@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}, address = {Grenoble}, volume = {23}, year = {2004-2005}, doi = {10.5802/tsg.231}, zbl = {1103.53019}, mrnumber = {2270223}, language = {fr}, url = {http://www.numdam.org/articles/10.5802/tsg.231/} }
TY - JOUR AU - Djadli, Zindine TI - Opérateurs géométriques et géométrie conforme JO - Séminaire de théorie spectrale et géométrie PY - 2004-2005 SP - 49 EP - 103 VL - 23 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.231/ DO - 10.5802/tsg.231 LA - fr ID - TSG_2004-2005__23__49_0 ER -
Djadli, Zindine. Opérateurs géométriques et géométrie conforme. Séminaire de théorie spectrale et géométrie, Volume 23 (2004-2005), pp. 49-103. doi : 10.5802/tsg.231. http://www.numdam.org/articles/10.5802/tsg.231/
[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 | Zbl
[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 | Zbl
[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
[Ada88] David R. Adams. A sharp inequality of J. Moser for higher order derivatives. Ann. of Math. (2), 128(2) :385–398, 1988. | MR | Zbl
[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 | Zbl
[AL99a] Thierry Aubin and Yan Yan Li. On the best Sobolev inequality. J. Math. Pures Appl. (9), 78(4) :353–387, 1999. | MR | Zbl
[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
[AM99] Antonio Ambrosetti and Andrea Malchiodi. A multiplicity result for the Yamabe problem on . J. Funct. Anal., 168(2) :529–561, 1999. | MR | Zbl
[Aub70a] Thierry Aubin. Métriques riemanniennes et courbure. J. Differential Geometry, 4 :383–424, 1970. | MR | Zbl
[Aub70b] Thierry Aubin. Sur la fonction exponentielle. C. R. Acad. Sci. Paris Sér. A-B, 270 :A1514–A1516, 1970. | MR | Zbl
[Aub74] Thierry Aubin. Fonction de Green et valeurs propres du laplacien. J. Math. Pures Appl. (9), 53 :347–371, 1974. | MR | Zbl
[Aub75a] Thierry Aubin. Équations différentielles non linéaires. Bull. Sci. Math. (2), 99(4) :201–210, 1975. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[Aub75e] Thierry Aubin. Problèmes isopérimétriques et espaces de Sobolev. C. R. Acad. Sci. Paris Sér. A-B, 280(5) :Aii, A279–A281, 1975. | MR | Zbl
[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 | Zbl
[Aub76b] Thierry Aubin. Espaces de Sobolev sur les variétés riemanniennes. Bull. Sci. Math. (2), 100(2) :149–173, 1976. | MR | Zbl
[Aub76c] Thierry Aubin. Problèmes isopérimétriques et espaces de Sobolev. J. Differential Geometry, 11(4) :573–598, 1976. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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. | Numdam | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[Aub94b] Thierry Aubin. Sur le problème de la courbure scalaire prescrite. Bull. Sci. Math., 118(5) :465–474, 1994. | MR | Zbl
[Aub98] Thierry Aubin. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. | MR | Zbl
[Aub01] Thierry Aubin. A course in differential geometry, volume 27 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[BM01] Massimiliano Berti and Andrea Malchiodi. Non-compactness and multiplicity results for the Yamabe problem on . J. Funct. Anal., 180(1) :210–241, 2001. | MR | Zbl
[BO91] Thomas P. Branson and Bent Orsted. Explicit functional determinants in four dimensions. Proc. Amer. Math. Soc., 113(3) :669–682, 1991. | MR | Zbl
[Bou81] Jean-Pierre Bourguignon. Les variétés de dimension à signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math., 63(2) :263–286, 1981. | EuDML | MR | Zbl
[Bra85] Thomas P. Branson. Differential operators canonically associated to a conformal structure. Math. Scand., 57(2) :293–345, 1985. | EuDML | MR | Zbl
