Geometry & Topology

Ricci flow on asymptotically Euclidean manifolds

Yu Li

Abstract

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.

Article information

Source
Geom. Topol., Volume 22, Number 3 (2018), 1837-1891.

Dates
Revised: 15 May 2017
Accepted: 15 June 2017
First available in Project Euclid: 31 March 2018

https://projecteuclid.org/euclid.gt/1522461628

Digital Object Identifier
doi:10.2140/gt.2018.22.1837

Mathematical Reviews number (MathSciNet)
MR3780446

Zentralblatt MATH identifier
06864268

Keywords
Ricci flow mass

Citation

Li, Yu. Ricci flow on asymptotically Euclidean manifolds. Geom. Topol. 22 (2018), no. 3, 1837--1891. doi:10.2140/gt.2018.22.1837. https://projecteuclid.org/euclid.gt/1522461628

References

• R A Adams, J J F Fournier, Sobolev spaces, 2nd edition, Pure and Applied Mathematics 140, Elsevier, Amsterdam (2003)
• R Arnowitt, S Deser, C W Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. 122 (1961) 997–1006
• T Aubin, Some nonlinear problems in Riemannian geometry, Springer (1998)
• S Axler, P Bourdon, W Ramey, Harmonic function theory, Graduate Texts in Mathematics 137, Springer (1992)
• 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) 313–349
• R Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986) 661–693
• L Bessières, G Besson, S Maillot, Ricci flow on open $3$–manifolds and positive scalar curvature, Geom. Topol. 15 (2011) 927–975
• X Cao, Q S Zhang, The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math. 228 (2011) 2891–2919
• A Chau, L-F Tam, C Yu, Pseudolocality for the Ricci flow and applications, Canad. J. Math. 63 (2011) 55–85
• B-L Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009) 363–382
• 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, Mathematical Surveys and Monographs 144, Amer. Math. Soc., Providence, RI (2008)
• 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, Mathematical Surveys and Monographs 163, Amer. Math. Soc., Providence, RI (2010)
• B Chow, P Lu, L Ni, Hamilton's Ricci flow, Graduate Studies in Mathematics 77, Amer. Math. Soc., Providence, RI (2006)
• 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) 1265–1267
• T H Colding, W P Minicozzi, II, Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008) 2537–2586
• X Dai, L Ma, Mass under the Ricci flow, Comm. Math. Phys. 274 (2007) 65–80
• M Feldman, T Ilmanen, D Knopf, Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209
• D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, 2nd edition, Grundl. Math. Wissen. 224, Springer (1983)
• L Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061–1083
• R S Hamilton, The formation of singularities in the Ricci flow, from “Surveys in differential geometry, II” (S-T Yau, editor), International Press, Cambridge, MA (1995) 7–136
• R Haslhofer, A mass-decreasing flow in dimension three, Math. Res. Lett. 19 (2012) 927–938
• H-J Hein, C LeBrun, Mass in Kähler geometry, Comm. Math. Phys. 347 (2016) 183–221
• G Huisken, T Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001) 353–437
• B Kleiner, J Lott, Notes on Perelman's papers, Geom. Topol. 12 (2008) 2587–2855
• B Kleiner, J Lott, Singular Ricci flows, I, preprint (2014)
• J M Lee, T H Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987) 37–91
• P Li, S-T Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986) 153–201
• D McFeron, G Székelyhidi, On the positive mass theorem for manifolds with corners, Comm. Math. Phys. 313 (2012) 425–443
• J Morgan, G Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs 3, Amer. Math. Soc., Providence, RI (2007)
• O Munteanu, J Wang, Smooth metric measure spaces with non-negative curvature, Comm. Anal. Geom. 19 (2011) 451–486
• G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)
• G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)
• G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)
• O S Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal. 42 (1981) 110–120
• R M Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, from “Topics in calculus of variations” (M Giaquinta, editor), Lecture Notes in Math. 1365, Springer (1989) 120–154
• R Schoen, S-T Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979) 45–76
• R Schoen, S-T Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979) 159–183
• R Schoen, S-T Yau, Lectures on differential geometry, International Press, Cambridge, MA (1994)
• 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) Available at \setbox0\makeatletter\@url http://www.math.msu.edu/~xwang/perel.pdf {\unhbox0
• W-X Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989) 303–394
• G Tian, J Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2005) 357–415
• P Topping, Lectures on the Ricci flow, London Math. Soc. Lecture Note Series 325, Cambridge Univ. (2006)
• E Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981) 381–402
• Q S Zhang, Strong noncollapsing and uniform Sobolev inequalities for Ricci flow with surgeries, Pacific J. Math. 239 (2009) 179–200
• Q S Zhang, Extremal of log Sobolev inequality and $W$ entropy on noncompact manifolds, J. Funct. Anal. 263 (2012) 2051–2101