Remarks on compact quasi-Einstein manifolds with boundary

Diógenes, R.; Gadelha, T.; Ribeiro, E., Jr

doi:10.1090/proc/15708

Remarks on compact quasi-Einstein manifolds with boundary
HTML articles powered by AMS MathViewer

by R. Diógenes, T. Gadelha and E. Ribeiro Jr PDF

Proc. Amer. Math. Soc. 150 (2022), 351-363 Request permission

Abstract:

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,\,g,\,u)$ of dimension $n\geq 4$ with boundary $\partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere $\Bbb {S}^n_+,$ or $g=dt^{2}+\psi ^{2}(t)g_{L}$ and $u=u(t),$ where $g_{L}$ is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension $n=3$ is also discussed.

References

Lucas Ambrozio, On static three-manifolds with positive scalar curvature, J. Differential Geom. 107 (2017), no. 1, 1–45. MR 3698233, DOI 10.4310/jdg/1505268028
D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206 (French). MR 889476, DOI 10.1007/BFb0075847
H. Baltazar, On critical point equation of compact manifolds with zero radial Weyl curvature, Geom. Dedicata 202 (2019), 337–355. MR 4001820, DOI 10.1007/s10711-018-0417-3
H. Baltazar, R. Diógenes, and E. Ribeiro Jr., Volume functional of compact 4-manifolds with a prescribed boundary metric, J. Geom. Anal. 31 (2021), no. 5, 4703–4720. MR 4244882, DOI 10.1007/s12220-020-00452-9
Halyson Baltazar and Ernani Ribeiro Jr., Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary, Pacific J. Math. 297 (2018), no. 1, 29–45. MR 3864225, DOI 10.2140/pjm.2018.297.29
A. Barros, R. Batista, and E. Ribeiro Jr., Bounds on volume growth of geodesic balls for Einstein warped products, Proc. Amer. Math. Soc. 143 (2015), no. 10, 4415–4422. MR 3373940, DOI 10.1090/S0002-9939-2015-12606-8
R. Batista, R. Diógenes, M. Ranieri, and E. Ribeiro Jr., Critical metrics of the volume functional on compact three-manifolds with smooth boundary, J. Geom. Anal. 27 (2017), no. 2, 1530–1547. MR 3625163, DOI 10.1007/s12220-016-9730-y
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
Christoph Böhm, Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math. 134 (1998), no. 1, 145–176. MR 1646591, DOI 10.1007/s002220050261
Miguel Brozos-Vázquez, Eduardo García-Río, Peter Gilkey, and Xabier Valle-Regueiro, Half conformally flat generalized quasi-Einstein manifolds of metric signature $(2,2)$, Internat. J. Math. 29 (2018), no. 1, 1850002, 25. MR 3756414, DOI 10.1142/S0129167X18500027
Huai-Dong Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1–38. MR 2648937
Huai-Dong Cao and Qiang Chen, On Bach-flat gradient shrinking Ricci solitons, Duke Math. J. 162 (2013), no. 6, 1149–1169. MR 3053567, DOI 10.1215/00127094-2147649
Huai-Dong Cao, Giovanni Catino, Qiang Chen, Carlo Mantegazza, and Lorenzo Mazzieri, Bach-flat gradient steady Ricci solitons, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 125–138. MR 3148109, DOI 10.1007/s00526-012-0575-3
Jeffrey Case, Yu-Jen Shu, and Guofang Wei, Rigidity of quasi-Einstein metrics, Differential Geom. Appl. 29 (2011), no. 1, 93–100. MR 2784291, DOI 10.1016/j.difgeo.2010.11.003
Jeffrey S. Case, The nonexistence of quasi-Einstein metrics, Pacific J. Math. 248 (2010), no. 2, 277–284. MR 2741248, DOI 10.2140/pjm.2010.248.277
Jeffrey S. Case, Smooth metric measure spaces, quasi-Einstein metrics, and tractors, Cent. Eur. J. Math. 10 (2012), no. 5, 1733–1762. MR 2959681, DOI 10.2478/s11533-012-0091-x
Giovanni Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012), no. 3-4, 751–756. MR 2945582, DOI 10.1007/s00209-011-0888-5
