Ricci flow coupled with harmonic map flow

Müller, Reto

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 1, pp. 101-142.

Nous étudions un système d’équations consistant en un couplage entre le flot de Ricci et le flot harmonique d’une fonction $\phi$ allant de $M$ dans une variété cible $N$,

$$\frac{\partial }{\partial t}g=-2\mathrm{Rc}+2\alpha \nabla \phi \otimes \nabla \phi ,\frac{\partial }{\partial t}\phi ={\tau }_g\phi ,$$

où $\alpha$ est une constante de couplage strictement positive (et pouvant dépendre du temps). De manière surprenante, ce système couplé peut être moins singulier que le flot de Ricci ou le flot harmonique si ceux-ci sont considérés de manière isolée. En particulier, on peut toujours montrer que la fonction $\phi$ ne se concentre pas le long de ce système à condition de prendre $\alpha$ assez grand. De plus, il est suffisant de borner la courbure de $(M,g(t\left)\right)$ le long du flot pour obtenir le contrôle de $\phi$ et de toutes ses dérivées si $\alpha \ge \underline{\alpha }>0$. À part ces phénomènes nouveaux, ce flot possède certaines propriétés analogues à celles du flot de Ricci. En particulier, il est possible de montrer la monotonie d’une énergie, d'une entropie et d'une fonctionnelle volume réduit. On utilise la monotonie de ces quantités pour montrer l'absence de solutions en « accordéon » et l'absence d'effondrement en temps fini le long du flot.

We investigate a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to some closed target manifold $N$,

$$\frac{\partial }{\partial t}g=-2\mathrm{Rc}+2\alpha \nabla \phi \otimes \nabla \phi ,\frac{\partial }{\partial t}\phi ={\tau }_g\phi ,$$

where $\alpha$ is a (possibly time-dependent) positive coupling constant. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of $\phi$ a-priori by choosing $\alpha$ large enough. Moreover, it suffices to bound the curvature of $(M,g(t\left)\right)$ to also obtain control of $\phi$ and all its derivatives if $\alpha \ge \underline{\alpha }>0$. Besides these new phenomena, the flow shares many good properties with the Ricci flow. In particular, we can derive the monotonicity of an energy, an entropy and a reduced volume functional. We then apply these monotonicity results to rule out non-trivial breathers and geometric collapsing at finite times.

Classification : 53C21, 53C43, 53C44, 58E20
Keywords: Ricci flow, harmonic map flow
Mot clés : flot de Ricci, flot harmonique

[1] S. Bando, Real analyticity of solutions of Hamilton's equation, Math. Z. 195 (1987), 93-97. | MR | Zbl

[2] S. Bernstein, Sur la généralisation du problème de Dirichlet II, Math. Ann. 69 (1910), 82-136. | JFM | MR

[3] H.-D. Cao & X.-P. Zhu, A complete proof of the Poincaré and geometrization conjectures-application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math. 10 (2006), 165-492. | MR | Zbl

[4] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo & L. Ni, The Ricci flow: Techniques and applications: Part I: Geometric aspects, Mathematical Surveys and Monographs 135, Amer. Math. Soc., 2007. | MR | Zbl

[5] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo & L. Ni, The Ricci flow: Techniques and applications: Part II: Analytic aspects, Mathematical Surveys and Monographs 144, Amer. Math. Soc., 2008. | MR | Zbl

[6] B. Chow & D. Knopf, The Ricci flow: An introduction, Mathematical Surveys and Monographs 110, Amer. Math. Soc., 2004. | MR | Zbl

[7] B. Chow, P. Lu & L. Ni, Hamilton's Ricci flow, Graduate Studies in Math. 77, Amer. Math. Soc., 2006. | MR | Zbl

[8] T. H. Colding & W. P. Minicozzi Ii, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman, J. Amer. Math. Soc. 18 (2005), 561-569. | MR | Zbl

[9] T. H. Colding & W. P. Minicozzi Ii, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008), 2537-2586. | MR | Zbl

[10] D. M. Deturck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), 157-162. | MR | Zbl

[11] J. Eells & L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. | MR | Zbl

[12] J. Eells & L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385-524. | MR | Zbl

[13] J. Eells & J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. | MR | Zbl

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

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

[16] R. S. Hamilton, The formation of singularities in the Ricci flow, in Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995. | MR | Zbl

[17] P. Hartman & A. Wintner, On the local behavior of solutions of non-parabolic partial differential equations, Amer. J. Math. 75 (1953), 449-476. | MR | Zbl

[18] J. Jost, Riemannian geometry and geometric analysis, third éd., Universitext, Springer, 2002. | MR | Zbl

[19] B. Kleiner & J. Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008), 2587-2855. | MR | Zbl

[20] T. Lamm, Biharmonic maps, Thèse, Albert-Ludwig-Universität Freiburg im Breisgau, 2005. | Zbl

[21] A. Lichnerowicz, Propagateurs et commutateurs en relativité générale, Publ. Math. I.H.É.S. 10 (1961). | Numdam | Zbl

[22] B. List, Evolution of an extended Ricci flow system, Thèse, Albert-Einstein-Institut, Berlin, 2005. | Zbl

[23] J. Lott, On the long-time behavior of type-III Ricci flow solutions, Math. Ann. 339 (2007), 627-666. | MR | Zbl

[24] C. Mantegazza & L. Martinazzi, A note on quasilinear parabolic equations on manifolds, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci. | Numdam | MR | Zbl

[25] J. Morgan & G. Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3, Amer. Math. Soc., 2007. | MR | Zbl

[26] J. Morgan & G. Tian, Completion of the proof of the geometrization conjecture, preprint arXiv:0809.4040.

[27] R. Müller, Differential Harnack inequalities and the Ricci flow, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2006. | MR | Zbl

[28] R. Müller, The Ricci flow coupled with harmonic map heat flow, Thèse, ETH Zürich, 2009. | Zbl

[29] R. Müller, Monotone volume formulas for geometric flows, J. reine angew. Math. 643 (2010), 39-57. | MR | Zbl

[30] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20-63. | MR | Zbl

[31] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint arXiv:math/0211159. | Zbl

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

[33] G. Perelman, Ricci flow with surgery on three-manifolds, preprint arXiv:math/0303109. | Zbl

[34] H. Poincaré, Cinquième complément à l'Analysis Situs, Rend. Circ. Mat. Palermo 18 (1904), 45-110. | JFM

[35] M. Reed & B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press Inc., 1978. | MR | Zbl

[36] O. S. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981), 110-120. | MR | Zbl

[37] O. C. Schnürer, F. Schulze & M. Simon, Stability of Euclidean space under Ricci flow, Comm. Anal. Geom. 16 (2008), 127-158. | MR | Zbl

[38] W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), 223-301. | MR | Zbl

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

[40] M. Struwe, Geometric evolution problems, in Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), IAS/Park City Math. Ser. 2, Amer. Math. Soc., 1996, 257-339. | Zbl

[41] T. C. Tao, Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective, preprint arXiv:math/0610903.

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

[43] P. Topping, Lectures on the Ricci flow, London Math. Soc. Lecture Note Series 325, Cambridge Univ. Press, 2006. | Zbl

[44] M. B. Williams, Results on coupled Ricci and harmonic map flows, preprint arXiv:1012.0291.