We prove a Kazdan–Warner type identity involving the 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.
Nous prouvons une identité de type Kazdan–Warner reliant la -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.
@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},
}
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] 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] 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] A variational characterization for , Calc. Var. PDE, Volume 20 (2004) no. 4, pp. 399-402
[4] 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] 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 curvature equation on , in preparation
[7] 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] A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geometry, Volume 63 (2003) no. 1, pp. 131-154
[10] Prescribing Gaussian curvature on , Duke Math. J., Volume 61 (1990), pp. 679-703
[11] Curvature functions on compact 2-manifolds, Ann. of Math., Volume 99 (1974), pp. 14-47
[12] Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry, Volume 10 (1975), pp. 113-134
[13] Refined asymptotics for constant scalar curvature metrics with isolated singularities, Invent. Math., Volume 135 (1999) no. 2, pp. 233-272
[14] 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] The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, Volume 6 (1971), pp. 247-258
[17] Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J., Volume 26 (1977) no. 3, pp. 459-472
[18] 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] Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., Volume 101 (2000) no. 2, pp. 283-316
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