Opérateurs géométriques et géométrie conforme
Séminaire de théorie spectrale et géométrie, Tome 23 (2004-2005) , pp. 49-103.
DOI : https://doi.org/10.5802/tsg.231
Classification : 35B33,  35J35,  53A30,  53C21
@article{TSG_2004-2005__23__49_0,
     author = {Djadli, Zindine},
     title = {Op\'erateurs g\'eom\'etriques et g\'eom\'etrie conforme},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {49--103},
     publisher = {Institut Fourier},
     volume = {23},
     year = {2004-2005},
     doi = {10.5802/tsg.231},
     mrnumber = {2270223},
     zbl = {1103.53019},
     language = {fr},
     url = {www.numdam.org/item/TSG_2004-2005__23__49_0/}
}
Djadli, Zindine. Opérateurs géométriques et géométrie conforme. Séminaire de théorie spectrale et géométrie, Tome 23 (2004-2005) , pp. 49-103. doi : 10.5802/tsg.231. http://www.numdam.org/item/TSG_2004-2005__23__49_0/

[AB99] Thierry Aubin and Abbas Bahri. Une remarque sur l’indice et la norme infinie des solutions d’équations elliptiques surlinéaires. Ricerche Mat., 48(suppl.) :117–128, 1999. Papers in memory of Ennio De Giorgi (Italian). | Zbl 0932.35083

[AB97a] T. Aubin and A. Bahri. Méthodes de topologie algébrique pour le problème de la courbure scalaire prescrite. J. Math. Pures Appl. (9), 76(6) :525–549, 1997. | MR 1465609 | Zbl 0886.58109

[AB97b] T. Aubin and A. Bahri. Une hypothèse topologique pour le problème de la courbure scalaire prescrite. J. Math. Pures Appl. (9), 76(10) :843–850, 1997. | MR 1489940 | Zbl 0916.58041

[Ada88] David R. Adams. A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. (2), 128(2) :385–398, 1988. | MR 960950 | Zbl 0672.31008

[Ahl87] Lars V. Ahlfors. Lectures on quasiconformal mappings. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987. With the assistance of Clifford J. Earle, Jr., Reprint of the 1966 original. | MR 883205 | Zbl 0605.30002

[AL99a] Thierry Aubin and Yan Yan Li. On the best Sobolev inequality. J. Math. Pures Appl. (9), 78(4) :353–387, 1999. | MR 1696357 | Zbl 0944.46027

[AL99b] Thierry Aubin and Yan Yan Li. Sur la meilleure constante dans l’inégalité de Sobolev. C. R. Acad. Sci. Paris Sér. I Math., 328(2) :135–138, 1999. | Zbl 0934.58036

[AM99] Antonio Ambrosetti and Andrea Malchiodi. A multiplicity result for the Yamabe problem on S n . J. Funct. Anal., 168(2) :529–561, 1999. | MR 1719213 | Zbl 0949.53028

[Aub74] Thierry Aubin. Fonction de Green et valeurs propres du laplacien. J. Math. Pures Appl. (9), 53 :347–371, 1974. | MR 358875 | Zbl 0295.35016

[Aub78] Thierry Aubin. Sur les meilleures constantes dans le théorème d’inclusion de Sobolev. C. R. Acad. Sci. Paris Sér. A-B, 287(11) :A795–A797, 1978. | MR 517491 | Zbl 0425.53021

[Aub79] Thierry Aubin. Meilleures constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal., 32(2) :148–174, 1979. | MR 534672 | Zbl 0411.46019

[Aub80] Thierry Aubin. Un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire sur la sphère. In Analysis on manifolds (Conf., Univ. Metz, Metz, 1979) (French), volume 80 of Astérisque, pages 4, 57–62. Soc. Math. France, Paris, 1980. | MR 620170 | Zbl 0465.58025

[Aub86] Thierry Aubin. Le problème de Yamabe concernant la courbure scalaire. In Differential geometry, Peñíscola 1985, volume 1209 of Lecture Notes in Math., pages 66–72. Springer, Berlin, 1986. | MR 863745 | Zbl 0614.53031

[Aub98] Thierry Aubin. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. | MR 1636569 | Zbl 0896.53003

[Aub01] Thierry Aubin. A course in differential geometry, volume 27 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. | MR 1799532 | Zbl 0966.53001

[Aub70a] Thierry Aubin. Métriques riemanniennes et courbure. J. Differential Geometry, 4 :383–424, 1970. | MR 279731 | Zbl 0212.54102

[Aub75a] Thierry Aubin. Équations différentielles non linéaires. Bull. Sci. Math. (2), 99(4) :201–210, 1975. | MR 431292 | Zbl 0321.35039