[Bra87] Thomas P. Branson. Group representations arising from Lorentz conformal geometry. J. Funct. Anal., 74(2) :199–291, 1987. | MR | Zbl
[Bra95] Thomas P. Branson. Sharp inequalities, the functional determinant, and the complementary series. Trans. Amer. Math. Soc., 347(10) :3671–3742, 1995. | MR | Zbl
[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 | Zbl
[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 | Zbl
[CGW94] S.-Y. A. Chang, M. Gursky, and T. Wolff. Lack of compactness in conformal metrics with curvature. J. Geom. Anal., 4(2) :143–153, 1994. | MR | Zbl
[CGY93] Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang. The scalar curvature equation on - and -spheres. Calc. Var. Partial Differential Equations, 1(2) :205–229, 1993. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Numdam | MR | Zbl
[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 | Numdam | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[CQ96] Sun-Yung A. Chang and Jie Qing. Zeta functional determinants on manifolds with boundary. Math. Res. Lett., 3(1) :1–17, 1996. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[CX96] Roger Chen and Xingwang Xu. Compactness of isospectral conformal metrics and isospectral potentials on a -manifold. Duke Math. J., 84(1) :131–154, 1996. | MR | Zbl
[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 | Zbl
[CY87] Sun-Yung Alice Chang and Paul C. Yang. Prescribing Gaussian curvature on . Acta Math., 159(3-4) :215–259, 1987. | MR | Zbl
[CY88] Sun-Yung A. Chang and Paul C. Yang. Conformal deformation of metrics on . J. Differential Geom., 27(2) :259–296, 1988. | MR | Zbl
[CY89a] Sun-Yung A. Chang and Paul C. Yang. Compactness of isospectral conformal metrics on . Comment. Math. Helv., 64(3) :363–374, 1989. | EuDML | MR | Zbl
[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 | Zbl
[CY90] Sun-Yung A. Chang and Paul C.-P. Yang. Isospectral conformal metrics on -manifolds. J. Amer. Math. Soc., 3(1) :117–145, 1990. | MR | Zbl
[CY91a] Sun-Yung A. Chang and Paul C. Yang. A perturbation result in prescribing scalar curvature on . Duke Math. J., 64(1) :27–69, 1991. | MR | Zbl
[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 | Zbl
[CY93] Sun-Yung Alice Chang and Paul C. Yang. Addendum to : “A perturbation result in prescribing scalar curvature on ” [Duke Math. J. 64 (1991), no. 1, 27–69 ; MR 92m :53063]. Duke Math. J., 71(1) :333–335, 1993. | Zbl
[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 | Zbl
[CY95b] Sun-Yung A. Chang and Paul C. Yang. Extremal metrics of zeta function determinants on -manifolds. Ann. of Math. (2), 142(1) :171–212, 1995. | MR | Zbl
[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 | Zbl
[CY97b] Sun-Yung A. Chang and Paul C. Yang. On uniqueness of solutions of th order differential equations in conformal geometry. Math. Res. Lett., 4(1) :91–102, 1997. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[Der83] Andrzej Derdziński. Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math., 49(3) :405–433, 1983. | EuDML | Numdam | MR | Zbl
[DHL00] Zindine Djadli, Emmanuel Hebey, and Michel Ledoux. Paneitz-type operators and applications. Duke Math. J., 104(1) :129–169, 2000. | MR | Zbl
[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 | MR | Zbl
[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 | MR | Zbl
[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 | Zbl
[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 | Numdam | MR | Zbl
[DMa] Zindine Djadli and Andrea Malchiodi. Existence of conformal metrics with constant -curvature. préprint, ArXiv : math.AP/0410141, À paraître dans Annals of Mathematics.
