Comptes Rendus
Partial Differential Equations
A Kazdan–Warner type identity for the σk curvature
[Une identité de type Kazdan–Warner pour la σk-courbure]
Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 475-478.

Nous prouvons une identité de type Kazdan–Warner reliant la σk-courbure et un champ de vecteurs conforme sur une variété compacte. Notre méthode permet aussi de fournir une preuve unifiée pour les conditions nécessaires dans le problème de Christoffel–Minkowski.

We prove a Kazdan–Warner type identity involving the σk curvature and a conformal Killing vector field on a compact manifold. Our method also works to provide a unified proof for the necessary conditions in the Christoffel–Minkowski problem.

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2006.01.023

Zheng-Chao Han 1

1 Department of Mathematics, Rutgers University, 110, Frelinghuysen Road, Piscataway, NJ 08854, USA
@article{CRMATH_2006__342_7_475_0,
     author = {Zheng-Chao Han},
     title = {A {Kazdan{\textendash}Warner} type identity for the $ {\sigma }_{k}$ curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {475--478},
     publisher = {Elsevier},
     volume = {342},
     number = {7},
     year = {2006},
     doi = {10.1016/j.crma.2006.01.023},
     language = {en},
}
TY  - JOUR
AU  - Zheng-Chao Han
TI  - A Kazdan–Warner type identity for the $ {\sigma }_{k}$ curvature
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 475
EP  - 478
VL  - 342
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2006.01.023
LA  - en
ID  - CRMATH_2006__342_7_475_0
ER  - 
%0 Journal Article
%A Zheng-Chao Han
%T A Kazdan–Warner type identity for the $ {\sigma }_{k}$ curvature
%J Comptes Rendus. Mathématique
%D 2006
%P 475-478
%V 342
%N 7
%I Elsevier
%R 10.1016/j.crma.2006.01.023
%G en
%F CRMATH_2006__342_7_475_0
Zheng-Chao Han. A Kazdan–Warner type identity for the $ {\sigma }_{k}$ curvature. Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 475-478. doi : 10.1016/j.crma.2006.01.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.023/

[1] J.P. Bourguignon Invariants intégraux fonctionnels pour des équations aux dérivées partielles d'origine géométrique, Peñíscola, 1985 (Lecture Notes in Math.), Volume vol. 1209, Springer, Berlin (1986), pp. 100-108

[2] J.P. Bourguignon; J.P. Ezin Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc., Volume 301 (1987) no. 2, pp. 723-736

[3] S. Brendle; J. Viaclovsky A variational characterization for σn/2, Calc. Var. PDE, Volume 20 (2004) no. 4, pp. 399-402

[4] S.-Y. Chang Conformal invariants and partial differential equations, Bull. Amer. Math. Soc. (N.S.), Volume 42 (2005) no. 3, pp. 365-393 (Colloquium Lecture Notes, AMS, Phoenix 2004)

[5] S.-Y. Chang; P. Yang The Inequality of Moser and Trudinger and applications to conformal geometry, Comm. Pure Appl. Math., Volume LVI (August 2003) no. 8, pp. 1135-1150 (Special issue dedicated to the memory of Jurgen K. Moser)

[6] S.-Y.A. Chang, Z.-C. Han, P. Yang, A priori estimates for solutions of the prescribed σ2 curvature equation on S4, in preparation

[7] B. Guan; P. Guan Convex hypersurfaces of prescribed curvatures, Ann. of Math., Volume 156 (2002), pp. 655-673

[8] P. Guan, C.S. Lin, G. Wang, Schouten tensor and some topological properties, Comm. Anal. Geom., in press

[9] M. Gursky; J. Viaclovsky A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geometry, Volume 63 (2003) no. 1, pp. 131-154

[10] Z.-C. Han Prescribing Gaussian curvature on S2, Duke Math. J., Volume 61 (1990), pp. 679-703

[11] J.L. Kazdan; F. Warner Curvature functions on compact 2-manifolds, Ann. of Math., Volume 99 (1974), pp. 14-47

[12] J.L. Kazdan; F. Warner Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry, Volume 10 (1975), pp. 113-134

[13] N. Korevaar; R. Mazzeo; F. Pacard; R. Schoen Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., Volume 135 (1999) no. 2, pp. 233-272

[14] J. Lelong-Ferrand Transformations conformes et quasi-conformes des variétés riemanniennes compactes (démonstration de la conjecture de Lichnerowicz), Acad. Roy. Belg., Cl. Sci. Mémoire XXXIX, Volume 5 (1971)

[15] YanYan Li, On some conformally invariant fully nonlinear equations, in: Proceedings of the International Congress of Mathematicians, vol. 3, Beijing, 2002, pp. 177–184