[Aub76a] Thierry Aubin. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. (9), 55(3) :269–296, 1976. | MR 431287 | Zbl 0336.53033

[Aub82a] Thierry Aubin. Best constants in the Sobolev imbedding theorem : the Yamabe problem. In Seminar on Differential Geometry, volume 102 of Ann. of Math. Stud., pages 173–184. Princeton Univ. Press, Princeton, N.J., 1982. | MR 645736 | Zbl 0483.53041

[Aub94a] Thierry Aubin. Prescribed scalar curvature and the method of isometry-concentration. In Partial differential equations of elliptic type (Cortona, 1992), Sympos. Math., XXXV, pages 37–45. Cambridge Univ. Press, Cambridge, 1994. | MR 1297772 | Zbl 0818.53061

[Aub70b] Thierry Aubin. Sur la fonction exponentielle. C. R. Acad. Sci. Paris Sér. A-B, 270 :A1514–A1516, 1970. | MR 271870 | Zbl 0197.47802

[Aub75b] Thierry Aubin. Étude d’un certain type d’équations différentielles non linéaires. C. R. Acad. Sci. Paris Sér. A-B, 280(7) :Aiii, A455–A457, 1975. | MR 383466 | Zbl 0295.35019

[Aub76b] Thierry Aubin. Espaces de Sobolev sur les variétés riemanniennes. Bull. Sci. Math. (2), 100(2) :149–173, 1976. | MR 488125 | Zbl 0328.46030

[Aub82b] Thierry Aubin. Nonlinear analysis on manifolds. Monge-Ampère equations, volume 252 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1982. | MR 681859 | Zbl 0512.53044

[Aub94b] Thierry Aubin. Sur le problème de la courbure scalaire prescrite. Bull. Sci. Math., 118(5) :465–474, 1994. | MR 1305205 | Zbl 0828.53035

[Aub75c] Thierry Aubin. Inégalités concernant la première valeur propre non nulle du laplacien pour certaines variétés riemanniennes. C. R. Acad. Sci. Paris Sér. A-B, 281(22) :Aii, A979–A982, 1975. | MR 413203 | Zbl 0323.35058

[Aub76c] Thierry Aubin. Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geometry, 11(4) :573–598, 1976. | MR 448404 | Zbl 0371.46011

[Aub75d] Thierry Aubin. Le problème de Yamabe concernant la courbure scalaire. C. R. Acad. Sci. Paris Sér. A-B, 280 :Aii, A721–A724, 1975. | MR 365407 | Zbl 0299.53026

[Aub75e] Thierry Aubin. Problèmes isopérimétriques et espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B, 280(5) :Aii, A279–A281, 1975. | MR 407905 | Zbl 0295.53024

[BC91] A. Bahri and J.-M. Coron. The scalar-curvature problem on the standard three-dimensional sphere. J. Funct. Anal., 95(1) :106–172, 1991. | MR 1087949 | Zbl 0722.53032

[BCY92a] Thomas P. Branson, Sun-Yung A. Chang, and Paul C. Yang. Estimates and extremals for zeta function determinants on four-manifolds. Comm. Math. Phys., 149(2) :241–262, 1992. | MR 1186028 | Zbl 0761.58053

[BCY92b] Thomas P. Branson, Sun-Yung A. Chang, and Paul C. Yang. Estimates and extremals for zeta function determinants on four-manifolds. Comm. Math. Phys., 149(2) :241–262, 1992. | MR 1186028 | Zbl 0761.58053

[Bes87] Arthur L. Besse. Einstein manifolds, volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1987. | MR 867684 | Zbl 0613.53001

[BGM71] Marcel Berger, Paul Gauduchon, and Edmond Mazet. Le spectre d’une variété riemannienne. Lecture Notes in Mathematics, Vol. 194. Springer-Verlag, Berlin, 1971. | MR 282313 | Zbl 0223.53034

[BL96] D. Bakry and M. Ledoux. Sobolev inequalities and Myers’s diameter theorem for an abstract Markov generator. Duke Math. J., 85(1) :253–270, 1996. | MR 1412446 | Zbl 0870.60071

[BM01] Massimiliano Berti and Andrea Malchiodi. Non-compactness and multiplicity results for the Yamabe problem on S n . J. Funct. Anal., 180(1) :210–241, 2001. | MR 1814428 | Zbl 0979.53038

[BO91] Thomas P. Branson and Bent Orsted. Explicit functional determinants in four dimensions. Proc. Amer. Math. Soc., 113(3) :669–682, 1991. | MR 1050018 | Zbl 0762.47019

[Bou81] Jean-Pierre Bourguignon. Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math., 63(2) :263–286, 1981. | EuDML 142798 | MR 610539 | Zbl 0456.53033

