Abstract
A fundamental question in Riemannian geometry is to find canonical metrics on a given smooth manifold. In the 1980s, R.S. Hamilton proposed an approach to this question based on parabolic partial differential equations. The goal is to start from a given initial metric and deform it to a canonical metric by means of an evolution equation. There are various natural evolution equations for Riemannian metrics, including the Ricci flow and the conformal Yamabe flow. In this survey, we discuss the global behavior of the solutions to these equations. In particular, we describe how these techniques can be used to prove the Differentiable Sphere Theorem.
Similar content being viewed by others
References
Andrews B., Nguyen H.: Four-manifolds with 1/4-pinched flag curvatures. Asian J. Math. 13, 251–270 (2009)
Aubin T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., (9) 55L, 269–296 (1976)
A. Bahri, Proof of the Yamabe conjecture, without the positive mass theorem, for locally conformally flat manifolds, In: Einstein Metrics and Yang–Mills Connections, (eds. T. Mabuchi and S. Mukai), Lecture Notes in Pure and Appl. Math., 145, Marcel Dekker, New York, 1993, pp. 1–26.
Berger M.: Les variétés Riemanniennes 1/4-pincées. Ann. Scuola Norm. Sup. Pisa (3) 14, 161–170 (1960)
Besson G., (2006) Preuve de la conjecture de Poincaré en déformant la métrique par la courbure de Ricci (d’après G. Perelman), Séminaire Bourbaki, 2004/2005. Astérisque 307: 309–347
Böhm C., Wilking B.: Manifolds with positive curvature operator are space forms. Ann. of Math. (2) 167, 1079–1097 (2008)
Bourguignon J.-P., Ricci curvature and Einstein metrics, In: Global Differential Geometry and Global Analysis, Lecture Notes in Math., 838, Springer-Verlag, 1981, pp. 42–63
Brendle S.: Convergence of the Yamabe flow for arbitrary initial energy. J. Differential Geom. 69, 217–278 (2005)
Brendle S.: A short proof for the convergence of the Yamabe flow on S n. Pure Appl. Math. Q. 3, 499–512 (2007)
Brendle S.: Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170, 541–576 (2007)
Brendle S.: Blow-up phenomena for the Yamabe equation. J. Amer. Math. Soc. 21, 951–979 (2008)
Brendle S.: A general convergence result for the Ricci flow in higher dimensions. Duke Math. J. 145, 585–601 (2008)
Brendle S.: A generalization of Hamilton’s differential Harnack inequality for the Ricci flow. J. Differential Geom. 82, 207–227 (2009)
S. Brendle, Ricci Flow and the Sphere Theorem, Grad. Stud. Math., 111, Amer. Math. Soc., Providence, RI, 2010.
Brendle S., Huisken G., Sinestrari C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158, 537–551 (2011)
Brendle S., Marques F.C.: Blow-up phenomena for the Yamabe equation II. J. Differential Geom. 81, 225–250 (2009)
Brendle S., Schoen R.: Classification of manifolds with weakly 1/4-pinched curvatures. Acta Math. 200, 1–13 (2008)
Brendle S., Schoen R.: Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc. 22, 287–307 (2009)
Brendle S., Schoen R.: Curvature, sphere theorems, and the Ricci flow. Bull. Amer. Math. Soc. (N.S.) 48, 1–32 (2011)
Chen H.: Pointwise 1/4-pinched 4-manifolds. Ann. Global Anal. Geom. 9, 161–176 (1991)
Chen X.: Weak limits of Riemannian metrics in surfaces with integral curvature bounds. Calc. Var. Partial Differential Equations 6, 189–226 (1998)
Chen X.: Calabi flow in Riemann surface revisited: a new point of view. Internat. Math. Res. Notices 2001, 275–297 (2001)
Chen X., Lu P., Tian G.: A note on uniformization of Riemann surfaces by Ricci flow. Proc. Amer. Math. Soc. 134, 3391–3393 (2006)
Chow B.: The Ricci flow on the 2-sphere. J. Differential Geom. 33, 325–334 (1991)
Chow B.: The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature. Comm. Pure Appl. Math. 45, 1003–1014 (1992)
Daskalopoulos P., Hamilton R.S.: Geometric estimates for the logarithmic fast diffusion equation. Comm. Anal. Geom. 12, 143–164 (2004)
DeTurck D.M.: Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18, 157–162 (1983)
Ecker K.: Heat equations in geometry and topology. Jahresber. Deutsch. Math.-Verein. 110, 117–141 (2008)
Fraser A.M.: Fundamental groups of manifolds with positive isotropic curvature. Ann. of Math. (2) 158, 345–354 (2003)
Hamilton R.S.: Three-manifolds with positive Ricci curvature, J. Differential Geom. 17, 255–306 (1982)
Hamilton R.S.: Four-manifolds with positive curvature operator. J. Differential Geom. 24, 153–179 (1986)
Hamilton R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–262 (1988)
R.S. Hamilton, Lectures on geometric flows, unpublished manuscript, 1989.