[16] M. Obata The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, Volume 6 (1971), pp. 247-258

[17] R. Reilly Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., Volume 26 (1977) no. 3, pp. 459-472

[18] R. Schoen The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math., Volume XLI (1988), pp. 317-392

[19] J. Viaclovsky Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., Volume 101 (2000) no. 2, pp. 283-316

  • Willian Tokura; Marcelo Barboza; Elismar Batista; Priscila Kai On almost quotient Yamabe solitons, Glasgow Mathematical Journal, Volume 66 (2024) no. 3, p. 479 | DOI:10.1017/s0017089524000119
  • Shujun Shi; Peihe Wang; Tian Wu; Hua Zhu An Obata-type formula and the Liouville-type theorem for a class of K-Hessian equations on the sphere, Proceedings of the American Mathematical Society, Volume 152 (2024) no. 8, p. 3537 | DOI:10.1090/proc/16857
  • Hao Fang; Biao Ma; Wei Wei A Liouville's theorem for some Monge-Ampère type equations, Journal of Functional Analysis, Volume 285 (2023) no. 4, p. 109973 | DOI:10.1016/j.jfa.2023.109973
  • Willian Isao Tokura; Elismar Dias Batista; Priscila Marques Kai; Marcelo Bezerra Barboza Triviality results for quasi k-Yamabe solitons, Archiv der Mathematik, Volume 119 (2022) no. 6, p. 623 | DOI:10.1007/s00013-022-01795-1
  • YanYan Li; Han Lu; Siyuan Lu On the σ2-Nirenberg problem on S2, Journal of Functional Analysis, Volume 283 (2022) no. 10, p. 109606 | DOI:10.1016/j.jfa.2022.109606
  • Jonah A. J. Duncan; Luc Nguyen Local pointwise second derivative estimates for strong solutions to the σk-Yamabe equation on Euclidean domains, Calculus of Variations and Partial Differential Equations, Volume 60 (2021) no. 5 | DOI:10.1007/s00526-021-02051-0
  • YanYan Li; Luc Nguyen; Bo Wang The axisymmetric σ-Nirenberg problem, Journal of Functional Analysis, Volume 281 (2021) no. 9, p. 109198 | DOI:10.1016/j.jfa.2021.109198
  • Xiaoxiao Zhang; Aijin Lin Positive solutions of p-th Yamabe type equations on graphs, Frontiers of Mathematics in China, Volume 13 (2018) no. 6, p. 1501 | DOI:10.1007/s11464-018-0734-8
  • Yanyan Li; Paolo Mastrolia; Dario D. Monticelli On conformally invariant equations on, Nonlinear Analysis: Theory, Methods Applications, Volume 95 (2014), p. 339 | DOI:10.1016/j.na.2013.09.016
  • GIOVANNI CATINO; CARLO MANTEGAZZA; LORENZO MAZZIERI ON THE GLOBAL STRUCTURE OF CONFORMAL GRADIENT SOLITONS WITH NONNEGATIVE RICCI TENSOR, Communications in Contemporary Mathematics, Volume 14 (2012) no. 06, p. 1250045 | DOI:10.1142/s0219199712500459
  • Yanyan Li; Paolo Mastrolia; Dario D. Monticelli On conformally invariant equations on -II. Exponential invariance, Nonlinear Analysis: Theory, Methods Applications, Volume 75 (2012) no. 13, p. 5194 | DOI:10.1016/j.na.2012.04.036
  • Sun-Yung Alice Chang; Zheng-Chao Han; Paul Yang On the prescribing σ 2 curvature equation on S4, Calculus of Variations and Partial Differential Equations, Volume 40 (2011) no. 3-4, p. 539 | DOI:10.1007/s00526-010-0350-2
  • Bin Guo; Zheng-Chao Han; Haizhong Li Two Kazdan–Warner-type identities for the renormalized volume coefficients and the Gauss–Bonnet curvatures of a Riemannian metric, Pacific Journal of Mathematics, Volume 251 (2011) no. 2, p. 257 | DOI:10.2140/pjm.2011.251.257
  • Zheng-Chao Han; YanYan Li; Eduardo V. Teixeira Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities, Inventiones mathematicae, Volume 182 (2010) no. 3, p. 635 | DOI:10.1007/s00222-010-0274-7
  • Philippe Delanoë On the local Nirenberg problem forσk-type curvatures, Pacific Journal of Mathematics, Volume 234 (2008) no. 2, p. 289 | DOI:10.2140/pjm.2008.234.289

Cité par 15 documents. Sources : Crossref

Commentaires - Politique