Conformal 4 -dimensional geometry: What analysis has to teach us

Christophe Margerin

Séminaire Bourbaki (2004-2005)

  • Volume: 47, page 415-468
  • ISSN: 0303-1179

Abstract

top
Starting at a 4 -dimensional generalization of Polyakov’s formula for regularized determinants, we present the ideas, tools, and results which led Chang S.-Y. A., M. Gursky and Yang P. to a sharp L 2 conformal sphere theorem. This includes the solution of Yamabe type problems for quadratic polynomials in the Ricci curvature. On our way we introduce “Conformal Pairs”, in particular the (4th order) Paneitz operator and its associated Q -curvature, and discuss how they relate to more classical 4 -dimensional conformal geometry. Elaborating on an argument due to M. Gursky and J. Viaclovsky, we also give a completely different, more direct and natural, proof of the main sphere theorem, and discuss further some related constructions of metrics with constant Q -curvature.

How to cite

top

Margerin, Christophe. "Géométrie conforme en dimension $4$ : ce que l’analyse nous apprend." Séminaire Bourbaki 47 (2004-2005): 415-468. <http://eudml.org/doc/252171>.

@article{Margerin2004-2005,
abstract = {Cet article présente les idées, les outils et les résultats qui ont permis à Chang S.-Y. A., M. Gursky et Yang P. de donner une caractérisation intégrale conforme de la sphère standard en dimension 4. Nous démarrons avec une généralisation à cette dimension de la formule de Polyakov pour les déterminants régularisés, que nous utilisons ensuite pour résoudre des problèmes du type “Yamabe” pour des polynômes quadratiques en la courbure de Ricci. Nous introduisons au passage le concept de paire conforme, en particulier l’opérateur (du quatrième ordre) de Paneitz et sa courbure $Q$ associée, et nous discutons leurs relations à la géométrie conforme classique. On trouvera aussi une preuve d’un esprit différent du théorème principal : beaucoup plus courte et naturelle, elle généralise un argument dû à M. Gursky et J. Viaclovsky qui l’a largement inspirée. On y donne enfin quelques constructions de métriques de courbure $Q$ constante, conséquence des arguments développés précédemment.},
author = {Margerin, Christophe},
journal = {Séminaire Bourbaki},
keywords = {conformal geometry; dimension $4$; pinching theorem; sphere theorem; conformal pairs; Paneitz operator; $Q$-curvature},
language = {fre},
pages = {415-468},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Géométrie conforme en dimension $4$ : ce que l’analyse nous apprend},
url = {http://eudml.org/doc/252171},
volume = {47},
year = {2004-2005},
}

TY - JOUR
AU - Margerin, Christophe
TI - Géométrie conforme en dimension $4$ : ce que l’analyse nous apprend
JO - Séminaire Bourbaki
PY - 2004-2005
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 47
SP - 415
EP - 468
AB - Cet article présente les idées, les outils et les résultats qui ont permis à Chang S.-Y. A., M. Gursky et Yang P. de donner une caractérisation intégrale conforme de la sphère standard en dimension 4. Nous démarrons avec une généralisation à cette dimension de la formule de Polyakov pour les déterminants régularisés, que nous utilisons ensuite pour résoudre des problèmes du type “Yamabe” pour des polynômes quadratiques en la courbure de Ricci. Nous introduisons au passage le concept de paire conforme, en particulier l’opérateur (du quatrième ordre) de Paneitz et sa courbure $Q$ associée, et nous discutons leurs relations à la géométrie conforme classique. On trouvera aussi une preuve d’un esprit différent du théorème principal : beaucoup plus courte et naturelle, elle généralise un argument dû à M. Gursky et J. Viaclovsky qui l’a largement inspirée. On y donne enfin quelques constructions de métriques de courbure $Q$ constante, conséquence des arguments développés précédemment.
LA - fre
KW - conformal geometry; dimension $4$; pinching theorem; sphere theorem; conformal pairs; Paneitz operator; $Q$-curvature
UR - http://eudml.org/doc/252171
ER -