Hamilton R.S.: The Harnack estimate for the Ricci flow. J. Differential Geom. 37, 225–243 (1993)
R.S. Hamilton, The formation of singularities in the Ricci flow, In: Surveys in Differential Geometry, vol. II, Int. Press, Cambridge, MA, 1995, pp. 7–136.
Hamilton R.S.: Four-manifolds with positive isotropic curvature. Comm. Anal. Geom. 5, 1–92 (1997)
Huisken G.: Ricci deformation of the metric on a Riemannian manifold. J. Differential Geom. 21, 47–62 (1985)
Klingenberg W.: Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung. Comment. Math. Helv. 35, 47–54 (1961)
Leeb B.: Geometrization of 3-dimensional manifolds and Ricci flow: on Perelman’s proof of the conjectures of Poincaré and Thurston. Boll. Unione Mat. Ital. (9) 1, 41–55 (2008)
C. Margerin, Pointwise pinched manifolds are space forms, In: Geometric Measure Theory and the Calculus of Variations, Arcata, 1984, Proc. Sympos. Pure Math., 44, Amer. Math. Soc., Providence, RI, 1986, pp. 307–328.
Margerin C.: A sharp characterization of the smooth 4-sphere in curvature terms. Comm. Anal. Geom. 6, 21–65 (1998)
Micallef M.J., Moore J.D.: Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann. of Math. (2) 127, 199–227 (1988)
S. Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, In: Geometric Measure Theory and the Calculus of Variations, Arcata, 1984, Proc. Sympos. Pure Math., 44, Amer. Math. Soc., Providence, RI, 1986, pp. 343–352.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159.
G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math/0303109.
G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245.
Schoen R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20, 479–495 (1984)
R. Schoen, On the number of constant scalar curvature metrics in a conformal class, In: Differential Geometry, (eds. H.B. Lawson, Jr., and K. Tenenblat), Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., 1991, pp. 311–320.
Schoen R., Yau S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979)
Schoen R., Yau S.-T.: Conformally flat manifolds, Kleinian groups. and scalar curvature. Invent. Math. 92, 47–71 (1988)
Schwetlick H., Struwe M.: Convergence of the Yamabe flow for large energies. J. Reine Angew. Math. 562, 59–100 (2003)
Simon L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. of. Math. (2) 118, 525–571 (1983)
Struwe M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187, 511–517 (1984)
Struwe M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1, 247–274 (2002)
Trudinger N.S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, 265–274 (1968)
J. Wolf, Spaces of Constant Curvature. Fifth ed., Publish or Perish, Houston, TX, 1984.
Yamabe H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math J. 12, 21–37 (1960)
Ye R.: Global existence and convergence of Yamabe flow. J. Differential Geom. 39, 35–50 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Hiraku Nakajima
This article is based on the 9th Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on June 4, 2011.
About this article
Cite this article
Brendle, S. Evolution equations in Riemannian geometry. Jpn. J. Math. 6, 45–61 (2011). https://doi.org/10.1007/s11537-011-1115-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11537-011-1115-1