[DMb] Zindine Djadli and Andrea Malchiodi. A fourth order uniformization theorem on some four manifolds with large total -curvature. C. R. Acad. Sci. Paris, Ser. I 340 (2005), 341-346. | MR | Zbl
[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 | Zbl
[DMOA] Zindine Djadli, Andrea Malchiodi, and Mohameden Ould Ahmedou. The prescribed boundary men curvature problem on . préprint. | MR | Zbl
[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 | Numdam | MR | Zbl
[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 | Zbl
[EG] José Escobar and G. Garcia. Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary. préprint. | MR | Zbl
[ES86] José F. Escobar and Richard M. Schoen. Conformal metrics with prescribed scalar curvature. Invent. Math., 86(2) :243–254, 1986. | EuDML | MR | Zbl
[Esc88] José Escobar. Sharp constant in a sobolev trace inequality. Indiana University Journal, (37) :687–698, 1988. | MR | Zbl
[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 | Zbl
[Esc96] José Escobar. Conformal metrics with prescribed mean curvature on the boundary. Calculus of Variations and Partial Differential Equations, (4) :559–592, 1996. | MR | Zbl
[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 | Zbl
[Gur98] Matthew J. Gursky. The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics. Ann. of Math. (2), 148(1) :315–337, 1998. | MR | Zbl
[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 | Zbl
[Gur00a] Matthew J. Gursky. Four-manifolds with and Einstein constants of the sphere. Math. Ann., 318(3) :417–431, 2000. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[Ili82] Saïd Ilias. Sur une inégalité de Sobolev. C. R. Acad. Sci. Paris Sér. I Math., 294(22) :731–734, 1982. | MR | Zbl
[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 | Zbl
[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 | Zbl
[Ili96] Saïd Ilias. Inégalités de Sobolev et résultats d’isolement pour les applications harmoniques. J. Funct. Anal., 139(1) :182–195, 1996. | Zbl
[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 | Zbl
[KW74a] Jerry L. Kazdan and F. W. Warner. Curvature functions for compact -manifolds. Ann. of Math. (2), 99 :14–47, 1974. | MR | Zbl
[KW74b] Jerry L. Kazdan and F. W. Warner. Curvature functions for open -manifolds. Ann. of Math. (2), 99 :203–219, 1974. | MR | Zbl
[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 | Zbl
[KW75b] Jerry L. Kazdan and F. W. Warner. Scalar curvature and conformal deformation of Riemannian structure. J. Differential Geometry, 10 :113–134, 1975. | MR | Zbl
[LeB86] Claude LeBrun. On the topology of self-dual -manifolds. Proc. Amer. Math. Soc., 98(4) :637–640, 1986. | MR | Zbl
[LeB95] Claude LeBrun. Einstein metrics and Mostow rigidity. Math. Res. Lett., 2(1) :1–8, 1995. | MR | Zbl
[Li93] Yan Yan Li. Prescribing scalar curvature on and related problems. J. Funct. Anal., 118(1) :43–118, 1993. | MR | Zbl
[Li95] Yan Yan Li. Prescribing scalar curvature on and related problems. I. J. Differential Equations, 120(2) :319–410, 1995. | MR | Zbl
[Li96] Yanyan Li. Prescribing scalar curvature on and related problems. II. Existence and compactness. Comm. Pure Appl. Math., 49(6) :541–597, 1996. | MR | Zbl
[Li99] Yan Yan Li. Harnack type inequality : the method of moving planes. Comm. Math. Phys., 200(2) :421–444, 1999. | MR | Zbl
[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 | Zbl
[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 | Zbl
[Lic58] André Lichnerowicz. Géométrie des groupes de transformations. Travaux et Recherches Mathématiques, III. Dunod, Paris, 1958. | MR | Zbl
[Lie83] Elliott H. Lieb. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. of Math. (2), 118(2) :349–374, 1983. | MR | Zbl
[Lin98] Chang-Shou Lin. A classification of solutions of a conformally invariant fourth order equation in . Comment. Math. Helv., 73(2) :206–231, 1998. | MR | Zbl
[Lin00] Chang-Shou Lin. Topological degree for mean field equations on . Duke Math. J., 104(3) :501–536, 2000. | MR | Zbl
[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 | Zbl
