cosmological parameters are dressed, Phys. Rev. Lett. 90 (2003), 31101–1–4.

[10] C. Buzzanca, The Laplacian of Lichnerowicz on tensors, Boll. Un. Mat. Ital. B (6), 3 (1984), 531–541. | MR | Zbl

[11] H-D. Cao, R. S. Hamilton and T. Ilmanen, Gaussian densities and stability for some Ricci solitons, arXiv:math.DG/0404165.

[12] M. Carfora, Fokker-Planck dynamics and entropies for the normalized Ricci flow, Adv. Theor. Math. Phys. 11 (2007), 635–681. See also arXiv:math.DG/0507309 v3. | MR | Zbl

[13] M. Carfora and A. Marzuoli, Model geometries in the space of Riemannian structures and Hamilton’s flow, Classical Quantum Gravity 5 (1988), 659–693. | MR | Zbl

[14] M. Carfora and A. Marzuoli, Smoothing out spatially closed cosmologies, Phys. Rev. Lett. 53 (1984), 2445. | MR

[15] M. Carfora and K. Piotrkowska, Renormalization group approach to relativistic cosmology , Phys. Rev. D 52 (1995), 4393. | MR

[16] B. Chow and S.-C. Chu, A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow, Math. Res. Lett. 2 (1995), 701–718. | MR | Zbl

[17] B. Chow and S.-C. Chu, A geometric approach to the linear trace Harnack inequality for the Ricci flow, Math. Res. Lett. 3 (1996), 549–568. | MR | Zbl

[18] B. Chow and D. Knopf, “The Ricci Flow: an Introduction”, Mathematical Surveys and Monographs, Vol. 110, American Mathematical Society, Providence, R.I., 2004. | MR | Zbl

[19] B. Chow, S-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu and L. Ni, “The Ricci Flow: Techniques and Applications. Part I: Geometric Aspects”, Mathematical Surveys and Monographs, Vol. 135, American Mathematical Society, Providence, R.I., 2007. | Zbl

[20] B. Chow and R. S. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math. 129 (1997), 213–238. | MR | Zbl

[21] B. Chow, P. Lu and L. Ni, “Hamilton’s Ricci Flow”, Graduate Studies in Mathematics, Vol. 77, American Mathematical Society, Providence, R.I., Science Press, New York, 2006. | Zbl

[22] D. Deturck, Deforming metrics in the direction of their Ricci tensor, J. Differential Geom. 18 (1983), 157–162. | MR | Zbl

[23] Y. Ding, A remark on degenerate singularities in three dimensional Ricci flow, Pacific J. Math. 240 (2009), 289–308. | MR | Zbl

[24] D. Ebin, The manifolds of Riemannian metrics, Global Analysis, Proc. Sympos. Pure Math. 15 (1968), 11–40. | MR

[25] K. Ecker, D. Knopf, L. Ni and P. Topping, Local monotonicity and mean value formulas for evolving Riemannian manifolds, J. reine angew. Math. (Crelle) 618 (2008), 89–130. | MR | Zbl

[26] D. H. Friedan, Nonlinear models in $2+\epsilon$ dimensions, Ann. Physics 163 (1985), 318–419. | MR | Zbl

[27] D. Garfinkle and J. Isenberg, The modelling of degenerate neck pinch singularities in Ricci flow by Bryant solitons, J. Math. Phys. 49, 073505 (2008), doi:10.1063/1.2948953. | MR | Zbl

[28] N. Garofalo and E. Lanconelli, Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients, Math. Ann. 283 (1989), 211–239. | EuDML | MR | Zbl

[29] P. B. Gilkey, “Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem” (2nd. ed.), CRC Press, Boca Raton, Florida, 1994. | MR

[30] P. B. Gilkey, J. Leahy and JH. Park, “Spinors, Spectral Geometry, and Riemannian Submersions”, Published on the EMIS server at: http://cdns.emis.de/monographs/GLP/index.html. Originally published as Lecture Notes Series, Vol. 40, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, 1998.

