Dans cette Note on étudie le flot de Q-courbure sur dans le cas d'une non-linéarité indéfinie. Le résultat montre que le problème de la Q-courbure imposée sur a une solution à condition que la Q-courbure non négative imposée f ait une partie strictement positive et des points critiques non dégénérés tels que aux points selles et une condition supplémentaire du type condition non triviale sur le degré.
In this Note, we study Q-curvature flow on with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on has a solution provided the prescribed non-negative Q-curvature f has its positive part, which possesses non-degenerate critical points such that at the saddle points and an extra condition such as a nontrivial degree counting condition.
Accepté le :
Publié le :
Li Ma 1 ; B. Liu 1
@article{CRMATH_2010__348_7-8_403_0, author = {Li Ma and B. Liu}, title = {Q-curvature flow with indefinite nonlinearity}, journal = {Comptes Rendus. Math\'ematique}, pages = {403--406}, publisher = {Elsevier}, volume = {348}, number = {7-8}, year = {2010}, doi = {10.1016/j.crma.2010.02.014}, language = {en}, }
Li Ma; B. Liu. Q-curvature flow with indefinite nonlinearity. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 403-406. doi : 10.1016/j.crma.2010.02.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.02.014/
[1] Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer, Berlin, 1998 MR1636569 (99i:58001)
[2] Q-curvature flow on 4-manifolds, Calc. Var., Volume 27 (2006) no. 1, pp. 75-104
[3] Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. of Math., Volume 138 (1993), pp. 213-242
[4] Global existence and convergence for a higher order flow in conformal geometry, Ann. of Math., Volume 158 (2003), pp. 323-343
[5] Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math., Volume 142 (1995), pp. 171-212
[6] Curvature flow to Nirenberg problem, Arch. Math., Volume 242 (2010) (online)
[7] A classification of solutions of conformally invariant fourth order equation in , Comment. Math. Helv., Volume 73 (1998), pp. 206-231 (MR1611691, Zbl0933.35057)
[8] Three remarks on mean field equations, Pacific J. Math., Volume 242 (2009), pp. 167-171
[9] Q-curvature flow on , J. Differential Geom., Volume 73 (2006), pp. 1-44
[10] Curvature flows on surfaces, Ann. Sc. Norm. Sup. Pisa, Ser. V, Volume 1 (2002), pp. 247-274
[11] On conformal deformation of metrics on , J. Funct. Anal., Volume 157 (1998), pp. 292-325
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☆ The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20090002110019.
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