New rigidity results for critical metrics of some quadratic curvature functionals
-
Marco Bernardini
Abstract
We prove a new rigidity result for metrics defined on closed smooth n-manifolds that are critical for the quadratic functional 𝔉t, which depends on the Ricci curvature Ric and the scalar curvature R, and that satisfy a pinching condition of the form Sec > ε R, where ε is a function of t and n, while Sec denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying Sec >
References
[1] M. T. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds. Calc. Var. Partial Differential Equations 5 (1997), 199–269. MR1438146 Zbl 0889.5802710.1007/s005260050066Search in Google Scholar
[2] M. Berger, Quelques formules de variation pour une structure riemannienne. Ann. Sci. École Norm. Sup. (4) 3 (1970), 285–294. MR278238 Zbl 0204.5480210.24033/asens.1194Search in Google Scholar
[3] A. L. Besse, Einstein manifolds. Springer 1987. MR867684 Zbl 0613.5300110.1007/978-3-540-74311-8Search in Google Scholar
[4] S. Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric. Duke Math. J. 151 (2010), 1–21. MR2573825 Zbl 1189.5304210.1215/00127094-2009-061Search in Google Scholar
[5] G. Catino, Critical metrics of the L2-norm of the scalar curvature. Proc. Amer. Math. Soc. 142 (2014), 3981–3986. MR3251738 Zbl 1300.5304410.1090/S0002-9939-2014-12238-6Search in Google Scholar
[6] G. Catino, Some rigidity results on critical metrics for quadratic functionals. Calc. Var. Partial Differential Equations 54 (2015), 2921–2937. MR3412398 Zbl 1327.5305310.1007/s00526-015-0889-zSearch in Google Scholar
[7] G. Catino, Rigidity of positively curved shrinking Ricci solitons in dimension four. Geom. Flows 4 (2019), 1–8. MR3946291 Zbl 1440.5305010.1515/geofl-2019-0001Search in Google Scholar
[8] G. Catino, P. Mastrolia, A perspective on canonical Riemannian metrics. Springer 2020. MR4207078 Zbl 1478.5300310.1007/978-3-030-57185-6Search in Google Scholar
[9] M. J. Gursky, J. A. Viaclovsky, A new variational characterization of three-dimensional space forms. Invent. Math. 145 (2001), 251–278. MR1872547 Zbl 1006.5800810.1007/s002220100147Search in Google Scholar
[10] M. J. Gursky, J. A. Viaclovsky, Rigidity and stability of Einstein metrics for quadratic curvature functionals. J. Reine Angew. Math. 700 (2015), 37–91. MR3318510 Zbl 1327.5305810.1515/crelle-2013-0024Search in Google Scholar
[11] Z. Hu, H. Li, A new variational characterization of n-dimensional space forms. Trans. Amer. Math. Soc. 356 (2004), 3005–3023. MR2052939 Zbl 1058.5302910.1090/S0002-9947-03-03486-XSearch in Google Scholar
[12] F. Lamontagne, Critical metrics for the L(2)-norm of the curvature tensor. PhD thesis, State University of New York at Stony Brook, 1993.Search in Google Scholar
[13] F. Lamontagne, Une remarque sur la norme L2 du tenseur de courbure. C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 237–240. MR1288410 Zbl 0806.53052Search in Google Scholar
[14] J. M. Lee, Introduction to Riemannian manifolds. Springer 2018. MR3887684 Zbl 1409.5300110.1007/978-3-319-91755-9Search in Google Scholar
[15] E. Ribeiro, Jr., Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor. Ann. Mat. Pura Appl. (4) 195 (2016), 2171–2181. MR3558323 Zbl 1354.5305410.1007/s10231-016-0557-8Search in Google Scholar
[16] S. Tanno, Deformations of Riemannian metrics on 3-dimensional manifolds. Tohoku Math. J. (2) 27 (1975), 437–444. MR436209 Zbl 0312.5305210.2748/tmj/1203529253Search in Google Scholar
[17] D. Yang, Rigidity of Einstein 4-manifolds with positive curvature. Invent. Math. 142 (2000), 435–450. MR1794068 Zbl 0981.5302510.1007/PL00005792Search in Google Scholar
Appendix A
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
Articles in the same Issue
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