[31] P. B. Gilkey, The heat content asymptotics for variable geometries, J. Phys. A: Math. Gen. 32 (1999), 2825–2834. | MR | Zbl

[32] P. B. Gilkey, K. Kirsten and JH. Park, Heat trace asymptotics of a time-dependent process, J. Phys. A: Math. Gen. 34 (2001), 1153–1168. | MR | Zbl

[33] C. Guenther, J. Isenberg and D. Knopf, Stability of the Ricci flow at Ricci-flat metrics, Commun. Anal. Geom. 10 (2002), 741–777. | MR | Zbl

[34] C. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal. 12 (2002), 425–436. | MR | Zbl

[35] C. Guenther, J. Isenberg and D. Knopf, Linear stability of homogeneous Ricci solitons, Int. Math. Res. Not. (2006), Art. ID 96253, 30 pp. | MR

[36] H-L. Gu and X-P. Zhu, The existence of type II singularities for the Ricci flow on $S^{n+1}$, Comm. Anal. Geom. 16 (2008), 467–494. | MR | Zbl

[37] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306. | MR | Zbl

[38] R. S. Hamilton, Four manifolds with positive curvature operator, J. Differential Geom. 24 (1986), 153–179. | MR | Zbl

[39] R. S. Hamilton, The formation of singularities in the Ricci flow, In: “Surveys in Differential Geometry”, Vol. 2, International Press, 1995, 7–136. | MR | Zbl

[40] G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 17 (1985), 47–62. | MR | Zbl

[41] H. Koch and T. Lamm, Geometric flows with rough initial data, arXiv:0902.1488. | MR | Zbl

[42] D. Knopf, Estimating the trace-free Ricci tensor in Ricci flow, Proc. Amer. Math. Soc. 137 (2009), 3099–3103. | MR | Zbl

[43] T. Ivey, Ricci solitons on compact three-manifolds, Differential Geom. Appl. 3 (1991), 301–307. | MR | Zbl

[44] A. Lichnerowicz, Propagateurs et commutateurs en relativité générale, Inst. Hautes Études Sci. Publ. Math. 10 (1961), 56 pp. | EuDML | Numdam | MR | Zbl

[45] J. Lott, Renormalization group flow for general sigma models, Comm. Math. Phys. 107 (1986), 165–176. | MR | Zbl

[46] L. Ni, A note on Perelman’s LYH inequality, Comm. Anal. Geom. 14 (2006), 883–905. | MR | Zbl

[47] L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow, J. Differential Geom. 75 (2007), 303–358. | MR | Zbl

[48] T. Oliynyk, V. Suneeta and E. Woolgar, A gradient flow for worldsheet nonlinear sigma models, Nucl. Phys. B 739 (2006), 441-458. | MR | Zbl

[49] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159

[50] G. Perelman, Ricci flow with surgery on Three-Manifolds, math.DG/0303109. | Zbl

[51] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, math.DG/0307245. | Zbl

[52] O. C. Schnurer, F. Schulze and M. Simon, Stability of Euclidean space under Ricci flow, Comm. Anal. Geom. 16 (2008), 127–158. | MR | Zbl

[53] N. Sesum, Curvature tensor under the Ricci flow, Amer. J. Math. 127 (2005), 1315–1324. | MR | Zbl

[54] N. Sesum, Linear and dynamical stability of Ricci flat metrics, Duke Math. J. 133 (2006), 1–26. | MR | Zbl

[55] M. Simon, Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 10 (2002), 1033–1074. | MR | Zbl

[56] W. P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381. | MR

[57] W. P. Thurston, “Three-dimensional Geometry and Topology”, Vol. 1, S. Levy (ed.), Princeton Math. Series, Vol. 35, Princeton Univ. Press, Princeton NJ, 1997. | MR | Zbl

[58] P. Topping, “Lectures on the Ricci Flow”, London Math. Soc. Lecture Notes Series, Vol. 325, Cambridge Univ. Press, 2006. | MR | Zbl

[59] R. Ye, Ricci flow, Einstein metrics and space forms, Trans. Amer. Math. Soc. 338 (1993), 871–896. | MR | Zbl