[Bra85] Thomas P. Branson. Differential operators canonically associated to a conformal structure. Math. Scand., 57(2) :293–345, 1985. | EuDML 166957 | MR 832360 | Zbl 0596.53009

[Bra87] Thomas P. Branson. Group representations arising from Lorentz conformal geometry. J. Funct. Anal., 74(2) :199–291, 1987. | MR 904819 | Zbl 0643.58036

[Bra95] Thomas P. Branson. Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc., 347(10) :3671–3742, 1995. | MR 1316845 | Zbl 0848.58047

[CC86] Lennart Carleson and Sun-Yung A. Chang. On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. (2), 110(2) :113–127, 1986. | MR 878016 | Zbl 0619.58013

[CC01] Sun-Yung Alice Chang and Wenxiong Chen. A note on a class of higher order conformally covariant equations. Discrete Contin. Dynam. Systems, 7(2) :275–281, 2001. | MR 1808400 | Zbl 1014.35025

[CGW94] S.-Y. A. Chang, M. Gursky, and T. Wolff. Lack of compactness in conformal metrics with L d/2 curvature. J. Geom. Anal., 4(2) :143–153, 1994. | MR 1277502 | Zbl 0810.53027

[CGY93] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang. The scalar curvature equation on 2- and 3-spheres. Calc. Var. Partial Differential Equations, 1(2) :205–229, 1993. | MR 1261723 | Zbl 0822.35043

[CGY99] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang. Regularity of a fourth order nonlinear PDE with critical exponent. Amer. J. Math., 121(2) :215–257, 1999. | MR 1680337 | Zbl 0921.35032

[CGY02] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang. An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math. (2), 155(3) :709–787, 2002. | MR 1923964 | Zbl 1031.53062

[CGY03] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang. A conformally invariant sphere theorem in four dimensions. Publ. Math. Inst. Hautes Études Sci., (98) :105–143, 2003. | EuDML 104193 | Numdam | MR 2031200 | Zbl 1066.53079

[CGYre] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang. A conformally invariant sphere theorem in four dimensions. Publ. Math. Inst. Hautes Études Sci., A paraître. | EuDML 104193 | Numdam | MR 2031200 | Zbl 1066.53079

[Cha87] Sun-Yung A. Chang. Extremal functions in a sharp form of Sobolev inequality. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 715–723, Providence, RI, 1987. Amer. Math. Soc. | MR 934274 | Zbl 0692.46029

[Cha96] Sun-Yung Alice Chang. The Moser-Trudinger inequality and applications to some problems in conformal geometry. In Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), volume 2 of IAS/Park City Math. Ser., pages 65–125. Amer. Math. Soc., Providence, RI, 1996. | MR 1369587 | Zbl 0924.58106

[Cha97] Sun-Yung Alice Chang. On zeta functional determinant. In Partial differential equations and their applications (Toronto, ON, 1995), volume 12 of CRM Proc. Lecture Notes, pages 25–50. Amer. Math. Soc., Providence, RI, 1997. With notes taken by Jie Qing. | MR 1479236 | Zbl 0888.58071

[Cha99] Sun-Yung A. Chang. On a fourth-order partial differential equation in conformal geometry. In Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Math., pages 127–150. Univ. Chicago Press, Chicago, IL, 1999. | MR 1743859 | Zbl 0982.53036

[CL03] Chiun-Chuan Chen and Chang-Shou Lin. Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math., 56(12) :1667–1727, 2003. | MR 2001443 | Zbl 1032.58010

[CQ96] Sun-Yung A. Chang and Jie Qing. Zeta functional determinants on manifolds with boundary. Math. Res. Lett., 3(1) :1–17, 1996. | MR 1393378 | Zbl 0865.58048

[CQ97a] Sun-Yung A. Chang and Jie Qing. The zeta functional determinants on manifolds with boundary. I. The formula. J. Funct. Anal., 147(2) :327–362, 1997. | MR 1454485 | Zbl 0914.58039

[CQ97b] Sun-Yung A. Chang and Jie Qing. The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set. J. Funct. Anal., 147(2) :363–399, 1997. | MR 1454486 | Zbl 0914.58040

[CQY] Sun-Yung A. Chang, Jie Qing, and Paul C. Yang. On the topology of conformally compact einstein 4-manifolds. J. Reine Angew. Math., A paraître.

