Ricci flow on asymptotically Euclidean manifolds

Li, Yu¹

¹ Mathematics Department, Stony Brook University, Stony Brook, NY, United States

Geometry & topology, Tome 22 (2018) no. 3, pp. 1837-1891.

Résumé

In this paper, we prove that if an asymptotically Euclidean manifold with nonnegative scalar curvature has long-time existence of Ricci flow, the ADM mass is nonnegative. We also give an independent proof of the positive mass theorem in dimension three.

Détail

DOI : 10.2140/gt.2018.22.1837

Classification : 53C44, 83C99
Keywords: Ricci flow, mass

Bibliographie

[1] R A Adams, J J F Fournier, Sobolev spaces, 140, Elsevier (2003)

[2] R Arnowitt, S Deser, C W Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122 (1961) 997 | DOI

[3] T Aubin, Some nonlinear problems in Riemannian geometry, Springer (1998) | DOI

[4] S Axler, P Bourdon, W Ramey, Harmonic function theory, 137, Springer (1992) | DOI

[5] S Bando, A Kasue, H Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989) 313 | DOI

[6] R Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986) 661 | DOI

[7] L Bessières, G Besson, S Maillot, Ricci flow on open 3–manifolds and positive scalar curvature, Geom. Topol. 15 (2011) 927 | DOI

[8] X Cao, Q S Zhang, The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math. 228 (2011) 2891 | DOI

[9] A Chau, L F Tam, C Yu, Pseudolocality for the Ricci flow and applications, Canad. J. Math. 63 (2011) 55 | DOI

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

[11] 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, II : Analytic aspects, 144, Amer. Math. Soc. (2008) | DOI

[12] 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, III : Geometric-analytic aspects, 163, Amer. Math. Soc. (2010) | DOI

[13] B Chow, P Lu, L Ni, Hamilton’s Ricci flow, 77, Amer. Math. Soc. (2006) | DOI

[14] B Chow, P Lu, B Yang, Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons, C. R. Math. Acad. Sci. Paris 349 (2011) 1265 | DOI

[15] T H Colding, W P Minicozzi Ii, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008) 2537 | DOI

[16] X Dai, L Ma, Mass under the Ricci flow, Comm. Math. Phys. 274 (2007) 65 | DOI

[17] M Feldman, T Ilmanen, D Knopf, Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169 | DOI

[18] D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, 224, Springer (1983) | DOI

[19] L Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061 | DOI

[20] R S Hamilton, The formation of singularities in the Ricci flow, from: "Surveys in differential geometry, II" (editor S T Yau), International Press (1995) 7

[21] R Haslhofer, A mass-decreasing flow in dimension three, Math. Res. Lett. 19 (2012) 927 | DOI

[22] H J Hein, C Lebrun, Mass in Kähler geometry, Comm. Math. Phys. 347 (2016) 183 | DOI

[23] G Huisken, T Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001) 353 | DOI

[24] B Kleiner, J Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008) 2587 | DOI

[25] B Kleiner, J Lott, Singular Ricci flows, I, preprint (2014)

[26] J M Lee, T H Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987) 37 | DOI

[27] P Li, S T Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986) 153 | DOI

[28] D Mcferon, G Székelyhidi, On the positive mass theorem for manifolds with corners, Comm. Math. Phys. 313 (2012) 425 | DOI

[29] J Morgan, G Tian, Ricci flow and the Poincaré conjecture, 3, Amer. Math. Soc. (2007)

[30] O Munteanu, J Wang, Smooth metric measure spaces with non-negative curvature, Comm. Anal. Geom. 19 (2011) 451 | DOI

[31] G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)

[32] G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)

[33] G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)

[34] O S Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981) 110 | DOI

[35] R M Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, from: "Topics in calculus of variations" (editor M Giaquinta), Lecture Notes in Math. 1365, Springer (1989) 120 | DOI

[36] R Schoen, S T Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979) 45 | DOI

[37] R Schoen, S T Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979) 159 | DOI

[38] R Schoen, S T Yau, Lectures on differential geometry, International Press (1994)

[39] N Sesum, G Tian, X Wang, Notes on Perelman’s paper on the entropy formula for the Ricci flow and its geometric applications, preprint (2004)

[40] W X Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303 | DOI

[41] G Tian, J Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2005) 357 | DOI

[42] P Topping, Lectures on the Ricci flow, 325, Cambridge Univ. (2006) | DOI

[43] E Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981) 381 | DOI

[44] Q S Zhang, Strong noncollapsing and uniform Sobolev inequalities for Ricci flow with surgeries, Pacific J. Math. 239 (2009) 179 | DOI

[45] Q S Zhang, Extremal of log Sobolev inequality and W entropy on noncompact manifolds, J. Funct. Anal. 263 (2012) 2051 | DOI