Uniqueness of instantaneously complete Ricci flows

Topping, Peter M¹

¹ Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Geometry & topology, Tome 19 (2015) no. 3, pp. 1477-1492.

Résumé

We prove uniqueness of instantaneously complete Ricci flows on surfaces. We do not require any bounds of any form on the curvature or its growth at infinity, nor on the metric or its growth (other than that implied by instantaneous completeness). Coupled with earlier work, this completes the well-posedness theory for instantaneously complete Ricci flows on surfaces.

Détail

DOI : 10.2140/gt.2015.19.1477

Classification : 35K55, 53C44, 58J35
Keywords: Ricci flow, logarithmic fast diffusion equation

Bibliographie

[1] P Albin, C L Aldana, F Rochon, Ricci flow and the determinant of the Laplacian on non-compact surfaces, Comm. PDE 38 (2013) 711

[2] B L Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009) 363

[3] B L Chen, X P Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006) 119

[4] B Chow, The Ricci flow on the $2$–sphere, J. Differential Geom. 33 (1991) 325

[5] M R R Coffey, Ricci flow and metric geometry, PhD thesis, University of Warwick (2015)

[6] P Daskalopoulos, C E Kenig, Degenerate diffusions, EMS Tracts in Mathematics 1, Eur. Math. Soc. (2007)

[7] P Daskalopoulos, M A Del Pino, On a singular diffusion equation, Comm. Anal. Geom. 3 (1995) 523

[8] D Deturck, Collected papers on Ricci flow, Series in Geometry and Topology 37, International Press (2003)

[9] E Dibenedetto, D J Diller, About a singular parabolic equation arising in thin film dynamics and in the Ricci flow for complete $\mathbf{R}^2$, from: "Partial differential equations and applications", Lecture Notes in Pure and Appl. Math. 177, Dekker (1996) 103

[10] G Giesen, Instantaneously complete Ricci flows on surfaces, PhD thesis, University of Warwick (2012)

[11] G Giesen, P M Topping, Ricci flow of negatively curved incomplete surfaces, Calc. Var. Partial Differential Equations 38 (2010) 357

[12] G Giesen, P M Topping, Existence of Ricci flows of incomplete surfaces, Comm. Partial Differential Equations 36 (2011) 1860

[13] G Giesen, P M Topping, Ricci flows with unbounded curvature, Math. Z. 273 (2013) 449

[14] G Giesen, P M Topping, Ricci flows with bursts of unbounded curvature, (2014)

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

[16] R S Hamilton, The Ricci flow on surfaces, from: "Mathematics and general relativity", Contemp. Math. 71, Amer. Math. Soc. (1988) 237

[17] R S Hamilton, The formation of singularities in the Ricci flow, from: "Proc. Conf. Geometry and Topology", Surveys in Differential Geometry 2, Int. Press (1995) 7

[18] L Ji, R Mazzeo, N Sesum, Ricci flow on surfaces with cusps, Math. Ann. 345 (2009) 819

[19] B Kotschwar, An energy approach to the problem of uniqueness for the Ricci flow, Comm. Anal. Geom. 22 (2014) 149

[20] G Perelman, The entropy formula for the Ricci flow and its geometric applications,

[21] T Richard, Canonical smoothing of compact Alexandrov surfaces via Ricci flow, (2012)

[22] A Rodriguez, J L Vazquez, J R Esteban, The maximal solution of the logarithmic fast diffusion equation in two space dimensions, Adv. Differential Equations 2 (1997) 867

[23] W X Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223

[24] P M Topping, Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics, J. Eur. Math. Soc. 12 (2010) 1429

[25] P M Topping, Uniqueness and nonuniqueness for Ricci flow on surfaces: Reverse cusp singularities, Int. Math. Res. Not. 2012 (2012) 2356

[26] P M Topping, Remarks on Hamilton's compactness theorem for Ricci flow, J. Reine Angew. Math. 692 (2014) 173

[27] J L Vázquez, Smoothing and decay estimates for nonlinear diffusion equations, Oxford Lecture Series Math. Appl. 33, Oxford University Press (2006)

[28] S T Yau, Remarks on conformal transformations, J. Differential Geometry 8 (1973) 369