[CQY00] Sun-Yung A. Chang, Jie Qing, and Paul C. Yang. Compactification of a class of conformally flat 4-manifold. Invent. Math., 142(1) :65–93, 2000. | MR 1784799 | Zbl 0990.53026

[CX96] Roger Chen and Xingwang Xu. Compactness of isospectral conformal metrics and isospectral potentials on a 4-manifold. Duke Math. J., 84(1) :131–154, 1996. | MR 1394750 | Zbl 0856.53032

[CXY98] Sun-Yung A. Chang, Xingwang Xu, and Paul C. Yang. A perturbation result for prescribing mean curvature. Math. Ann., 310(3) :473–496, 1998. | MR 1612266 | Zbl 0893.35033

[CY87] Sun-Yung Alice Chang and Paul C. Yang. Prescribing Gaussian curvature on S 2 . Acta Math., 159(3-4) :215–259, 1987. | MR 908146 | Zbl 0636.53053

[CY88] Sun-Yung A. Chang and Paul C. Yang. Conformal deformation of metrics on S 2 . J. Differential Geom., 27(2) :259–296, 1988. | MR 925123 | Zbl 0649.53022

[CY90] Sun-Yung A. Chang and Paul C.-P. Yang. Isospectral conformal metrics on 3-manifolds. J. Amer. Math. Soc., 3(1) :117–145, 1990. | MR 1015647 | Zbl 0701.58056

[CY93] Sun-Yung Alice Chang and Paul C. Yang. Addendum to : “A perturbation result in prescribing scalar curvature on S n ” [Duke Math. J. 64 (1991), no. 1, 27–69 ; MR 92m :53063]. Duke Math. J., 71(1) :333–335, 1993. | Zbl 0796.53045

[CY99] Sun-Yung A. Chang and Paul C. Yang. On a fourth order curvature invariant. In Spectral problems in geometry and arithmetic (Iowa City, IA, 1997), volume 237 of Contemp. Math., pages 9–28. Amer. Math. Soc., Providence, RI, 1999. | MR 1710786 | Zbl 0982.53035

[CY00] Sun-Yung A. Chang and Paul C. Yang. Fourth order equations in conformal geometry. In Global analysis and harmonic analysis (Marseille-Luminy, 1999), volume 4 of Sémin. Congr., pages 155–165. Soc. Math. France, Paris, 2000. | MR 1822359 | Zbl 0997.35016

[CY89a] Sun-Yung A. Chang and Paul C. Yang. Compactness of isospectral conformal metrics on S 3 . Comment. Math. Helv., 64(3) :363–374, 1989. | EuDML 140160 | MR 998854 | Zbl 0679.53038

[CY91a] Sun-Yung A. Chang and Paul C. Yang. A perturbation result in prescribing scalar curvature on S n . Duke Math. J., 64(1) :27–69, 1991. | MR 1131392 | Zbl 0739.53027

[CY95a] Luis A. Caffarelli and Yi Song Yang. Vortex condensation in the Chern-Simons Higgs model : an existence theorem. Comm. Math. Phys., 168(2) :321–336, 1995. | MR 1324400 | Zbl 0846.58063

[CY97a] Sun-Yung A. Chang and Paul C. Yang. Determinants and extremal metrics in conformal geometry. In Geometry from the Pacific Rim (Singapore, 1994), pages 37–57. de Gruyter, Berlin, 1997. | MR 1468237 | Zbl 0898.58008

[CY89b] Sun-Yung A. Chang and Paul C. Yang. The conformal deformation equation and isospectral set of conformal metrics. In Recent developments in geometry (Los Angeles, CA, 1987), volume 101 of Contemp. Math., pages 165–178. Amer. Math. Soc., Providence, RI, 1989. | MR 1034980 | Zbl 0698.53024

[CY91b] Sun-Yung A. Chang and Paul C. Yang. Spectral invariants of conformal metrics. In Harmonic analysis (Sendai, 1990), ICM-90 Satell. Conf. Proc., pages 51–60. Springer, Tokyo, 1991. | MR 1261428 | Zbl 0804.53066

[CY95b] Sun-Yung A. Chang and Paul C. Yang. Extremal metrics of zeta function determinants on 4-manifolds. Ann. of Math. (2), 142(1) :171–212, 1995. | MR 1338677 | Zbl 0842.58011

[CY97b] Sun-Yung A. Chang and Paul C. Yang. On uniqueness of solutions of nth order differential equations in conformal geometry. Math. Res. Lett., 4(1) :91–102, 1997. | MR 1432813 | Zbl 0903.53027

[DD01] Zindine Djadli and Olivier Druet. Extremal functions for optimal Sobolev inequalities on compact manifolds. Calc. Var. Partial Differential Equations, 12(1) :59–84, 2001. | MR 1808107 | Zbl 0998.58008

[Der83] Andrzej Derdziński. Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math., 49(3) :405–433, 1983. | EuDML 89617 | Numdam | MR 707181 | Zbl 0527.53030

[DHL00] Zindine Djadli, Emmanuel Hebey, and Michel Ledoux. Paneitz-type operators and applications. Duke Math. J., 104(1) :129–169, 2000. | MR 1769728 | Zbl 0998.58009

[DJ02] Zindine Djadli and Antoinette Jourdain. Nodal solutions for scalar curvature type equations with perturbation terms on compact Riemannian manifolds. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 5(1) :205–226, 2002. | EuDML 194568 | MR 1881932 | Zbl 02217074

[Dja] Zindine Djadli. Existence result for the mean field problem on Riemann surfaces of all genuses. Préprint.

