Q-curvature flow with indefinite nonlinearity

Ma, Li; Liu, B.

doi:10.1016/j.crma.2010.02.014

Partial Differential Equations/Differential Geometry

Q-curvature flow with indefinite nonlinearity
[Flot de Q-courbure pour une non-linéarité indéfinie]

Présenté par : Paul Malliavin

Ma, Li ¹ ; Liu, B.¹

¹ Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Comptes Rendus. Mathématique, Tome 348 (2010) no. 7-8, pp. 403-406.

Résumé
Abstract

Dans cette Note on étudie le flot de Q-courbure sur $S^{4}$ dans le cas d'une non-linéarité indéfinie. Le résultat montre que le problème de la Q-courbure imposée sur $S^{4}$ 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 $Δ_{S^{4}} f \neq 0$ aux points selles et une condition supplémentaire du type condition non triviale sur le degré.

In this Note, we study Q-curvature flow on $S^{4}$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^{4}$ has a solution provided the prescribed non-negative Q-curvature f has its positive part, which possesses non-degenerate critical points such that $Δ_{S^{4}} f \neq 0$ at the saddle points and an extra condition such as a nontrivial degree counting condition.

Reçu le : 2008-12-09
Accepté le : 2010-02-03
Publié le : 2010-03-02

DOI : 10.1016/j.crma.2010.02.014

Affiliations des auteurs :

Ma, Li ¹ ; Liu, B. ¹

¹ Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

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Ma, Li; Liu, B. Q-curvature flow with indefinite nonlinearity. Comptes Rendus. Mathématique, Tome 348 (2010) no. 7-8, pp. 403-406. doi : 10.1016/j.crma.2010.02.014. http://www.numdam.org/articles/10.1016/j.crma.2010.02.014/

Bibliographie
Cité par

[1] Aubin, Th. Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer, Berlin, 1998 MR1636569 (99i:58001)

[2] Baird, Paul; Fardoun, Ali; Regbaoui, Rachid Q-curvature flow on 4-manifolds, Calc. Var., Volume 27 (2006) no. 1, pp. 75-104

[3] Beckner, W. Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. of Math., Volume 138 (1993), pp. 213-242

[4] Brendle, S. Global existence and convergence for a higher order flow in conformal geometry, Ann. of Math., Volume 158 (2003), pp. 323-343

[5] Chang, S.Y.A.; Yang, P. Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math., Volume 142 (1995), pp. 171-212

[6] Hong, M.C.; Ma, L. Curvature flow to Nirenberg problem, Arch. Math., Volume 242 (2010) (online)

[7] Lin, C.S. A classification of solutions of conformally invariant fourth order equation in $R^{n}$ , Comment. Math. Helv., Volume 73 (1998), pp. 206-231 (MR1611691, Zbl0933.35057)

[8] Ma, L. Three remarks on mean field equations, Pacific J. Math., Volume 242 (2009), pp. 167-171

[9] Malchiodi, A.; Struwe, M. Q-curvature flow on $S^{4}$ , J. Differential Geom., Volume 73 (2006), pp. 1-44

[10] Struwe, M. Curvature flows on surfaces, Ann. Sc. Norm. Sup. Pisa, Ser. V, Volume 1 (2002), pp. 247-274

[11] Wei, J.; Xu, X. On conformal deformation of metrics on $S^{n}$ , J. Funct. Anal., Volume 157 (1998), pp. 292-325

Cité par Sources :

^☆ The research is partially supported by the National Natural Science Foundation of China 10631020 and SRFDP 20090002110019.