References

top
  1. [A] D. R. Adams – “A sharp inequality of J. Moser for higher order derivatives”, Ann. of Math. (2) 128 (1988), no. 2, p. 385–398. Zbl0672.31008MR960950
  2. [CGY0] S.-Y. A. Chang, M. J. Gursky & P. C. Yang – “Regularity of a fourth order nonlinear PDE with critical exponent”, Amer. J. Math. 121 (1999), no. 2, p. 215–257. Zbl0921.35032MR1680337
  3. [CGY1] —, “An a priori estimate for a fully nonlinear equation on four-manifolds”, J. Anal. Math.87 (2002), p. 151–186. Zbl1067.58028MR1945280
  4. [CGY2] —, “An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature”, Ann. of Math. (2) 155 (2002), no. 3, p. 709–787. Zbl1031.53062MR1923964
  5. [CGY3] —, “A conformally invariant sphere theorem in four dimensions”, Publ. Math. Inst. Hautes Études Sci. (2003), no. 98, p. 105–143. Zbl1066.53079MR2031200
  6. [CQ] S.-Y. A. Chang & J. Qing – “The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set”, J. Funct. Anal. 147 (1997), no. 2, p. 363–399. Zbl0914.58040MR1454486
  7. [CY] S.-Y. A. Chang & P. C. Yang – “Extremal metrics of zeta function determinants on 4 -manifolds”, Ann. of Math. (2) 142 (1995), no. 1, p. 171–212. Zbl0842.58011MR1338677
  8. [G1] M. J. Gursky – “The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics”, Ann. of Math. (2) 148 (1998), no. 1, p. 315–337. Zbl0949.53025MR1652920
  9. [G2] —, “The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE”, Comm. Math. Phys. 207 (1999), no. 1, p. 131–143. Zbl0988.58013MR1724863
  10. [GV] M. J. Gursky & J. A. Viaclovsky – “A fully nonlinear equation on four-manifolds with positive scalar curvature”, J. Differential Geom. 63 (2003), no. 1, p. 131–154. Zbl1070.53018MR2015262
  11. [GW] P. Guan & G. Wang – “Local estimates for a class of fully nonlinear equations arising from conformal geometry”, Int. Math. Res. Not. (2003), no. 26, p. 1413–1432. Zbl1042.53021MR1976045
  12. [K] O. Kobayashi – “Scalar curvature of a metric with unit volume”, Math. Ann. 279 (1987), no. 2, p. 253–265. Zbl0611.53037MR919505
  13. [KMPS] N. Korevaar, R. Mazzeo, F. Pacard & R. Schoen – “Refined asymptotics for constant scalar curvature metrics with isolated singularities”, Invent. Math. 135 (1999), no. 2, p. 233–272. Zbl0958.53032MR1666838
  14. [L] Y. Y. Li – “Degree theory for second order nonlinear elliptic operators and its applications”, Comm. Partial Differential Equations 14 (1989), no. 11, p. 1541–1578. Zbl0702.35094MR1026774
  15. [LL] A. Li & Y. Li – “On some conformally invariant fully nonlinear equations”, Comm. Pure Appl. Math. 56 (2003), no. 10, p. 1416–1464. Zbl1155.35353MR1988895
  16. [M] C. Margerin – “A sharp characterization of the smooth 4 -sphere in curvature terms”, Comm. Anal. Geom. 6 (1998), no. 1, p. 21–65. Zbl0966.53022MR1619838
  17. [S] R. M. Schoen – “Analytic aspects of the harmonic map problem”, in Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, p. 321–358. Zbl0551.58011MR765241
  18. [SU] R. Schoen & K. Uhlenbeck – “A regularity theory for harmonic maps”, J. Differential Geom. 17 (1982), no. 2, p. 307–335. Zbl0521.58021MR664498
  19. [SY] J.-P. Sha & D. Yang – “Positive Ricci curvature on compact simply connected 4 -manifolds”, in Differential geometry : Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, p. 529–538. Zbl0788.53029MR1216644
  20. [Y] D. Yang – “ L p pinching and compactness theorems for compact Riemannian manifolds”, Forum Math. 4 (1992), no. 3, p. 323–333. Zbl0753.53027MR1164099
  21. [ABKS] K. Akutagawa, B. Botvinnik, O. Kobayashi & H. Seshadri – “The Weyl functional near the Yamabe invariant”, J. Geom. Anal. 13 (2003), no. 1, p. 1–20. Zbl1048.53020MR1967032
  22. [Be] W. Beckner – “Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality”, Ann. of Math. (2) 138 (1993), no. 1, p. 213–242. Zbl0826.58042MR1230930
  23. [BØ] T. P. Branson & B. Ørsted – “Explicit functional determinants in four dimensions”, Proc. Amer. Math. Soc. 113 (1991), no. 3, p. 669–682. Zbl0762.47019MR1050018
  24. [CY2] S.-Y. A. Chang & P. C. Yang – “Non-linear partial differential equations in conformal geometry”, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) (Beijing), Higher Ed. Press, 2002, p. 189–207. Zbl1036.53024MR1989185
  25. [FG1] C. Fefferman & C. R. Graham – “Conformal invariants”, Astérisque (1985), Numéro Hors Série, p. 95–116. Zbl0602.53007MR837196
  26. [FG2] —, “ Q -curvature and Poincaré metrics”, Math. Res. Lett. 9 (2002), no. 2-3, p. 139–151. Zbl1016.53031MR1909634
  27. [FH] C. Fefferman & K. Hirachi – “Ambient metric construction of Q -curvature in conformal and CR geometries”, Math. Res. Lett. 10 (2003), no. 5-6, p. 819–831. Zbl1166.53309MR2025058
  28. [FP] P. M. Fitzpatrick & J. Pejsachowicz – “An extension of the Leray-Schauder degree for fully nonlinear elliptic problems”, in Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983), Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, p. 425–438. Zbl0628.47039MR843576
  29. [GJMS] C. R. Graham, R. Jenne, L. J. Mason & G. A. J. Sparling – “Conformally invariant powers of the Laplacian. I. Existence”, J. London Math. Soc. (2) 46 (1992), no. 3, p. 557–565. Zbl0726.53010MR1190438
  30. [GP] A. R. Gover & L. J. Peterson – “Conformally invariant powers of the Laplacian, Q -curvature, and tractor calculus”, Comm. Math. Phys. 235 (2003), no. 2, p. 339–378. Zbl1022.58014MR1969732
  31. [GV2] M. J. Gursky & J. A. Viaclovsky – “Fully nonlinear equations on Riemannian manifolds with negative curvature”, Indiana Univ. Math. J. 52 (2003), no. 2, p. 399–419. Zbl1036.53025MR1976082
  32. [GZ] C. R. Graham & M. Zworski – “Scattering matrix in conformal geometry”, Invent. Math. 152 (2003), no. 1, p. 89–118. Zbl1030.58022MR1965361
  33. [OPS1] B. Osgood, R. Phillips, & P. Sarnak – “Compact isospectral sets of surfaces”, J. Funct. Anal. 80 (1988), no. 1, p. 212–234. Zbl0653.53021MR960229
  34. [OPS2] —, “Extremals of determinants of Laplacians”, J. Funct. Anal. 80 (1988), no. 1, p. 148–211. Zbl0653.53022MR960228

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.