[Dja99a] Zindine Djadli. Nonlinear elliptic equations involving critical Sobolev exponent on compact Riemannian manifolds in presence of symmetries. Rev. Mat. Complut., 12(1) :201–229, 1999. | EuDML 44435 | MR 1698904 | Zbl 0968.58016

[Dja99b] Zindine Djadli. Nonlinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. Calc. Var. Partial Differential Equations, 8(4) :293–326, 1999. | MR 1700267 | Zbl 0953.58017

[DJLW99] Weiyue Ding, Jürgen Jost, Jiayu Li, and Guofang Wang. Existence results for mean field equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 16(5) :653–666, 1999. | EuDML 78478 | Numdam | MR 1712560 | Zbl 0937.35055

[DMa] Zindine Djadli and Andrea Malchiodi. Existence of conformal metrics with constant Q-curvature. préprint, ArXiv : math.AP/0410141, À paraître dans Annals of Mathematics.

[DMA02] Zindine Djadli, Andrea Malchiodi, and Mohameden Ould Ahmedou. Prescribing a fourth order conformal invariant on the standard sphere. I. A perturbation result. Commun. Contemp. Math., 4(3) :375–408, 2002. | MR 1918751 | Zbl 1023.58020

[DMb] Zindine Djadli and Andrea Malchiodi. A fourth order uniformization theorem on some four manifolds with large total Q-curvature. C. R. Acad. Sci. Paris, Ser. I 340 (2005), 341-346. | MR 2127107 | Zbl 1076.53046

[DMOA] Zindine Djadli, Andrea Malchiodi, and Mohameden Ould Ahmedou. The prescribed boundary men curvature problem on B 4 . préprint. | MR 2095819 | Zbl 1108.35070

[DMOA02] Zindine Djadli, Andrea Malchiodi, and Mohameden Ould Ahmedou. Prescribing a fourth order conformal invariant on the standard sphere, part ii : Blow-up analysis and applications. Annali della Scuola Normale Superiore di Pisa, 1(2) :387–434, 2002. | EuDML 84475 | Numdam | MR 1991145 | Zbl 02217249

[DMOA03] Zindine Djadli, Andrea Malchiodi, and Mohameden Ould Ahmedou. Prescribing scalar and boundary mean curvature on the three dimensional half sphere. J. Geom. Anal., 13(2) :233–267, 2003. | MR 1967027 | Zbl 1092.53028

[EG] José Escobar and G. Garcia. Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary. préprint. | MR 2054619 | Zbl 1056.53026

[ES86] José F. Escobar and Richard M. Schoen. Conformal metrics with prescribed scalar curvature. Invent. Math., 86(2) :243–254, 1986. | EuDML 143395 | MR 856845 | Zbl 0628.53041

[Esc88] José Escobar. Sharp constant in a sobolev trace inequality. Indiana University Journal, (37) :687–698, 1988. | MR 962929 | Zbl 0666.35014

[Esc92] José Escobar. Conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary. Annals of Mathematics, (136) :1–50, 1992. | MR 1173925 | Zbl 0766.53033

[Esc96] José Escobar. Conformal metrics with prescribed mean curvature on the boundary. Calculus of Variations and Partial Differential Equations, (4) :559–592, 1996. | MR 1416000 | Zbl 0867.53034

[Gur94] Matthew J. Gursky. Locally conformally flat four- and six-manifolds of positive scalar curvature and positive Euler characteristic. Indiana Univ. Math. J., 43(3) :747–774, 1994. | MR 1305946 | Zbl 0832.53032

[Gur98] Matthew J. Gursky. The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics. Ann. of Math. (2), 148(1) :315–337, 1998. | MR 1652920 | Zbl 0949.53025

[Gur99] Matthew J. Gursky. The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE. Comm. Math. Phys., 207(1) :131–143, 1999. | MR 1724863 | Zbl 0988.58013

[Gur00a] Matthew J. Gursky. Four-manifolds with δW + =0 and Einstein constants of the sphere. Math. Ann., 318(3) :417–431, 2000. | MR 1800764 | Zbl 1034.53032