[LP87] John M. Lee and Thomas H. Parker. The Yamabe problem. Bull. Amer. Math. Soc. (N.S.), 17(1) :37–91, 1987. | MR | Zbl
[Mal] Andrea Malchiodi. Compactness of solutions to some geometric fourth-order equations. J. Reine Angew. Math., à paraître. | MR | Zbl
[Mar98] Christophe Margerin. A sharp characterization of the smooth -sphere in curvature terms. Comm. Anal. Geom., 6(1) :21–65, 1998. | MR | Zbl
[Mos71] J. Moser. A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J., 20 :1077–1092, 1970/71. | MR | Zbl
[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 | Zbl
[MP94] Carlo Marchioro and Mario Pulvirenti. Mathematical theory of incompressible nonviscous fluids, volume 96 of Applied Mathematical Sciences. Springer-Verlag, New York, 1994. | MR | Zbl
[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 | Zbl
[Oki95a] Kate Okikiolu. The Campbell-Hausdorff theorem for elliptic operators and a related trace formula. Duke Math. J., 79(3) :687–722, 1995. | MR | Zbl
[Oki95b] Kate Okikiolu. The multiplicative anomaly for determinants of elliptic operators. Duke Math. J., 79(3) :723–750, 1995. | MR | Zbl
[Oki01] K. Okikiolu. Critical metrics for the determinant of the Laplacian in odd dimensions. Ann. of Math. (2), 153(2) :471–531, 2001. | MR | Zbl
[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 | Zbl
[OPS88a] B. Osgood, R. Phillips, and P. Sarnak. Compact isospectral sets of surfaces. J. Funct. Anal., 80(1) :212–234, 1988. | MR | Zbl
[OPS88b] B. Osgood, R. Phillips, and P. Sarnak. Extremals of determinants of Laplacians. J. Funct. Anal., 80(1) :148–211, 1988. | MR | Zbl
[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
[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 | Zbl
[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 | Zbl
[RS71] D. B. Ray and I. M. Singer. -torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7 :145–210, 1971. | MR | Zbl
[Sch84] Richard Schoen. Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom., 20(2) :479–495, 1984. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[Sob38] S.L. Sobolev. Sur un théorème d’analyse fonctionnelle. Math. Sb., 46 :471–496, 1938. | Zbl
[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 | Zbl
[Str88] Michael Struwe. The existence of surfaces of constant mean curvature with free boundaries. Acta Math., 160(1-2) :19–64, 1988. | MR | Zbl
[Str90] Michael Struwe. Variational methods. Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[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
[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 | Zbl
[SY81] Richard Schoen and Shing Tung Yau. Proof of the positive mass theorem. II. Comm. Math. Phys., 79(2) :231–260, 1981. | MR | Zbl
[SY88] R. Schoen and S.-T. Yau. Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math., 92(1) :47–71, 1988. | MR | Zbl
[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 | Zbl
[SZ96] Richard Schoen and Dong Zhang. Prescribed scalar curvature on the -sphere. Calc. Var. Partial Differential Equations, 4(1) :1–25, 1996. | MR | Zbl
[Tal76] Giorgio Talenti. Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4), 110 :353–372, 1976. | MR | Zbl
[Tar96] Gabriella Tarantello. Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys., 37(8) :3769–3796, 1996. | MR | Zbl
[Tru67] Neil S. Trudinger. On imbeddings into Orlicz spaces and some applications. J. Math. Mech., 17 :473–483, 1967. | MR | Zbl
[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 | Zbl
[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 | Zbl
[Via00] Jeff A. Viaclovsky. Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J., 101(2) :283–316, 2000. | MR | Zbl
[WX99] Juncheng Wei and Xingwang Xu. Classification of solutions of higher order conformally invariant equations. Math. Ann., 313(2) :207–228, 1999. | MR | Zbl
[Yam60] Hidehiko Yamabe. On a deformation of Riemannian structures on compact manifolds. Osaka Math. J., 12 :21–37, 1960. | MR | Zbl
[Yan00] Yisong Yang. On a system of nonlinear elliptic equations arising in theoretical physics. J. Funct. Anal., 170(1) :1–36, 2000. | MR | Zbl
[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 | Zbl
Cited by Sources: