## Ambient obstruction flow

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- 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

## Additional Information

**Christopher Lopez**- Affiliation: Department of Mathematics, 419 Rowland Hall , University of California, Irvine , Irvine, California 92697-3875
- Address at time of publication: Department of Mathematics, South Hall, Room 6502, University of California, Santa Barbara, Santa Barbara, California 93106-3080
- MR Author ID: 951587
- Email: clopez00@math.ucsb.edu
- Received by editor(s): October 26, 2015
- Received by editor(s) in revised form: September 11, 2016, and October 18, 2016
- Published electronically: January 18, 2018
- Additional Notes: The author was supported by an NSF AGEP Supplement under DMS-1301864.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 4111-4145 - MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/tran/7106
- MathSciNet review: 3811522