[Gur00b] Matthew J. Gursky. Some local and non-local variational problems in Riemannian geometry. In Global analysis and harmonic analysis (Marseille-Luminy, 1999), volume 4 of Sémin. Congr., pages 167–177. Soc. Math. France, Paris, 2000. | MR 1822360 | Zbl 0996.53027

[GV01] Matthew J. Gursky and Jeff A. Viaclovsky. A new variational characterization of three-dimensional space forms. Invent. Math., 145(2) :251–278, 2001. | MR 1872547 | Zbl 1006.58008

[GV03] Matthew J. Gursky and Jeff A. Viaclovsky. A fully nonlinear equation on four-manifolds with positive scalar curvature. J. Differential Geom., 63(1) :131–154, 2003. | MR 2015262 | Zbl 1070.53018

[Ili82] Saïd Ilias. Sur une inégalité de Sobolev. C. R. Acad. Sci. Paris Sér. I Math., 294(22) :731–734, 1982. | MR 667236 | Zbl 0504.53038

[Ili83] Saïd Ilias. Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes. Ann. Inst. Fourier (Grenoble), 33(2) :151–165, 1983. | Numdam | MR 699492 | Zbl 0528.53040

[Ili93] Saïd Ilias. Un nouveau résultat de pincement de la première valeur propre du laplacien et conjecture du diamètre pincé. Ann. Inst. Fourier (Grenoble), 43(3) :843–863, 1993. | Numdam | MR 1242618 | Zbl 0783.53024

[Ili96] Saïd Ilias. Inégalités de Sobolev et résultats d’isolement pour les applications harmoniques. J. Funct. Anal., 139(1) :182–195, 1996. | Zbl 0868.53027

[KV95] Maxim Kontsevich and Simeon Vishik. Geometry of determinants of elliptic operators. In Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993), volume 131 of Progr. Math., pages 173–197. Birkhäuser Boston, Boston, MA, 1995. | MR 1373003 | Zbl 0920.58061

[KW74a] Jerry L. Kazdan and F. W. Warner. Curvature functions for compact 2-manifolds. Ann. of Math. (2), 99 :14–47, 1974. | MR 343205 | Zbl 0273.53034

[KW75a] Jerry L. Kazdan and F. W. Warner. Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. of Math. (2), 101 :317–331, 1975. | MR 375153 | Zbl 0297.53020

[KW74b] Jerry L. Kazdan and F. W. Warner. Curvature functions for open 2-manifolds. Ann. of Math. (2), 99 :203–219, 1974. | MR 343206 | Zbl 0278.53031

[KW75b] Jerry L. Kazdan and F. W. Warner. Scalar curvature and conformal deformation of Riemannian structure. J. Differential Geometry, 10 :113–134, 1975. | MR 365409 | Zbl 0296.53037

[LeB86] Claude LeBrun. On the topology of self-dual 4-manifolds. Proc. Amer. Math. Soc., 98(4) :637–640, 1986. | MR 861766 | Zbl 0606.53029

[LeB95] Claude LeBrun. Einstein metrics and Mostow rigidity. Math. Res. Lett., 2(1) :1–8, 1995. | MR 1312972 | Zbl 0974.53035

[Li93] Yan Yan Li. Prescribing scalar curvature on S 3 ,S 4 and related problems. J. Funct. Anal., 118(1) :43–118, 1993. | MR 1245597 | Zbl 0790.53040

[Li95] Yan Yan Li. Prescribing scalar curvature on S n and related problems. I. J. Differential Equations, 120(2) :319–410, 1995. | MR 1347349 | Zbl 0827.53039

[Li96] Yanyan Li. Prescribing scalar curvature on S n and related problems. II. Existence and compactness. Comm. Pure Appl. Math., 49(6) :541–597, 1996. | MR 1383201 | Zbl 0849.53031

[Li99] Yan Yan Li. Harnack type inequality : the method of moving planes. Comm. Math. Phys., 200(2) :421–444, 1999. | MR 1673972 | Zbl 0928.35057

[Li00a] Peter Li. Curvature and function theory on Riemannian manifolds. In Surveys in differential geometry, Surv. Differ. Geom., VII, pages 375–432. International Press, Somerville, MA, 2000. | MR 1919432 | Zbl 1066.53084

[Li00b] Yan Yan Li. Best Sobolev inequalities on Riemannian manifolds. In Differential equations and mathematical physics (Birmingham, AL, 1999), volume 16 of AMS/IP Stud. Adv. Math., pages 273–278. Amer. Math. Soc., Providence, RI, 2000. | MR 1764757 | Zbl 01780469

[Lic58] André Lichnerowicz. Géométrie des groupes de transformations. Travaux et Recherches Mathématiques, III. Dunod, Paris, 1958. | MR 124009 | Zbl 0096.16001

