Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
conformally flat; 4-manifold; variational characterization
conformally flat; 4-manifold; variational characterization
Summary:
In this paper, we give a new variational characterization of certain 4-manifolds. More precisely, let $R$ and $Ric$ denote the scalar curvature and Ricci curvature respectively of a Riemannian metric, we prove that if $(M^{4},g)$ is compact and locally conformally flat and $g$ is the critical point of the functional $$ F(g)=\int _{M^{4}}(aR^{2}+b|Ric|^{2})\,\mathrm {d}v_{g}\,,$$ where $$(a,b)\in \mathbb {R}^{2}\setminus L_{1}\cup L_{2}$$ $$L_{1}\colon 3a+b=0\,;\quad L_{2}\colon 6a-b+1=0\,,$$ then $(M^{4},g)$ is either scalar flat or a space form.
References:
[1] Berger, M.: Riemannian geometry during the second half of the twentieth century. 2000, University Lecture Series, Vol. 17, Amer. Math. Soc., Providence RI, MR 1729907 | Zbl 0944.53001
[2] Besse, A. L.: Einstein Manifolds. 1987, Erg. Math. Grenzgebiete 10, Springer-Verlag, Heidberg-Berlin-New York, MR 0867684 | Zbl 0613.53001
[3] Chang, S.-Y. A., Gursky, M. J., Yang, P.: An equation of Monge-Ampere type in conformal georemtry and four manifolds of positive Ricci curvature. Ann. Math., 155, 3, 2002, 709-787, DOI 10.2307/3062131 | MR 1923964
[4] Gursky, M. J.: Locally conformally flat 4 and 6 manifolds of positive scalar curvature and positive Euler characteristic. Indiana Univ. Math. J., 43, 1994, 747-774, DOI 10.1512/iumj.1994.43.43033 | MR 1305946
[5] Gursky, M. J.: Manifolds with $\delta W^{+}=0$ and Einstein constant of the sphere. Math. Ann., 318, 3, 2000, 417-431, DOI 10.1007/s002080000130 | MR 1800764 | Zbl 1034.53032
[6] Gursky, M. J., Viaclovsky, J. A.: A new variational characterization of three-dimensional space forms. Invent. Math., 145, 2001, 251-278, DOI 10.1007/s002220100147 | MR 1872547 | Zbl 1006.58008
[7] Hu, Z. J., Li, H. Z.: A new variational characterization of n-dimensional space forms. Trans. Amer. Math. Soc., 356, 8, 2004, 3005-3023, DOI 10.1090/S0002-9947-03-03486-X | MR 2052939 | Zbl 1058.53029
[8] Lanczos, C.: A remarkable property of the riemann-Christoffel tensor in four dimentions. Ann. Math., 39, 4, 1938, 842-850, DOI 10.2307/1968467 | MR 1503440
[9] LeBrun, C., Maskit, B.: On optimal 4-manifolds metrics. J. Georem. Anal., 18, 2, 2008, 537-564, DOI 10.1007/s12220-008-9019-x | MR 2393270
[10] Reilly, R. C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom., MR49:6102, 8, 3, 1973, 465-477, MR 0341351 | Zbl 0277.53030
[11] Rosenberg, S.: The Variation of the de Rham zeta function. Trans. Amer. Math. Soc., 299, 2, 1987, 535-557, DOI 10.1090/S0002-9947-1987-0869220-4 | MR 0869220 | Zbl 0615.53033
[12] Schoen, R.: Variation theory for the total scalar curvature functional for Riemannian metrics and related topics. Lecture Notes in Math. 1365, Topics in Calculus of Variations, Montecatini. Terme Springer. Verlag, 1987, 120-154, MR 0994021
[13] Wu, Faen: On the variation of a metric and it's application. Acta. Math. Sinica, 26, 10, 2010, 2003-2014, DOI 10.1007/s10114-010-7470-7 | MR 2718097
Search
Partner of
