Some rigidity results on q-solitons

References

[1] R. Arnowitt, S. Deser, C. W. Misner, The dynamics of general relativity. In: Gravitation: An introduction to current research, 227–265, Wiley, New York-London 1962. MR143629 Zbl 0115.43103Search in Google Scholar

[2] R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Z. 9 (1921), 110–135. MR1544454 JFM 48.1035.0110.1007/BF01378338Search in Google Scholar

[3] E. Bahuaud, C. Guenther, J. Isenberg, R. Mazzeo, Well-posedness of nonlinear flows on manifolds of bounded geometry. Ann. Global Anal. Geom. 65 (2024), Paper No. 25, 39 pages. MR4742191 Zbl 0786236810.1007/s10455-023-09940-xSearch in Google Scholar

[4] E. Bahuaud, D. Helliwell, Short-time existence for some higher-order geometric flows. Comm. Partial Differential Equations 36 (2011), 2189–2207. MR2852074 Zbl 1242.5307910.1080/03605302.2011.593015Search in Google Scholar

[5] E. Bahuaud, D. Helliwell, Uniqueness for some higher-order geometric flows. Bull. Lond. Math. Soc. 47 (2015), 980–995. MR3431578 Zbl 1338.5309210.1112/blms/bdv076Search in Google Scholar

[6] J.-P. Bourguignon, J.-P. Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Amer. Math. Soc. 301 (1987), 723–736. MR882712 Zbl 0622.5302310.1090/S0002-9947-1987-0882712-7Search in Google Scholar

[7] H.-D. Cao, Recent progress on Ricci solitons. In: Recent advances in geometric analysis, volume 11 of Adv. Lect. Math., 1–38, International Press, Somerville, MA 2010. MR2648937 Zbl 1201.53046Search in Google Scholar

[8] H.-D. Cao, Q. Chen, On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162 (2013), 1149–1169. MR3053567 Zbl 1277.53036)10.1215/00127094-2147649Search in Google Scholar

[9] B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, L. Ni, The Ricci flow: techniques and applications. Part I, volume 135 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2007. MR2302600 Zbl 1157.5303410.1090/surv/144Search in Google Scholar

[10] A. W. Cunha, E. Griffin, On non-compact gradient solitons. Ann. Global Anal. Geom. 63 (2023), Paper No. 27, 20 pages. MR4595986 Zbl 1517.5308610.1007/s10455-023-09904-1Search in Google Scholar

[11] S. Das, S. Kar, Bach flows of product manifolds. Int. J. Geom. Methods Mod. Phys. 9 (2012), 1250039, 18 pages. MR2948858 Zbl 1258.5307010.1142/S0219887812500399Search in Google Scholar

[12] C. Fefferman, C. R. Graham, The ambient metric, volume 178 of Annals of Mathematics Studies. Princeton Univ. Press 2012. MR2858236 Zbl 1243.5300410.23943/princeton/9780691153131.001.0001Search in Google Scholar

[13] A. Ghosh, On Bach almost solitons. Beitr. Algebra Geom. 63 (2022), 45–54. MR4393949 Zbl 1489.5306910.1007/s13366-021-00565-4Search in Google Scholar

[14] E. Griffin, Gradient ambient obstruction solitons on homogeneous manifolds. Ann. Global Anal. Geom. 60 (2021), 469–499. MR4304859 Zbl 1486.5304810.1007/s10455-021-09784-3Search in Google Scholar

[15] E. Griffin, R. Poddar, R. Sharma, W. Wylie, Extended solitons of the ambient obstruction flow. Preprint 2024, arXiv:2405.1587010.1007/s12220-025-02002-7Search in Google Scholar

[16] H. Hiramatu, Riemannian manifolds admitting a projective vector field. Kodai Math. J. 3 (1980), 397–406. MR604484 Zbl 0454.5302910.2996/kmj/1138036262Search in Google Scholar

[17] P. T. Ho, Bach flow. J. Geom. Phys. 133 (2018), 1–9. MR3850251 Zbl 1398.5307410.1016/j.geomphys.2018.07.008Search in Google Scholar

[18] S. Kim, Rigidity of noncompact complete Bach-flat manifolds. J. Geom. Phys. 60 (2010), 637–642. MR2602377 Zbl 1188.5303410.1016/j.geomphys.2009.12.014Search in Google Scholar

[19] O. Nouhaud, Transformations infinitésimales harmoniques. C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A573–A576. MR290289 Zbl 0242.53013Search in Google Scholar

[20] P. Petersen, W. Wylie, Rigidity of gradient Ricci solitons. Pacific J. Math. 241 (2009), 329–345. MR2507581 Zbl 1176.5304810.2140/pjm.2009.241.329Search in Google Scholar

[21] P. Petersen, W. Wylie, Rigidity of homogeneous gradient soliton metrics and related equations. Differential Geom. Appl. 84 (2022), Paper No. 101929, 29 pages. MR4457372 Zbl 1506.5306410.1016/j.difgeo.2022.101929Search in Google Scholar

[22] R. Sharma, Characterizations of weakly conformally flat and quasi Einstein manifolds. J. Geom. 114 (2023), Paper No. 17, 9 pages. MR4615516 Zbl 1520.5303210.1007/s00022-023-00680-8Search in Google Scholar

[23] R. Sharma, S. Deshmukh, Ricci almost solitons with associated projective vector field. Adv. Geom. 22 (2022), 1–8. MR4371940 Zbl 1486.5306510.1515/advgeom-2021-0034Search in Google Scholar

[24] S. E. Stepanov, I. G. Shandra, Geometry of infinitesimal harmonic transformations. Ann. Global Anal. Geom. 24 (2003), 291–299. MR1996772 Zbl 1035.5309010.1023/A:1024753028255Search in Google Scholar

[25] S. E. Stepanov, V. N. Shelepova, A remark on Ricci solitons. Math. Notes 86 (2009), 447–450. MR2591387 Zbl 1182.5306210.1134/S0001434609090193Search in Google Scholar

[26] S. Tanno, Promenades on spheres. A sketch book of eight scenes on spheres. Lecture notes, Dept. Math., Tokyo Inst. Tech., 1996.Search in Google Scholar

[27] Y. Tashiro, Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117 (1965), 251–275. MR174022 Zbl 0136.1770110.1090/S0002-9947-1965-0174022-6Search in Google Scholar

[28] A. Thompson, Bach flow of simply connected nilmanifolds. Adv. Geom. 24 (2024), 127–139. MR4694258 Zbl 1539.5311510.1515/advgeom-2023-0032Search in Google Scholar

[29] D. J. Welsh, Jr., Manifolds that admit parallel vector fields. Illinois J. Math. 30 (1986), 9–18. MR822382 Zbl 0581.5302710.1215/ijm/1256044750Search in Google Scholar

[30] K. Yano, Integral formulas in Riemannian geometry. Dekker 1970. MR284950 Zbl 0213.23801Search in Google Scholar