[Lie83] Elliott H. Lieb. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2), 118(2) :349–374, 1983. | MR 717827 | Zbl 0527.42011

[Lin98] Chang-Shou Lin. A classification of solutions of a conformally invariant fourth order equation in R n . Comment. Math. Helv., 73(2) :206–231, 1998. | MR 1611691 | Zbl 0933.35057

[Lin00] Chang-Shou Lin. Topological degree for mean field equations on S 2 . Duke Math. J., 104(3) :501–536, 2000. | MR 1781481 | Zbl 0964.35038

[Lio85] P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana, 1(1) :145–201, 1985. | MR 834360 | Zbl 0704.49005

[LP87] John M. Lee and Thomas H. Parker. The Yamabe problem. Bull. Amer. Math. Soc. (N.S.), 17(1) :37–91, 1987. | MR 888880 | Zbl 0633.53062

[Mal] Andrea Malchiodi. Compactness of solutions to some geometric fourth-order equations. J. Reine Angew. Math., à paraître. | MR 2248155 | Zbl 1098.53032

[Mar98] Christophe Margerin. A sharp characterization of the smooth 4-sphere in curvature terms. Comm. Anal. Geom., 6(1) :21–65, 1998. | MR 1619838 | Zbl 0966.53022

[Mos71] J. Moser. A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J., 20 :1077–1092, 1970/71. | MR 301504 | Zbl 0213.13001

[MP49] S. Minakshisundaram and AA. Pleijel. Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canadian J. Math., 1 :242–256, 1949. | MR 31145 | Zbl 0041.42701

[MP94] Carlo Marchioro and Mario Pulvirenti. Mathematical theory of incompressible nonviscous fluids, volume 96 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994. | MR 1245492 | Zbl 0789.76002

[MS67] H. P. McKean, Jr. and I. M. Singer. Curvature and the eigenvalues of the Laplacian. J. Differential Geometry, 1(1) :43–69, 1967. | MR 217739 | Zbl 0198.44301

[Oki01] K. Okikiolu. Critical metrics for the determinant of the Laplacian in odd dimensions. Ann. of Math. (2), 153(2) :471–531, 2001. | MR 1829756 | Zbl 0985.58013

[Oki95a] Kate Okikiolu. The Campbell-Hausdorff theorem for elliptic operators and a related trace formula. Duke Math. J., 79(3) :687–722, 1995. | MR 1355181 | Zbl 0854.35137

[Oki95b] Kate Okikiolu. The multiplicative anomaly for determinants of elliptic operators. Duke Math. J., 79(3) :723–750, 1995. | MR 1355182 | Zbl 0851.58048

[Ono82] E. Onofri. On the positivity of the effective action in a theory of random surfaces. Comm. Math. Phys., 86(3) :321–326, 1982. | MR 677001 | Zbl 0506.47031

[OPS88a] B. Osgood, R. Phillips, and P. Sarnak. Compact isospectral sets of surfaces. J. Funct. Anal., 80(1) :212–234, 1988. | MR 960229 | Zbl 0653.53021

[OPS88b] B. Osgood, R. Phillips, and P. Sarnak. Extremals of determinants of Laplacians. J. Funct. Anal., 80(1) :148–211, 1988. | MR 960228 | Zbl 0653.53022

[Pan83] S. Paneitz. A quartic conformally covariant diferential operator for arbitrary pseudo-riemannian manifolds. Préprint, 1883.

[Pol81] A. M. Polyakov. Quantum geometry of bosonic strings. Phys. Lett. B, 103(3) :207–210, 1981. | MR 623209

[Pol96] Alexander Polden. Curves and surfaces of least total curvature and fourth-order flows. Dissertation - Universität Tubingen, 1996.

[Ric94] Ken Richardson. Critical points of the determinant of the Laplace operator. J. Funct. Anal., 122(1) :52–83, 1994. | MR 1274583 | Zbl 0805.58063

[Ros97] Steven Rosenberg. The Laplacian on a Riemannian manifold, volume 31 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. An introduction to analysis on manifolds. | MR 1462892 | Zbl 0868.58074

[RS71] D. B. Ray and I. M. Singer. R-torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7 :145–210, 1971. | MR 295381 | Zbl 0239.58014

[Sch84] Richard Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom., 20(2) :479–495, 1984. | MR 788292 | Zbl 0576.53028

[Sch89] Richard M. Schoen. Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In Topics in calculus of variations (Montecatini Terme, 1987), volume 1365 of Lecture Notes in Math., pages 120–154. Springer, Berlin, 1989. | MR 994021 | Zbl 0702.49038

