Ambient obstruction flow

Lopez, Christopher

doi:10.1090/tran/7106

Ambient obstruction flow
HTML articles powered by AMS MathViewer

by Christopher Lopez PDF

Trans. Amer. Math. Soc. 370 (2018), 4111-4145 Request permission

Abstract:

We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud–Helliwell which is based on the Fefferman–Graham ambient obstruction tensor. First, we obtain local L2 smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimates for the curvature tensor. We use the pointwise smoothing estimates to show that the curvature must blow up for a finite time singular solution. We also use the pointwise smoothing estimates to prove a compactness theorem for a sequence of solutions with bounded C0 curvature norm and injectivity radius bounded from below at one point. Finally, we use the compactness theorem to obtain a singularity model from a finite time singular solution and to characterize the behavior at infinity of a nonsingular solution.

References

Michael T. Anderson and Piotr T. Chruściel, Asymptotically simple solutions of the vacuum Einstein equations in even dimensions, Comm. Math. Phys. 260 (2005), no. 3, 557–577. MR 2183957, DOI 10.1007/s00220-005-1424-4
Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
Eric Bahuaud and Dylan Helliwell, Short-time existence for some higher-order geometric flows, Comm. Partial Differential Equations 36 (2011), no. 12, 2189–2207. MR 2852074, DOI 10.1080/03605302.2011.593015
Eric Bahuaud and Dylan Helliwell, Uniqueness for some higher-order geometric flows, Bull. Lond. Math. Soc. 47 (2015), no. 6, 980–995. MR 3431578, DOI 10.1112/blms/bdv076
Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
V. Bour, Fourth Order Curvature Flows and Geometric Applications, arXiv:1012.0342v1
Thomas P. Branson, The functional determinant, Lecture Notes Series, vol. 4, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. MR 1325463
Thomas P. Branson and Bent Ørsted, Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc. 113 (1991), no. 3, 669–682. MR 1050018, DOI 10.1090/S0002-9939-1991-1050018-8
Huai-Dong Cao and Xi-Ping 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), no. 2, 165–492. MR 2233789, DOI 10.4310/AJM.2006.v10.n2.a2
S.-Y. Alice Chang, Michael Eastwood, Bent Ørsted, and Paul C. Yang, What is $Q$-curvature?, Acta Appl. Math. 102 (2008), no. 2-3, 119–125. MR 2407525, DOI 10.1007/s10440-008-9229-z
X. X. Chen and W. Y. He, On the Calabi flow, Amer. J. Math. 130 (2008), no. 2, 539–570. MR 2405167, DOI 10.1353/ajm.2008.0018
Xiuxiong Chen and Weiyong He, The Calabi flow on Kähler surfaces with bounded Sobolev constant (I), Math. Ann. 354 (2012), no. 1, 227–261. MR 2957626, DOI 10.1007/s00208-011-0723-7
B. Chow et al., The Ricci flow: techniques and applications. Part I, Mathematical Surveys and Monographs, 135, Amer. Math. Soc., Providence, RI, 2007. MR2302600 (2008f:53088)
B. Chow et al., The Ricci flow: techniques and applications. Part II, Mathematical Surveys and Monographs, 144, Amer. Math. Soc., Providence, RI, 2008. MR2365237
B. Chow and D. Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs, 110, Amer. Math. Soc., Providence, RI, 2004. MR2061425 (2005e:53101)
Charles Fefferman and C. Robin Graham, Conformal invariants, Astérisque Numéro Hors Série (1985), 95–116. The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837196
Charles Fefferman and C. Robin Graham, The ambient metric, Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton, NJ, 2012. MR 2858236
A. Rod Gover and F. Leitner, A sub-product construction of Poincaré-Einstein metrics, Internat. J. Math. 20 (2009), no. 10, 1263–1287. MR 2574315, DOI 10.1142/S0129167X09005753
C. Robin Graham and Kengo Hirachi, The ambient obstruction tensor and $Q$-curvature, AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lect. Math. Theor. Phys., vol. 8, Eur. Math. Soc., Zürich, 2005, pp. 59–71. MR 2160867, DOI 10.4171/013-1/3
C. Robin Graham, Ralph Jenne, Lionel J. Mason, and George A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), no. 3, 557–565. MR 1190438, DOI 10.1112/jlms/s2-46.3.557
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Richard S. Hamilton, The formation of singularities in the Ricci flow, in Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Int. Press, Cambridge, MA. MR1375255 (97e:53075)
Richard S. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math. 117 (1995), no. 3, 545–572. MR 1333936, DOI 10.2307/2375080
Emmanuel Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR 1481970, DOI 10.1007/BFb0092907
Ali Ulaş Özgür Kişisel, Özgür Sarıoğlu, and Bayram Tekin, Cotton flow, Classical Quantum Gravity 25 (2008), no. 16, 165019, 15. MR 2429736, DOI 10.1088/0264-9381/25/16/165019
Bruce Kleiner and John Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. MR 2460872, DOI 10.2140/gt.2008.12.2587
Brett Kotschwar, An energy approach to uniqueness for higher-order geometric flows, J. Geom. Anal. 26 (2016), no. 4, 3344–3368. MR 3544962, DOI 10.1007/s12220-015-9670-y
Ernst Kuwert and Reiner Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom. 10 (2002), no. 2, 307–339. MR 1900754, DOI 10.4310/CAG.2002.v10.n2.a4
Claude LeBrun, Curvature functionals, optimal metrics, and the differential topology of 4-manifolds, Different faces of geometry, Int. Math. Ser. (N. Y.), vol. 3, Kluwer/Plenum, New York, 2004, pp. 199–256. MR 2102997, DOI 10.1007/0-306-48658-X_{5}
John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
C. Mantegazza, Smooth geometric evolutions of hypersurfaces, Geom. Funct. Anal. 12 (2002), no. 1, 138–182. MR 1904561, DOI 10.1007/s00039-002-8241-0
Carlo Mantegazza and Luca Martinazzi, A note on quasilinear parabolic equations on manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 857–874. MR 3060703
John Morgan and Gang Tian, Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007. MR 2334563, DOI 10.1305/ndjfl/1193667709
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159
—, Ricci flow with surgery on three-manifolds, arXiv:math/0303109
—, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245
Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
Ronald J. Riegert, A nonlocal action for the trace anomaly, Phys. Lett. B 134 (1984), no. 1-2, 56–60. MR 729802, DOI 10.1016/0370-2693(84)90983-3
Jeffrey D. Streets, The gradient flow of $\int _M|\textrm {Rm}|^2$, J. Geom. Anal. 18 (2008), no. 1, 249–271. MR 2365674, DOI 10.1007/s12220-007-9000-0
Jeffrey Streets, The long time behavior of fourth order curvature flows, Calc. Var. Partial Differential Equations 46 (2013), no. 1-2, 39–54. MR 3016500, DOI 10.1007/s00526-011-0472-1