Some characterizations on critical metrics for quadratic curvature functions

Huang, Guangyue; Chen, Li

doi:10.1090/proc/13740

Some characterizations on critical metrics for quadratic curvature functions
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by Guangyue Huang and Li Chen PDF

Proc. Amer. Math. Soc. 146 (2018), 385-395 Request permission

Abstract:

Under some integral conditions, we classify closed $n$-dimensional manifolds of which the metrics are critical for quadratic curvature functions. Moreover, under some curvature conditions, we also obtain that a critical metric must be Einstein.

References

Michael T. Anderson, Extrema of curvature functionals on the space of metrics on $3$-manifolds, Calc. Var. Partial Differential Equations 5 (1997), no. 3, 199–269. MR 1438146, DOI 10.1007/s005260050066
Marcel Berger, Quelques formules de variation pour une structure riemannienne, Ann. Sci. École Norm. Sup. (4) 3 (1970), 285–294 (French). MR 278238
Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
Jean-Pierre Bourguignon, The “magic” of Weitzenböck formulas, Variational methods (Paris, 1988) Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 251–271. MR 1205158
Giovanni Catino, On conformally flat manifolds with constant positive scalar curvature, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2627–2634. MR 3477081, DOI 10.1090/proc/12925
Giovanni Catino, Some rigidity results on critical metrics for quadratic functionals, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2921–2937. MR 3412398, DOI 10.1007/s00526-015-0889-z
Andrzej Derdziński, On compact Riemannian manifolds with harmonic curvature, Math. Ann. 259 (1982), no. 2, 145–152. MR 656660, DOI 10.1007/BF01457307
Dennis DeTurck and Hubert Goldschmidt, Regularity theorems in Riemannian geometry. II. Harmonic curvature and the Weyl tensor, Forum Math. 1 (1989), no. 4, 377–394. MR 1016679, DOI 10.1515/form.1989.1.377
Georges de Rham, Sur la reductibilité d’un espace de Riemann, Comment. Math. Helv. 26 (1952), 328–344 (French). MR 52177, DOI 10.1007/BF02564308
Matthew J. Gursky and Jeff A. Viaclovsky, A new variational characterization of three-dimensional space forms, Invent. Math. 145 (2001), no. 2, 251–278. MR 1872547, DOI 10.1007/s002220100147
Matthew J. Gursky and Jeff A. Viaclovsky, Rigidity and stability of Einstein metrics for quadratic curvature functionals, J. Reine Angew. Math. 700 (2015), 37–91. MR 3318510, DOI 10.1515/crelle-2013-0024
Emmanuel Hebey and Michel Vaugon, Effective $L_p$ pinching for the concircular curvature, J. Geom. Anal. 6 (1996), no. 4, 531–553 (1997). MR 1601401, DOI 10.1007/BF02921622
Zejun Hu, Seiki Nishikawa, and Udo Simon, Critical metrics of the Schouten functional, J. Geom. 98 (2010), no. 1-2, 91–113. MR 2739189, DOI 10.1007/s00022-010-0057-8
Zejun Hu and Haizhong Li, A new variational characterization of $n$-dimensional space forms, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3005–3023. MR 2052939, DOI 10.1090/S0002-9947-03-03486-X
Guangyue Huang and Yong Wei, The classification of $(m,\rho )$-quasi-Einstein manifolds, Ann. Global Anal. Geom. 44 (2013), no. 3, 269–282. MR 3101859, DOI 10.1007/s10455-013-9366-0
Guangyue Huang and Bingqing Ma, Riemannian manifolds with harmonic curvature, Colloq. Math. 145 (2016), no. 2, 251–257. MR 3557136, DOI 10.4064/cm6826-4-2016
Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47–62. MR 806701
Jerry L. Kazdan, Unique continuation in geometry, Comm. Pure Appl. Math. 41 (1988), no. 5, 667–681. MR 948075, DOI 10.1002/cpa.3160410508
François Lamontagne, Une remarque sur la norme $L^2$ du tenseur de courbure, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 3, 237–240 (French, with English and French summaries). MR 1288410
François Lamontagne, A critical metric for the $L^2$-norm of the curvature tensor on $S^3$, Proc. Amer. Math. Soc. 126 (1998), no. 2, 589–593. MR 1425130, DOI 10.1090/S0002-9939-98-04171-9
Paolo Mastrolia, Dario D. Monticelli, and Marco Rigoli, A note on curvature of Riemannian manifolds, J. Math. Anal. Appl. 399 (2013), no. 2, 505–513. MR 2996728, DOI 10.1016/j.jmaa.2012.10.044
Masafumi Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213. MR 353216, DOI 10.2307/2373587
N. K. Smolentsev, Spaces of Riemannian metrics, Sovrem. Mat. Prilozh. 31, Geometriya (2005), 69–147 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 142 (2007), no. 5, 2436–2519. MR 2464555, DOI 10.1007/s10958-007-0185-3
Shükichi Tanno, Deformations of Riemannian metrics on 3-dimensional manifolds, Tohoku Math. J. (2) 27 (1975), no. 3, 437–444. MR 436209, DOI 10.2748/tmj/1203529253