Giovanni Catino, A note on four-dimensional (anti-)self-dual quasi-Einstein manifolds, Differential Geom. Appl. 30 (2012), no. 6, 660–664. MR 2996860, DOI 10.1016/j.difgeo.2012.09.005
G. Catino, P. Mastrolia, and D. D. Monticelli, Gradient Ricci solitons with vanishing conditions on Weyl, J. Math. Pures Appl. (9) 108 (2017), no. 1, 1–13 (English, with English and French summaries). MR 3660766, DOI 10.1016/j.matpur.2016.10.007
Giovanni Catino, Carlo Mantegazza, Lorenzo Mazzieri, and Michele Rimoldi, Locally conformally flat quasi-Einstein manifolds, J. Reine Angew. Math. 675 (2013), 181–189. MR 3021450, DOI 10.1515/crelle.2011.183
Giovanni Catino, Paolo Mastrolia, Dario D. Monticelli, and Marco Rigoli, On the geometry of gradient Einstein-type manifolds, Pacific J. Math. 286 (2017), no. 1, 39–67. MR 3582400, DOI 10.2140/pjm.2017.286.39
Qiang Chen and Chenxu He, On Bach flat warped product Einstein manifolds, Pacific J. Math. 265 (2013), no. 2, 313–326. MR 3096503, DOI 10.2140/pjm.2013.265.313
Justin Corvino, Michael Eichmair, and Pengzi Miao, Deformation of scalar curvature and volume, Math. Ann. 357 (2013), no. 2, 551–584. MR 3096517, DOI 10.1007/s00208-013-0903-8
R. Diógenes and T. Gadelha, Compact quasi-Einstein manifolds with boundary, to appear in Math. Nachr., arXiv:1911.10068 [math.DG].
Peter Petersen and William Wylie, On the classification of gradient Ricci solitons, Geom. Topol. 14 (2010), no. 4, 2277–2300. MR 2740647, DOI 10.2140/gt.2010.14.2277
Chenxu He, Peter Petersen, and William Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012), no. 2, 271–311. MR 2928714, DOI 10.4310/CAG.2012.v20.n2.a3
Chenxu He, Peter Petersen, and William Wylie, Warped product Einstein metrics over spaces with constant scalar curvature, Asian J. Math. 18 (2014), no. 1, 159–189. MR 3215345, DOI 10.4310/AJM.2014.v18.n1.a9
H. He, Critical metrics of the volume functional on three-dimensional manifolds, arXiv:2101.05621 [math.DG], 2021.
Paolo Mastrolia and Michele Rimoldi, Some triviality results for quasi-Einstein manifolds and Einstein warped products, Geom. Dedicata 169 (2014), 225–237. MR 3175246, DOI 10.1007/s10711-013-9852-3
Pengzi Miao and Luen-Fai Tam, On the volume functional of compact manifolds with boundary with constant scalar curvature, Calc. Var. Partial Differential Equations 36 (2009), no. 2, 141–171. MR 2546025, DOI 10.1007/s00526-008-0221-2
Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
M. Ranieri and E. Ribeiro Jr., Bach-flat noncompact steady quasi-Einstein manifolds, Arch. Math. (Basel) 108 (2017), no. 5, 507–519. MR 3638302, DOI 10.1007/s00013-016-1014-z
Ernani Ribeiro Jr. and Keti Tenenblat, Noncompact quasi-Einstein manifolds conformal to a Euclidean space, Math. Nachr. 294 (2021), no. 1, 132–144. MR 4245571, DOI 10.1002/mana.201900189
Michele Rimoldi, A remark on Einstein warped products, Pacific J. Math. 252 (2011), no. 1, 207–218. MR 2862148, DOI 10.2140/pjm.2011.252.207
Alex Sandro Santos, Critical metrics of the scalar curvature functional satisfying a vanishing condition on the Weyl tensor, Arch. Math. (Basel) 109 (2017), no. 1, 91–100. MR 3658306, DOI 10.1007/s00013-017-1030-7
Lin Feng Wang, On noncompact $\tau$-quasi-Einstein metrics, Pacific J. Math. 254 (2011), no. 2, 449–464. MR 2900025, DOI 10.2140/pjm.2011.254.449
Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. MR 2577473, DOI 10.4310/jdg/1261495336
Fei Yang and Liangdi Zhang, Rigidity of gradient shrinking Ricci solitons, Asian J. Math. 24 (2020), no. 4, 533–547. MR 4226660, DOI 10.4310/AJM.2020.v24.n4.a1