[Sch91a] Richard M. Schoen. On the number of constant scalar curvature metrics in a conformal class. In Differential geometry, volume 52 of Pitman Monogr. Surveys Pure Appl. Math., pages 311–320. Longman Sci. Tech., Harlow, 1991. | MR 1173050 | Zbl 0733.53021

[Sch91b] Richard M. Schoen. A report on some recent progress on nonlinear problems in geometry. In Surveys in differential geometry (Cambridge, MA, 1990), pages 201–241. Lehigh Univ., Bethlehem, PA, 1991. | MR 1144528 | Zbl 0791.53003

[Sob38] S.L. Sobolev. Sur un théorème d’analyse fonctionnelle. Math. Sb., 46 :471–496, 1938. | Zbl 0022.14803

[ST98] Michael Struwe and Gabriella Tarantello. On multivortex solutions in Chern-Simons gauge theory. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(1) :109–121, 1998. | MR 1619043 | Zbl 0912.58046

[Str88] Michael Struwe. The existence of surfaces of constant mean curvature with free boundaries. Acta Math., 160(1-2) :19–64, 1988. | MR 926524 | Zbl 0646.53005

[Str90] Michael Struwe. Variational methods. Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems. | MR 1078018 | Zbl 0746.49010

[SY81] Richard Schoen and Shing Tung Yau. Proof of the positive mass theorem. II. Comm. Math. Phys., 79(2) :231–260, 1981. | MR 612249 | Zbl 0494.53028

[SY88] R. Schoen and S.-T. Yau. Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math., 92(1) :47–71, 1988. | MR 931204 | Zbl 0658.53038

[SY94] R. Schoen and S.-T. Yau. Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu, Translated from the Chinese by Ding and S. Y. Cheng, Preface translated from the Chinese by Kaising Tso. | MR 1333601 | Zbl 0830.53001

[SY79a] R. Schoen and S. T. Yau. On the structure of manifolds with positive scalar curvature. Manuscripta Math., 28(1-3) :159–183, 1979. | MR 535700 | Zbl 0423.53032

[SY79b] Richard Schoen and Shing Tung Yau. On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys., 65(1) :45–76, 1979. | MR 526976 | Zbl 0405.53045

[SY79c] Richard Schoen and Shing Tung Yau. On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys., 65(1) :45–76, 1979. | MR 526976 | Zbl 0405.53045

[SY79d] Richard Schoen and Shing Tung Yau. Positivity of the total mass of a general space-time. Phys. Rev. Lett., 43(20) :1457–1459, 1979. | MR 547753

[SY79e] Richard M. Schoen and Shing Tung Yau. Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity. Proc. Nat. Acad. Sci. U.S.A., 76(3) :1024–1025, 1979. | MR 524327 | Zbl 0415.53029

[SZ96] Richard Schoen and Dong Zhang. Prescribed scalar curvature on the n-sphere. Calc. Var. Partial Differential Equations, 4(1) :1–25, 1996. | MR 1379191 | Zbl 0843.53037

[Tal76] Giorgio Talenti. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4), 110 :353–372, 1976. | MR 463908 | Zbl 0353.46018

[Tar96] Gabriella Tarantello. Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys., 37(8) :3769–3796, 1996. | MR 1400816 | Zbl 0863.58081

[Tru67] Neil S. Trudinger. On imbeddings into Orlicz spaces and some applications. J. Math. Mech., 17 :473–483, 1967. | MR 216286 | Zbl 0163.36402

[Tru68] Neil S. Trudinger. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3), 22 :265–274, 1968. | Numdam | MR 240748 | Zbl 0159.23801

[UV00] Karen K. Uhlenbeck and Jeff A. Viaclovsky. Regularity of weak solutions to critical exponent variational equations. Math. Res. Lett., 7(5-6) :651–656, 2000. | MR 1809291 | Zbl 0977.58020

[Via00] Jeff A. Viaclovsky. Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J., 101(2) :283–316, 2000. | MR 1738176 | Zbl 0990.53035

[WX99] Juncheng Wei and Xingwang Xu. Classification of solutions of higher order conformally invariant equations. Math. Ann., 313(2) :207–228, 1999. | MR 1679783 | Zbl 0940.35082

[Yam60] Hidehiko Yamabe. On a deformation of Riemannian structures on compact manifolds. Osaka Math. J., 12 :21–37, 1960. | MR 125546 | Zbl 0096.37201

[Yan00] Yisong Yang. On a system of nonlinear elliptic equations arising in theoretical physics. J. Funct. Anal., 170(1) :1–36, 2000. | MR 1736194 | Zbl 0942.35069

[YY80] Paul C. Yang and Shing Tung Yau. Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7(1) :55–63, 1980. | Numdam | MR 577325 | Zbl 0446.58017